@InBook{Macias2017,
  pages     = {3973--3997},
  title     = {{C}omparison and {C}omputational {P}erformance of {T}sunami-{H}y{SEA} and {MOST} {M}odels for {LANTEX} 2013 {S}cenario: {I}mpact {A}ssessment on {P}uerto {R}ico {C}oasts},
  publisher = {Springer International Publishing},
  year      = {2017},
  author    = {Mac{\'i}as, Jorge and Mercado, Aurelio and Gonz{\'a}lez Vida, Jos{\'e} M. and Ortega-Acosta, Sergio and Castro D{\'i}az, Manuel J.},
  editor    = {Geist, Eric-L. and Fritz, Hermann-M. and Rabinovich, Alexander-B. and Tanioka, Yuichiro},
  address   = {Cham},
  booktitle = {Global Tsunami Science: Past and Future, Volume I},
  url       = {http://dx.doi.org/10.1007/978-3-319-55480-8_16},
}

@Article{BIngham2d,
  author  = {Fern{\'a}ndez Nieto, E. D. and Gallardo, Jos{\'e}-M. and Vigneaux, Paul},
  title   = {{E}fficient numerical schemes for viscoplastic avalanches. {P}art 2: the 2{D} case},
  journal = {Journal of Computational Physics},
  year    = {2017},
  volume  = {Accepted},
}

@Article{Lynett_etal_2017,
  author  = {Patrick-J., Lynett and Kara, Gately and Rick, Wilson and Luis, Montoya and Diego, Arcas and Betul, Aytore and Yefei, Bai and Jeremy-D., Bricker and Castro D{\'i}az, Manuel J. and Kwok-Fai, Cheung and C.-Gabriel, David and Gozde-Guney, Dogan and Cipriano, Escalante and Gonz{\'a}lez-Vida, J.-M. and Stephan-T., Grilli and Troy, W.-Heitmann and Horrillo, Juan and Utku, K{\^a}noğlu and Rozita, Kian and James, T.-Kirby and Wenwen, Li and Mac{\'i}as, Jorge and Dmitry, J.-Nicolsky and Ortega-Acosta, Sergio and Alyssa, Pampell-Manis and Yong, Sung-Park and Volker, Roeber and Naeimeh, Sharghivand and Michael, Shelby and Fengyan, Shi and Babak, Tehranirad and Elena, Tolkova and Hon-Kie, Thio and Deniz, Velioğlu and Ahmet, Cevdet-Yal{\c{c}}ıner and Yoshiki, Yamazaki and Andrey, Zaytsev and Y.-J., Zhang},
  title   = {{I}nter-model analysis of tsunami-induced coastal currents},
  journal = {Ocean Modelling},
  year    = {2017},
  volume  = {114},
  pages   = {14 - 32},
  url     = {http://www.sciencedirect.com/science/article/pii/S1463500317300513},
}

@Article{RMHD,
  author  = {Castro D{\'i}az, Manuel J. and Gallardo, Jos{\'e}-M. and Marquina, Antonio},
  title   = {{J}acobian-free approximate solvers for hyperbolic systems: {A}pplication to relativistic magnetohydrodynamics},
  journal = {Computer Physics Communications},
  year    = {2017},
  volume  = {219},
  pages   = {108-120},
}

@Article{Macias_etal_2017_nthmp,
  author  = {Mac{\'i}as, Jorge and Castro D{\'i}az, Manuel J. and Ortega-Acosta, Sergio and Escalante, Cipriano and Gonz{\'a}lez-Vida, J.-M.},
  title   = {{P}erformance benchmarking of {T}sunami-{H}y{SEA} model for {NTHMP}'s inundation mapping activities},
  journal = {Accepted in Pure and Applied Geophysics},
  year    = {2017},
}

@InCollection{Barcenas_et_al_2017,
  author    = {B{\'a}rcenas, P. and Lobo, F.-J. and Fern{\'a}ndez-Salas, L.-M. and Ortega-S{\'a}nchez, M. and Mendes, I. and Mac{\'i}as, Jorge},
  title     = {{P}rodeltaic undulations and hyperpycnal flows ({I}): {M}orphological observations},
  booktitle = {Atlas of Bedforms in the Western Mediterranean},
  publisher = {Springer},
  year      = {2017},
  editor    = {J. Guill{\'e}n and J. Acosta and S. Bren{\'e} and F. Chiocci and A. Palanques},
  chapter   = {18},
  pages     = {107-112},
}

@InCollection{Lobo_et_al_2017,
  author    = {Lobo, F.-J. and B{\'a}rcenas, P. and Mendes, I. and Ortega-S{\'a}nchez, M. and Mac{\'i}as, Jorge and Fern{\'a}ndez-Salas, L.-M.},
  title     = {{P}rodeltaic undulations and hyperpycnal flows ({II}): {E}volutionary trends},
  booktitle = {Atlas of Bedforms in the Western Mediterranean},
  publisher = {Springer},
  year      = {2017},
  editor    = {J. Guill{\'e}n and J. Acosta and S. Bren{\'e} and F. Chiocci and A. Palanques},
  chapter   = {18},
  pages     = {113-120},
}

@Article{GV_etal_Lituya,
  author   = {Gonz{\'a}lez Vida, Jos{\'e} M. and Mac{\'i}as, Jorge and Castro D{\'i}az, Manuel J. and S{\'a}nchez-Linares, Carlos and de la Asunci{\'o}n, Marc and Ortega-Acosta, Sergio and Arcas, Diego},
  title    = {{T}he {L}ituya {B}ay landslide-generated mega-tsunami. {N}umerical simulation and sensitivity analysis},
  journal  = {Submitted to JGR},
  year     = {2017},
  keywords = {Lituya Bay, Landslide-HySEA model, numerical simulation},
}

@Article{Lastras_etal_2016,
  author  = {Lastras, G. and Iglesias, O. and Mac{\'i}as, Jorge and Gonz{\'a}lez-Vida, J.-M. and Campoamor, J.-L. and Costa, S. and Canals, M.},
  title   = {{T}sunamigenic potential of small adjacent submarine landslides: {A} study case in {E}ivissa {C}hannel, {W}estern {M}editerranean {S}ea},
  journal = {In progress},
  year    = {2017},
}

@Article{Approx.Osher,
  author   = {Castro D{\'i}az, Manuel J. and Gallardo, Jos{\'e}-M. and Marquina, Antonio},
  title    = {{A}pproximate {O}sher-{S}olomon schemes for hyperbolic systems},
  journal  = {Applied Mathematics and Computation},
  year     = {2016},
  volume   = {272},
  number   = {2},
  pages    = {347-368},
  month    = {January},
  abstract = {This paper is concerned with a new kind of Riemann solvers for hyperbolic systems, which can be applied both in the conservative and nonconservative cases. In particular, the proposed schemes constitute a simple version of the classical Osher–Solomon Riemann solver, and extend in some sense the schemes proposed in Dumbser and Toro (2011) [19,20]. The viscosity matrix of the numerical flux is constructed as a linear combination of functional evaluations of the Jacobian of the flux at several quadrature points. Some families of functions have been proposed to this end: Chebyshev polynomials and rational-type functions. Our schemes have been tested with different initial value Riemann problems for ideal gas dynamics, magnetohydrodynamics and multilayer shallow water equations. The numerical tests indicate that the proposed schemes are robust, stable and accurate with a satisfactory time step restriction, and provide an efficient alternative for approximating time-dependent solutions in which the spectral decomposition is computationally expensive.},
  keywords = {Hyperbolic systems, incomplete Riemann solvers, Osher-Solomon method, Euler equations, ideal magnetohydrodynamics, multilayer shallow water equations},
  url      = {http://www.sciencedirect.com/science/article/pii/S0096300315008851},
}

This paper is concerned with a new kind of Riemann solvers for hyperbolic systems, which can be applied both in the conservative and nonconservative cases. In particular, the proposed schemes constitute a simple version of the classical Osher–Solomon Riemann solver, and extend in some sense the schemes proposed in Dumbser and Toro (2011) [19,20]. The viscosity matrix of the numerical flux is constructed as a linear combination of functional evaluations of the Jacobian of the flux at several quadrature points. Some families of functions have been proposed to this end: Chebyshev polynomials and rational-type functions. Our schemes have been tested with different initial value Riemann problems for ideal gas dynamics, magnetohydrodynamics and multilayer shallow water equations. The numerical tests indicate that the proposed schemes are robust, stable and accurate with a satisfactory time step restriction, and provide an efficient alternative for approximating time-dependent solutions in which the spectral decomposition is computationally expensive.

@Article{Macias_etal_2016_LANTEX2013,
  author   = {Mac{\'i}as, Jorge and Mercado, A. and Gonz{\'a}lez-Vida, J.-M. and Ortega-Acosta, Sergio and Castro D{\'i}az, Manuel J.},
  title    = {{C}omparison and numerical performance of {T}sunami-{H}y{SEA} and {MOST} models for {LANTEX} 2013 scenario. {I}mpact assessment on {P}uerto {R}ico},
  journal  = {Pure and Applied Geophysics},
  year     = {2016},
  volume   = {173},
  number   = {12},
  pages    = {3973-3997},
  abstract = {HySEA tsunami model is used to simulate the Caribbean LANTEX 2013 scenario. The numerical simulation of the propagation and inundation phases is performed with a single integrated model but using different mesh resolutions and nested meshes. Special emphasis is put on assessing the most exposed coastal areas at Puerto Rico affected by this event. Some comparisons with MOST tsunami model available at UPR are made. Both models compare well for propagating tsunami waves in open sea, producing very similar results. The main discrepancies are observed in coastal areas, where maximum wave height provided by the propagation module of MOST is different from the one provided by HySEA. The main reason is that, while HySEA always compute inundation effects, MOST propagation module does not include runup physics. Henceforth, in near-shore shallow waters, HySEA should be compared with the inundation version of MOST. Nevertheless the most striking difference resides in computational time; HySEA is coded using the advantages of GPU architecture, and can produce a 4 hour simulation in a 60 arc-sec resolution for the whole Caribbean Sea in less than 4 min with a single GPU and as fast as 11 seconds with 32 GPUs. When details about the inundation must be simulated, a 1 arc-sec (approximately 30 m) inundation resolution mesh covering all of Puerto Rico, an island with dimensions of 160 km east-west and 56 km north-south, is used, and a three level nested meshes technique implemented. In this case approximately 11 hours of wall clock time are needed for a 2 hour simulation in a single GPU. When domain decomposition techniques are finally implemented by breaking up the computational domain into sub-domains and assigning a GPU to each subdomain (multiGPU HySEA version), the wall clock time should decrease significantly, allowing high-resolution inundation modeling in just a few hours and at a modest hardware cost compared with present tsunami models.
},
  keywords = {HySEA model, MOST model, tsunamis, numerical simulation, LANTEX 2013, Caribbean Sea, Puerto Rico},
  url      = {http://dx.doi.org/10.1007/s00024-016-1387-8},
}

HySEA tsunami model is used to simulate the Caribbean LANTEX 2013 scenario. The numerical simulation of the propagation and inundation phases is performed with a single integrated model but using different mesh resolutions and nested meshes. Special emphasis is put on assessing the most exposed coastal areas at Puerto Rico affected by this event. Some comparisons with MOST tsunami model available at UPR are made. Both models compare well for propagating tsunami waves in open sea, producing very similar results. The main discrepancies are observed in coastal areas, where maximum wave height provided by the propagation module of MOST is different from the one provided by HySEA. The main reason is that, while HySEA always compute inundation effects, MOST propagation module does not include runup physics. Henceforth, in near-shore shallow waters, HySEA should be compared with the inundation version of MOST. Nevertheless the most striking difference resides in computational time; HySEA is coded using the advantages of GPU architecture, and can produce a 4 hour simulation in a 60 arc-sec resolution for the whole Caribbean Sea in less than 4 min with a single GPU and as fast as 11 seconds with 32 GPUs. When details about the inundation must be simulated, a 1 arc-sec (approximately 30 m) inundation resolution mesh covering all of Puerto Rico, an island with dimensions of 160 km east-west and 56 km north-south, is used, and a three level nested meshes technique implemented. In this case approximately 11 hours of wall clock time are needed for a 2 hour simulation in a single GPU. When domain decomposition techniques are finally implemented by breaking up the computational domain into sub-domains and assigning a GPU to each subdomain (multiGPU HySEA version), the wall clock time should decrease significantly, allowing high-resolution inundation modeling in just a few hours and at a modest hardware cost compared with present tsunami models.

@Article{nhess-16-2593-2016,
  author  = {Molinari, I. and Tonini, R. and Lorito, S. and Piatanesi, A. and Romano, F. and Melini, D. and Hoechner, A. and Gonz{\'a}lez-Vida, J.-M. and Mac{\'i}as, Jorge and Castro D{\'i}az, Manuel J. and de-la-Asunci{\'o}n, M.},
  title   = {{F}ast evaluation of tsunami scenarios: uncertainty assessment for a {M}editerranean {S}ea database},
  journal = {Natural Hazards and Earth System Sciences},
  year    = {2016},
  volume  = {16},
  number  = {12},
  pages   = {2593--2602},
  url     = {http://www.nat-hazards-earth-syst-sci.net/16/2593/2016/},
}

@Article{fernandez-nieto_formal_2016,
  author  = {Fern{\'a}ndez Nieto, Enrique D. and Morales de Luna, Tom{\'a}s and Narbona-Reina, Gladys and Zabsonr{\'e}, Jean},
  title   = {{F}ormal deduction of the {S}aint-{V}enant-{E}xner model including arbitrarily sloping sediment beds and associated energy},
  journal = {ESAIM: Mathematical Modelling and Numerical Analysis},
  year    = {2016},
  month   = {mar},
  url     = {http://www.esaim-m2an.org/10.1051/m2an/2016018},
}

@Article{Sanchez_Linares2016,
  author  = {S{\'a}nchez-Linares, Carlos and de la Asunci{\'o}n, Marc and Castro D{\'i}az, Manuel J. and Gonz{\'a}lez Vida, Jos{\'e} M. and Mac{\'i}as, Jorge and Siddhartha, Mishra},
  title   = {{U}ncertainty quantification in tsunami modeling using multi-level {M}onte {C}arlo finite volume method},
  journal = {Journal of Mathematics in Industry},
  year    = {2016},
  volume  = {6},
  number  = {1},
  pages   = {5},
  url     = {http://dx.doi.org/10.1186/s13362-016-0022-8},
}

@Article{berthon2015efficient,
  author   = {Berthon, Christophe and Foucher, Fran{\c{c}}oise and Morales de Luna, Tom{\'a}s},
  title    = {{A}n efficient splitting technique for two-layer shallow-water model},
  journal  = {Numerical Methods for Partial Differential Equations},
  year     = {2015},
  abstract = {We consider the numerical approximation of the weak solutions of the two-layer shallow-water equations. The model under consideration is made of two usual one-layer shallow-water model coupled by nonconservative products. Because of the nonconservative products of the system, which couple both one-layer shallow-water subsystems, the usual numerical methods have to consider the full model. Of course, uncoupled numerical techniques, just involving finite volume schemes for the basic shallow-water equations, are very attractive since they are very easy to implement and they are costless. Recently, a stable layer splitting technique was introduced [Bouchut and Morales de Luna, M2AN Math Model Numer Anal 42 (2008), 683–698]. In the same spirit, we exhibit new splitting technique, which is proved to be well balanced and non-negative preserving. The main benefit issuing from the here derived uncoupled method is the ability to correctly approximate the solution of very severe benchmarks.},
}

We consider the numerical approximation of the weak solutions of the two-layer shallow-water equations. The model under consideration is made of two usual one-layer shallow-water model coupled by nonconservative products. Because of the nonconservative products of the system, which couple both one-layer shallow-water subsystems, the usual numerical methods have to consider the full model. Of course, uncoupled numerical techniques, just involving finite volume schemes for the basic shallow-water equations, are very attractive since they are very easy to implement and they are costless. Recently, a stable layer splitting technique was introduced [Bouchut and Morales de Luna, M2AN Math Model Numer Anal 42 (2008), 683–698]. In the same spirit, we exhibit new splitting technique, which is proved to be well balanced and non-negative preserving. The main benefit issuing from the here derived uncoupled method is the ability to correctly approximate the solution of very severe benchmarks.

@Article{Macias_etal2015_LANTEX2013,
  author   = {Mac{\'i}as, Jorge and Mercado, Aurelio and Gonz{\'a}lez-Vida, J.-M. and Ortega-Acosta, Sergio and Castro D{\'i}az, Manuel J.},
  title    = {{N}umerical {S}imulation of {LANTEX} 2013 {T}sunami {S}cenario with {H}y{SEA} {M}odel. {I}mpact {A}ssessment on {P}uerto {R}ico {C}oasts},
  year     = {2015},
  abstract = {HySEA tsunami model is used to simulate the Caribbean LANTEX 2013 scenario. The numerical simulation of the propagation and inundation phases is performed with a single integrated model but using different mesh resolutions and nested meshes. Special emphasis is put on assessing the most exposed coastal areas at Puerto Rico affected by this event. Some comparisons with MOST tsunami model available at UPR are made. Both models compare well for propagating tsunami waves in open sea, producing very similar results. The main discrepancies are observed in coastal areas, where maximum wave height provided by the propagation module of MOST is different from the one provided by HySEA. The main reason is that, while HySEA always compute inundation effects, MOST propagation module does not include runup physics. Henceforth, in near-shore shallow waters, HySEA should be compared with the inundation version of MOST. Nevertheless the most striking difference resides in computational time; HySEA is coded using the advantages of GPU architecture, and can produce a 4 hour simulation in a 60 arc-sec resolution for the whole Caribbean Sea in less than 4 min with a single GPU and as fast as 11 seconds with 32 GPUs. When details about the inundation must be simulated, a 1 arc-sec (approximately 30 m) inundation resolution mesh covering all of Puerto Rico, an island with dimensions of 160 km east-west and 56 km north-south, is used, and a three level nested meshes technique implemented. In this case approximately 11 hours of wall clock time are needed for a 2 hour simulation in a single GPU. When domain decomposition techniques are finally implemented by breaking up the computational domain into sub-domains and assigning a GPU to each subdomain (multiGPU HySEA version), the wall clock time should decrease significantly, allowing high-resolution inundation modeling in just a few hours and at a modest hardware cost compared with present tsunami models.},
  keywords = {HySEA model, LANTEX 2013, Tsunami simulation, numerical model},
}

HySEA tsunami model is used to simulate the Caribbean LANTEX 2013 scenario. The numerical simulation of the propagation and inundation phases is performed with a single integrated model but using different mesh resolutions and nested meshes. Special emphasis is put on assessing the most exposed coastal areas at Puerto Rico affected by this event. Some comparisons with MOST tsunami model available at UPR are made. Both models compare well for propagating tsunami waves in open sea, producing very similar results. The main discrepancies are observed in coastal areas, where maximum wave height provided by the propagation module of MOST is different from the one provided by HySEA. The main reason is that, while HySEA always compute inundation effects, MOST propagation module does not include runup physics. Henceforth, in near-shore shallow waters, HySEA should be compared with the inundation version of MOST. Nevertheless the most striking difference resides in computational time; HySEA is coded using the advantages of GPU architecture, and can produce a 4 hour simulation in a 60 arc-sec resolution for the whole Caribbean Sea in less than 4 min with a single GPU and as fast as 11 seconds with 32 GPUs. When details about the inundation must be simulated, a 1 arc-sec (approximately 30 m) inundation resolution mesh covering all of Puerto Rico, an island with dimensions of 160 km east-west and 56 km north-south, is used, and a three level nested meshes technique implemented. In this case approximately 11 hours of wall clock time are needed for a 2 hour simulation in a single GPU. When domain decomposition techniques are finally implemented by breaking up the computational domain into sub-domains and assigning a GPU to each subdomain (multiGPU HySEA version), the wall clock time should decrease significantly, allowing high-resolution inundation modeling in just a few hours and at a modest hardware cost compared with present tsunami models.

@Article{2015MarGR..36..309L,
  author   = {Lobo, F.-J. and Goff, J.-A. and Mendes, I. and B{\'a}rcenas, P. and Mart{\'i}n-Rosales, W. and Fern{\'a}ndez-Salas, L.-M. and Mac{\'i}as, Jorge and D{\'i}az-del-R{\'i}o, V.},
  title    = {{S}patial variability of prodeltaic undulations on the {G}uadalfeo {R}iver prodelta: support to the genetic interpretation as hyperpycnal flow deposits},
  journal  = {Marine Geophysical Research},
  year     = {2015},
  volume   = {36},
  pages    = {309-333},
  month    = {dec},
  keywords = {Alboran Sea, Guadalfeo River prodelta, Submarine undulations, Multibeam bathymetry, Geostatistics},
}

@Article{Barcenas_etal_2015,
  author   = {B{\'a}rcenas, P. and Lobo, F.-J. and Mac{\'i}as, Jorge and Fern{\'a}ndez-Salas, L.-M. and L{\'o}pez-Gonz{\'a}lez, N. and D{\'i}az-del-R{\'i}o, V.},
  title    = {{S}ubmarine deltaic geometries linked to steep, mountainous drainage basins in the northern shelf of the {A}lboran {S}ea: {F}illing the gaps in the spectrum of deltaic deposition},
  journal  = {Geomorphology},
  year     = {2015},
  volume   = {232},
  pages    = {125-144},
  month    = {March},
  abstract = {P. B ́arcenas, F.J. Lobo, J. Mac ́ıas, L.M. Fern ́andez-Salas, N. L ́opez-Gonz ́alez y V. D ́ıaz del R ́ıo },
  url      = {http://link.springer.com/article/10.1007%2Fs11001-014-9233-9},
}

P. B ́arcenas, F.J. Lobo, J. Mac ́ıas, L.M. Fern ́andez-Salas, N. L ́opez-Gonz ́alez y V. D ́ıaz del R ́ıo

@Article{Macias_etal_2015,
  author   = {Mac{\'i}as, Jorge and V{\'a}zquez, J.-T. and V{\'a}zquez, L.-M. and Fern{\'a}ndez-Salas, L.-M. and Gonz{\'a}lez-Vida, J.-M. and B{\'a}rcenas, P. and Castro D{\'i}az, Manuel J. and D{\'i}az-del-R{\'i}o, V. and Alonso, B.},
  title    = {{T}he {A}l-{B}oran{\'i} mass movement system and associated tsunami. {A} modeling approach},
  journal  = {Marine Geology},
  year     = {2015},
  volume   = {361},
  pages    = {79-95},
  month    = {March},
  abstract = { J. Mac ́ıas, J.T. V ́azquez, L.M. Fern ́andez-Salas, J.M. Gonz ́alez-Vida, P. B ́arcenas, Castro-D{\'i}az, Manuel-J. y V. D ́ıaz del R ́ıo
T ́ıtulo: },
  url      = {http://www.sciencedirect.com/science/article/pii/S0025322714003806},
}

J. Mac ́ıas, J.T. V ́azquez, L.M. Fern ́andez-Salas, J.M. Gonz ́alez-Vida, P. B ́arcenas, Castro-Díaz, Manuel-J. y V. D ́ıaz del R ́ıo T ́ıtulo:

@Article{RVM,
  author  = {Castro D{\'i}az, Manuel J. and Gallardo, Jos{\'e}-M. and Marquina, Antonio},
  title   = {{A} class of incomplete {R}iemann solvers based on uniform rational approximations to the absolute value function},
  journal = {Journal of Scientific Computing},
  year    = {2014},
  volume  = {60},
  pages   = {363-389},
}

@Article{LIBS3,
  author  = {Moros, J. and Serrano, J. and S{\'a}nchez-Linares, Carlos and Mac{\'i}as, Jorge and Laserna, J.-J.},
  title   = {{A}dvanced {R}ecognition of {E}xplosives in {T}races on {P}olymer {S}urfaces {U}sing {LIBS} and {S}upervised {L}earning {C}lassifiers},
  journal = {Analytica Chimica Acta},
  year    = {2014},
  volume  = {806},
  pages   = {107-116},
  month   = {Jan},
}

@Article{Avalanches-1d,
  author   = {Fern{\'a}ndez Nieto, E. D. and Gallardo, Jos{\'e}-M. and Vigneaux, Paul},
  title    = {{E}fficient numerical schemes for viscoplastic avalanches. {P}art 1: {T}he 1{D} case},
  journal  = {Journal of Computational Physics},
  year     = {2014},
  volume   = {264},
  number   = {1},
  pages    = {55-90},
  month    = {May},
  abstract = {This paper deals with the numerical resolution of a shallow water viscoplastic flow model. Viscoplastic materials are characterized by the existence of a yield stress: below a certain critical threshold in the imposed stress, there is no deformation and the material behaves like a rigid solid, but when that yield value is exceeded, the material flows like a fluid. In the context of avalanches, it means that after going down a slope, the material can stop and its free surface has a non-trivial shape, as opposed to the case of water (Newtonian fluid). The model involves variational inequalities associated with the yield threshold: finite-volume schemes are used together with duality methods (namely Augmented Lagrangian and Berm{\'u}dez–Moreno) to discretize the problem. To be able to accurately simulate the stopping behavior of the avalanche, new schemes need to be designed, involving the classical notion of well-balancing. In the present context, it needs to be extended to take into account the viscoplastic nature of the material as well as general bottoms with wet/dry fronts which are encountered in geophysical geometries. We derived such schemes and numerical experiments are presented to show their performances.},
  keywords = { Viscoplastic; Shallow water; Well-balanced; Finite volume; Variational inequality; Bingham},
}

This paper deals with the numerical resolution of a shallow water viscoplastic flow model. Viscoplastic materials are characterized by the existence of a yield stress: below a certain critical threshold in the imposed stress, there is no deformation and the material behaves like a rigid solid, but when that yield value is exceeded, the material flows like a fluid. In the context of avalanches, it means that after going down a slope, the material can stop and its free surface has a non-trivial shape, as opposed to the case of water (Newtonian fluid). The model involves variational inequalities associated with the yield threshold: finite-volume schemes are used together with duality methods (namely Augmented Lagrangian and Bermúdez–Moreno) to discretize the problem. To be able to accurately simulate the stopping behavior of the avalanche, new schemes need to be designed, involving the classical notion of well-balancing. In the present context, it needs to be extended to take into account the viscoplastic nature of the material as well as general bottoms with wet/dry fronts which are encountered in geophysical geometries. We derived such schemes and numerical experiments are presented to show their performances.

@Article{ramirez_etal_2014_1,
  author        = {Ram{\'i}rez-Romero, Eduardo and Vichi, Marcelo and Castro D{\'i}az, Manuel J. and Mac{\'i}as, Jorge and Mac{\'i}as, Diego-M. and Garc{\'i}a, Carlos-M. and Bruno, Miguel},
  title         = {{M}odeling biogeochemical seasonal cycle in the {S}trait of {G}ibraltar},
  journal       = {Journal of Marine Systems},
  year          = {2014},
  volume        = {139},
  pages         = {348–361},
  month         = {November},
  abstract      = {A physical-biological coupled model was used to estimate the effect of the physical processes
at the Strait of Gibraltar over the biogeochemical features of the Atlantic Inflow (AI) towards the
Mediterranean Sea. This work was focused on the seasonal variation of the biogeochemical patterns in
the AI and the role of the Strait; including primary production and phytoplankton features. As the
physical model is 1D (horizontal) and two-layer, different integration methods for the primary
production in the Biogeochemical Fluxes Model (BFM) have been evaluated. An approach based on the
integration of a production-irradiance function was the chosen method. Using this Plankton Functional
Type model (BFM), a simplified phytoplankton seasonal cycle in the AI was simulated. Main results
included a principal bloom in spring dominated by nanoflagellates, whereas minimum biomass (mostly
picophytoplankton) was simulated during summer. Physical processes occurring in the Strait could
trigger primary production and raised phytoplankton biomass (during spring and autumn), mainly due
to two combined effects. First, in the Strait a strong interfacial mixing (causing nutrient supply to the
upper layer) is produced, and, second, a shoaling of the surface Atlantic layer occurs eastward. Our
results show that these phenomena caused an integrated production of 105 g C m−2 year−1 in the
eastern side of the Strait, and would also modify the proportion of the different phytoplankton groups.
Nanoflagellates were favored during spring/autumn while picophytoplankton is more abundant in
summer. Finally, AI could represent a relevant source of nutrients and biomass to Alboran Sea,
fertilizing upper layer of this area with 4.95 megatons nitrate year-1 and 0.44 megatons C year-1. A
main advantage of this coupled model is the capability of solving relevant high-resolution processes as
the tidal forcing without an expensive performance, allowing to assess the effect of these phenomena
on the biogeochemical patterns at longer time scales.},
  impact_factor = {2.476},
  keywords      = {Strait of Gibraltar; Tidal mixing; Biogeochemical patterns; Atlantic Inflow; Alboran Sea},
  url           = {http://www.sciencedirect.com/science/article/pii/S0924796314001833},
}

A physical-biological coupled model was used to estimate the effect of the physical processes at the Strait of Gibraltar over the biogeochemical features of the Atlantic Inflow (AI) towards the Mediterranean Sea. This work was focused on the seasonal variation of the biogeochemical patterns in the AI and the role of the Strait; including primary production and phytoplankton features. As the physical model is 1D (horizontal) and two-layer, different integration methods for the primary production in the Biogeochemical Fluxes Model (BFM) have been evaluated. An approach based on the integration of a production-irradiance function was the chosen method. Using this Plankton Functional Type model (BFM), a simplified phytoplankton seasonal cycle in the AI was simulated. Main results included a principal bloom in spring dominated by nanoflagellates, whereas minimum biomass (mostly picophytoplankton) was simulated during summer. Physical processes occurring in the Strait could trigger primary production and raised phytoplankton biomass (during spring and autumn), mainly due to two combined effects. First, in the Strait a strong interfacial mixing (causing nutrient supply to the upper layer) is produced, and, second, a shoaling of the surface Atlantic layer occurs eastward. Our results show that these phenomena caused an integrated production of 105 g C m−2 year−1 in the eastern side of the Strait, and would also modify the proportion of the different phytoplankton groups. Nanoflagellates were favored during spring/autumn while picophytoplankton is more abundant in summer. Finally, AI could represent a relevant source of nutrients and biomass to Alboran Sea, fertilizing upper layer of this area with 4.95 megatons nitrate year-1 and 0.44 megatons C year-1. A main advantage of this coupled model is the capability of solving relevant high-resolution processes as the tidal forcing without an expensive performance, allowing to assess the effect of these phenomena on the biogeochemical patterns at longer time scales.

@Article{fernandez-nieto2014influence,
  author   = {Fern{\'a}ndez Nieto, E. D. and Lucas and Morales de Luna, Tom{\'a}s and Cordier, St{\'e}phane},
  title    = {{O}n the influence of the thickness of the sediment moving layer in the definition of the bedload transport formula in {E}xner systems},
  journal  = {Computers {\&} Fluids},
  year     = {2014},
  volume   = {91},
  pages    = {87--106},
  abstract = {In this paper we study Exner system and introduce a modified general definition for bedload transport flux. The new formulation has the advantage of taking into account the thickness of the sediment layer which avoids mass conservation problems in certain situations. Moreover, it reduces to a classical solid transport discharge formula in the case of quasi-uniform regime. We also present several numerical tests where we compare the proposed sediment transport formula with the classical formulation and we show the behavior of the new model in different configurations.},
  url      = {http://www.sciencedirect.com/science/article/pii/S0045793013004866},
}

In this paper we study Exner system and introduce a modified general definition for bedload transport flux. The new formulation has the advantage of taking into account the thickness of the sediment layer which avoids mass conservation problems in certain situations. Moreover, it reduces to a classical solid transport discharge formula in the case of quasi-uniform regime. We also present several numerical tests where we compare the proposed sediment transport formula with the classical formulation and we show the behavior of the new model in different configurations.

@Article{moralesdeluna2014relation,
  author   = {Morales de Luna, Tom{\'a}s and Castro D{\'i}az, Manuel J. and Par{\'e}s, Carlos},
  title    = {{R}elation between {PVM} schemes and simple {R}iemann solvers},
  journal  = {Numerical Methods for Partial Differential Equations},
  year     = {2014},
  volume   = {30(4)},
  pages    = {1315-1341},
  month    = {mar},
  abstract = {Approximate Riemann solvers (ARS) and polynomial viscosity matrix
	(PVM) methods constitute two general frameworks to derive numerical
	schemes for hyperbolic systems of Partial Differential Equations
	(PDE's). In this work, the relation between these two frameworks
	is analyzed: we show that every PVM method can be interpreted in
	terms of an approximate Riemann solver provided that it is based
	on a polynomial that interpolates the absolute value function at
	some points. Furthermore, the converse is true provided that the
	ARS satisfies a technical property to be specified. Besides its
	theoretical interest, this relation provides a useful tool to investigate
	the properties of some well-known numerical methods that are particular
	cases of PVM methods, as the analysis of some properties is easier
	for ARS methods. We illustrate this usefulness by analyzing the
	positivity-preservation property of some well-known numerical methods
	for the shallow water system. © 2014 Wiley Periodicals, Inc. Numer
	Methods Partial Differential Eq, 2014},
  keywords = {finite volume schemes, PVM schemes, Riemann solvers, Shallow water equations},
  url      = {http://onlinelibrary.wiley.com/doi/10.1002/num.21871/abstract},
}

Approximate Riemann solvers (ARS) and polynomial viscosity matrix (PVM) methods constitute two general frameworks to derive numerical schemes for hyperbolic systems of Partial Differential Equations (PDE's). In this work, the relation between these two frameworks is analyzed: we show that every PVM method can be interpreted in terms of an approximate Riemann solver provided that it is based on a polynomial that interpolates the absolute value function at some points. Furthermore, the converse is true provided that the ARS satisfies a technical property to be specified. Besides its theoretical interest, this relation provides a useful tool to investigate the properties of some well-known numerical methods that are particular cases of PVM methods, as the analysis of some properties is easier for ARS methods. We illustrate this usefulness by analyzing the positivity-preservation property of some well-known numerical methods for the shallow water system. © 2014 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2014

@Article{jabaloy_etal_2014,
  author        = {Jabaloy, A. and Lobo, F.-J. and Azor, A. and Mart{\'i}n-Rosales, W. and P{\'e}rez-Pe{\~n}a, J.-V. and B{\'a}rcenas, P. and Mac{\'i}as, Jorge and Fern{\'a}ndez-Salas, L.-M. and V{\'a}zquez-V{\'i}lchez, M.},
  title         = {{S}ix thousand years of coastline evolution in the {G}uadalfeo deltaic system (southern {I}berian {P}eninsula)},
  journal       = {Geomorphology},
  year          = {2014},
  volume        = {206},
  pages         = {374 - 391},
  impact_factor = {2.52},
  url           = {http://www.sciencedirect.com/science/article/pii/S0169555X13005254},
}

@Article{Lobo_etal_2014_MGR,
  author   = {Lobo, F.-J. and Goff, J.-A. and Mendes, I. and B{\'a}rcenas, P. and Fern{\'a}ndez-Salas, L.-M. and Mart{\'i}n-Rosales, W. and Mac{\'i}as, Jorge and D{\'i}az-del-R{\'i}o, V.},
  title    = {{S}patial variability of prodeltaic undulations on the {G}uadalfeo {R}iver prodelta: {N}ew clues for their genetic interpretation},
  journal  = {Marine Geophysical Research},
  year     = {2014},
  abstract = {Two fields of prodeltaic undulations located off the Guadalfeo River were studied by integrating surficial (multibeam bathymetry, backscatter, sediment samples) and sub-surface (seismic profiles, sediment cores) data. Our main motivation was to analyze the along- and across-shelf variability of the seafloor undulations, in order to obtain useful insights into genetic mechanisms. A geostatistical analysis was performed, based on the determination of characteristic parameters and derived relationships. The undulations occur over a concave-upward surface which shows a seaward-decreasing slope. Most of the undulations are symmetrical to asymmetrically-oriented toward the coast. Two main fields are correlated with the present and previous river mouths. The western field, associated with the modern river mouth, is highly symmetrical, with the higher undulations in an axial position and diminishing the width/height relationship both laterally and downslope. In contrast, the eastern field, associated with an historic river mouth, shows lower-amplitude undulations, the width/height changes are less pronounced, and the undulations are more elongated. The two undulation fields exhibit subseafloor reflections that are subparallel to the seafloor, with peaks that migrate upslope upward in the stratigraphic column and which appear to correlate with coarse-grained layers. We support the contention that prodeltaic undulations off the Guadalfeo River should be regarded as sediment waves. Assuming a sediment-wave process, a strong normal-to-contour sediment flows with a riverine origin (e.g., hyperpycnal flows) may have been active during undulation generation. Both morphometric parameters of the river basin and estimations of sediment concentration during exceptional flood events are in agreement with an episodic activity of high freshwater discharges. Most of the geomorphic parameters and stratigraphic observations indicate a change of sediment supply conditions related to the change in river mouth position, attributed to a temporal change in the activity of hyperpycnal flows.},
  keywords = {Alboran Sea - Guadalfeo River prodelta - Submarine undulations - Multibeam bathymetry - Geostatistics},
  url      = {http://link.springer.com/article/10.1007/s11001-014-9233-9},
}

Two fields of prodeltaic undulations located off the Guadalfeo River were studied by integrating surficial (multibeam bathymetry, backscatter, sediment samples) and sub-surface (seismic profiles, sediment cores) data. Our main motivation was to analyze the along- and across-shelf variability of the seafloor undulations, in order to obtain useful insights into genetic mechanisms. A geostatistical analysis was performed, based on the determination of characteristic parameters and derived relationships. The undulations occur over a concave-upward surface which shows a seaward-decreasing slope. Most of the undulations are symmetrical to asymmetrically-oriented toward the coast. Two main fields are correlated with the present and previous river mouths. The western field, associated with the modern river mouth, is highly symmetrical, with the higher undulations in an axial position and diminishing the width/height relationship both laterally and downslope. In contrast, the eastern field, associated with an historic river mouth, shows lower-amplitude undulations, the width/height changes are less pronounced, and the undulations are more elongated. The two undulation fields exhibit subseafloor reflections that are subparallel to the seafloor, with peaks that migrate upslope upward in the stratigraphic column and which appear to correlate with coarse-grained layers. We support the contention that prodeltaic undulations off the Guadalfeo River should be regarded as sediment waves. Assuming a sediment-wave process, a strong normal-to-contour sediment flows with a riverine origin (e.g., hyperpycnal flows) may have been active during undulation generation. Both morphometric parameters of the river basin and estimations of sediment concentration during exceptional flood events are in agreement with an episodic activity of high freshwater discharges. Most of the geomorphic parameters and stratigraphic observations indicate a change of sediment supply conditions related to the change in river mouth position, attributed to a temporal change in the activity of hyperpycnal flows.

@Conference{Bingham1,
  author    = {Gallardo, Jos{\'e}-M. and Fern{\'a}ndez Nieto, E. D. and Vigneaux, Paul},
  title     = {{A} finite volume/duality method for {B}ingham viscoplastic flow},
  booktitle = {Numerical methods for hyperbolic problems: Theory and applications},
  year      = {2013},
  editor    = {V{\'a}zquez-Cend{\'o}n, Elena and Hidalgo, Arturo and Garc{\'i}a-Navarro, Pilar and Cea, Luis},
  pages     = {121-126},
  publisher = {CRC Press/Balkema},
}

@Article{fernandez-nieto_multilayer_2013,
  author   = {Fern{\'a}ndez Nieto, E. D. and E.-H., Kon{\'e} and Morales de Luna, Tom{\'a}s and B{\"u}rger, R.},
  title    = {{A} multilayer shallow water system for polydisperse sedimentation},
  journal  = {Journal of Computational Physics},
  year     = {2013},
  volume   = {238},
  pages    = {281--314},
  month    = {apr},
  abstract = {This work considers the flow of a fluid containing one disperse substance consisting of small particles that belong to different species differing in size and density. The flow is modelled by combining a multilayer shallow water approach with a polydisperse sedimentation process. This technique allows one to keep information on the vertical distribution of the solid particles in the mixture, and thereby to model the segregation of the particle species from each other, and from the fluid, taking place in the vertical direction of the gravity body force only. This polydisperse sedimentation process is described by the well-known Masliyah–Lockett–Bassoon (MLB) velocity functions. The resulting multilayer sedimentation-flow model can be written as a hyperbolic system with nonconservative products. The definitions of the nonconservative products are related to the hydrostatic pressure and to the mass and momentum hydrodynamic transfer terms between the layers. For the numerical discretization a strategy of two steps is proposed, where the first one is also divided into two parts. In the first step, instead of approximating the complete model, we approximate a reduced model with a smaller number of unknowns. Then, taking advantage of the fact that the concentrations are passive scalars in the system, we approximate the concentrations of the different species by an upwind scheme related to the numerical flux of the total concentration. In the second step, the effect of the transference terms defined in terms of the MLB model is introduced. These transfer terms are approximated by using a numerical flux function used to discretize the 1D vertical polydisperse model, see B{\"u}rger et al. [ R. B{\"u}rger, A. Garc{\'i}a, K.H. Karlsen, J.D. Towers, A family of numerical schemes for kinematic flows with discontinuous flux, J. Eng. Math. 60 (2008) 387–425]. Finally, some numerical examples are presented. Numerical results suggest that the multilayer shallow water model could be adequate in situations where the settling takes place from a suspension that undergoes horizontal movement.},
  keywords = {Finite volume, Multilayer, Polydisperse, Sediment, Shallow water},
  url      = {http://www.sciencedirect.com/science/article/pii/S0021999112007395},
}

This work considers the flow of a fluid containing one disperse substance consisting of small particles that belong to different species differing in size and density. The flow is modelled by combining a multilayer shallow water approach with a polydisperse sedimentation process. This technique allows one to keep information on the vertical distribution of the solid particles in the mixture, and thereby to model the segregation of the particle species from each other, and from the fluid, taking place in the vertical direction of the gravity body force only. This polydisperse sedimentation process is described by the well-known Masliyah–Lockett–Bassoon (MLB) velocity functions. The resulting multilayer sedimentation-flow model can be written as a hyperbolic system with nonconservative products. The definitions of the nonconservative products are related to the hydrostatic pressure and to the mass and momentum hydrodynamic transfer terms between the layers. For the numerical discretization a strategy of two steps is proposed, where the first one is also divided into two parts. In the first step, instead of approximating the complete model, we approximate a reduced model with a smaller number of unknowns. Then, taking advantage of the fact that the concentrations are passive scalars in the system, we approximate the concentrations of the different species by an upwind scheme related to the numerical flux of the total concentration. In the second step, the effect of the transference terms defined in terms of the MLB model is introduced. These transfer terms are approximated by using a numerical flux function used to discretize the 1D vertical polydisperse model, see Bürger et al. [ R. Bürger, A. García, K.H. Karlsen, J.D. Towers, A family of numerical schemes for kinematic flows with discontinuous flux, J. Eng. Math. 60 (2008) 387–425]. Finally, some numerical examples are presented. Numerical results suggest that the multilayer shallow water model could be adequate in situations where the settling takes place from a suspension that undergoes horizontal movement.

@Article{wafCF2013,
  author   = {de la Asunci{\'o}n, Marc and Castro D{\'i}az, Manuel J. and Fern{\'a}ndez Nieto, E. D. and M. Mantas, Jos{\'e} and Ortega-Acosta, Sergio and Gonz{\'a}lez-Vida, J.-M.},
  title    = {{E}fficient {GPU} implementation of a two waves {TVD}-{WAF} method for the two-dimensional one layer shallow water system on structured meshes},
  journal  = {Computers {\&} Fluids},
  year     = {2013},
  volume   = {80},
  pages    = {441-452},
  abstract = {The numerical solutions of shallow water equations are useful for applications related to geophysical flows that usually take place in large computational domains and could require real time calculation. Therefore, parallel versions of accurate and efficient numerical solvers for high performance platforms are needed to be able to deal with these simulation scenarios in reasonable times. In this paper we present an efficient CUDA implementation of a first and second order HLL methods and a two-waves TVD-WAF one. We propose to write all these methods under a common framework, such as, their CUDA implementations share the same structure. In particular, the reformulation of TVD-WAF numerical flux and the improved definition of the flux limiter allows us to obtain a more robust solver in situations like wet/dry fronts. Finally, some numerical tests are presented showing that the TVD-WAF method is slightly slower that the first order HLL method and two times faster than the second order HLL method, but it provides numerical results almost as accurate as the second order HLL scheme.},
  url      = {http://dx.doi.org/10.1016/j.compfluid.2012.01.012},
}

The numerical solutions of shallow water equations are useful for applications related to geophysical flows that usually take place in large computational domains and could require real time calculation. Therefore, parallel versions of accurate and efficient numerical solvers for high performance platforms are needed to be able to deal with these simulation scenarios in reasonable times. In this paper we present an efficient CUDA implementation of a first and second order HLL methods and a two-waves TVD-WAF one. We propose to write all these methods under a common framework, such as, their CUDA implementations share the same structure. In particular, the reformulation of TVD-WAF numerical flux and the improved definition of the flux limiter allows us to obtain a more robust solver in situations like wet/dry fronts. Finally, some numerical tests are presented showing that the TVD-WAF method is slightly slower that the first order HLL method and two times faster than the second order HLL method, but it provides numerical results almost as accurate as the second order HLL scheme.

@Article{Laser_TALANTA_2013,
  author  = {Moros, J. and Serrano, J. and Galllego, F.-J. and Mac{\'i}as, Jorge and Laserna, J.-J.},
  title   = {{R}ecognition of explosives fingerprints on objects for courier services using machine learning methods and laser-induced breakdown spectroscopy},
  journal = {TALANTA},
  year    = {2013},
  volume  = {110},
  pages   = {108-117},
}

@Article{moralesdeluna13Reliability,
  author   = {Morales de Luna, Tom{\'a}s and Castro D{\'i}az, Manuel J. and Par{\'e}s, Carlos},
  title    = {{R}eliability of first order numerical schemes for solving shallow water system over abrupt topography},
  journal  = {Applied Mathematics and Computation},
  year     = {2013},
  volume   = {219},
  number   = {17},
  pages    = {9012--9032},
  abstract = {Abstract We compare some first order well-balanced numerical schemes for shallow water system with special interest in applications where there are abrupt variations of the topography. We show that the space step required to obtain a prescribed error depends on the method. Moreover, the solutions given by the numerical scheme can be significantly different if not enough space resolution is used. We shall pay special attention to the well-known hydrostatic reconstruction technique where it is shown that the effect of large bottom discontinuities might be missed and a modification is proposed to avoid this problem.},
}

Abstract We compare some first order well-balanced numerical schemes for shallow water system with special interest in applications where there are abrupt variations of the topography. We show that the space step required to obtain a prescribed error depends on the method. Moreover, the solutions given by the numerical scheme can be significantly different if not enough space resolution is used. We shall pay special attention to the well-known hydrostatic reconstruction technique where it is shown that the effect of large bottom discontinuities might be missed and a modification is proposed to avoid this problem.

@InBook{moralesdeluna2013sedimentNova,
  chapter   = {Mathematical Models for Simulation of Bedload and Suspension Sediment Transport},
  pages     = {239--261},
  title     = {{S}ediment transport: monitoring, modeling, and management},
  publisher = {Nova Publishers},
  year      = {2013},
  author    = {Morales de Luna, Tom{\'a}s},
  editor    = {Khan, Abdul-A. and Wu, Weiming},
  url       = {https://www.novapublishers.com/catalog/product_info.php?products_id=41654},
}

@Article{castro2012HLLC,
  author  = {Castro D{\'i}az, Manuel J. and Fern{\'a}ndez Nieto, E. D. and Morales de Luna, Tom{\'a}s and Narbona-Reina, Gladys and Par{\'e}s, Carlos},
  title   = {{A} {HLLC} scheme for nonconservative hyperbolic problems. {A}pplication to turbidity currents with sediment transport},
  journal = {ESAIM: Mathematical Modelling and Numerical Analysis},
  year    = {2012},
  volume  = {47},
  number  = {1},
  pages   = {1--32},
  url     = {http://www.esaim-m2an.org/action/displayAbstract?fromPage=online{\&}aid=8654736},
}

@Article{jpdc2012,
  author   = {de la Asunci{\'o}n, Marc and Mantas, Jos{\'e}-M. and Castro D{\'i}az, Manuel J. and Fern{\'a}ndez Nieto, E. D.},
  title    = {{A}n {MPI}-{CUDA} implementation of an improved {R}oe method for two-layer shallow water systems},
  journal  = {Journal of Parallel and Distributed Computing},
  year     = {2012},
  volume   = {72},
  number   = {9},
  pages    = {1065-1072},
  abstract = {The numerical solution of two-layer shallow water systems is required to simulate accurately stratified fluids, which are ubiquitous in nature: they appear in atmospheric flows, ocean currents, oil spills, etc. Moreover, the implementation of the numerical schemes to solve these models in realistic scenarios imposes huge demands of computing power. In this paper, we tackle the acceleration of these simulations in triangular meshes by exploiting the combined power of several CUDA-enabled GPUs in a GPU cluster. For that purpose, an improvement of a path conservative Roe-type finite volume scheme which is specially suitable for GPU implementation is presented, and a distributed implementation of this scheme which uses CUDA and MPI to exploit the potential of a GPU cluster is developed. This implementation overlaps MPI communication with CPU–GPU memory transfers and GPU computation to increase efficiency. Several numerical experiments, performed on a cluster of modern CUDA-enabled GPUs, show the efficiency of the distributed solver.},
  url      = {http://dx.doi.org/10.1016/j.jpdc.2011.07.012},
}

The numerical solution of two-layer shallow water systems is required to simulate accurately stratified fluids, which are ubiquitous in nature: they appear in atmospheric flows, ocean currents, oil spills, etc. Moreover, the implementation of the numerical schemes to solve these models in realistic scenarios imposes huge demands of computing power. In this paper, we tackle the acceleration of these simulations in triangular meshes by exploiting the combined power of several CUDA-enabled GPUs in a GPU cluster. For that purpose, an improvement of a path conservative Roe-type finite volume scheme which is specially suitable for GPU implementation is presented, and a distributed implementation of this scheme which uses CUDA and MPI to exploit the potential of a GPU cluster is developed. This implementation overlaps MPI communication with CPU–GPU memory transfers and GPU computation to increase efficiency. Several numerical experiments, performed on a cluster of modern CUDA-enabled GPUs, show the efficiency of the distributed solver.

@Book{Tsu_Alb_2012,
  title     = {{D}eslizamientos {S}ubmarinos y {T}sunamis en el {M}ar de {A}lbor{\'a}n. {U}n {E}jemplo de {M}odelizaci{\'o}n.},
  publisher = {Instituto Espa{\~n}ol de Oceanograf{\'i}a},
  year      = {2012},
  author    = {Mac{\'i}as, Jorge and Fern{\'a}ndez-Salas, L.-M. and Gonz{\'a}lez-Vida, J.-M. and V{\'a}zquez, J.-T. and Castro D{\'i}az, Manuel J. and B{\'a}rcenas, P. and D{\'i}az del R{\'i}o, V. and Morales de Luna, Tom{\'a}s and de la Asunci{\'o}n, Marc and Par{\'e}s, Carlos},
  volume    = {6},
  series    = {Temas de Oceanograf{\'i}a},
  address   = {Avda. del Brasil, 31. 28020 Madrid},
  comments  = {You can buy the printed version of this book at: https://www.libreriasl.com/catalogo/ver/id/487691/titulo/deslizamientos-submarinos-y-tsunamis-en-el-mar-de-alboran-un-ejemplo-de-modelizacion-numerica.html},
  keywords  = {Simulaci{\'o}n de Tsunamis, Tsunami de Albor{\'a}n, Deslizamientos submarinos},
  url       = {http://hdl.handle.net/10508/1073},
}

@InBook{castroCRC2012,
  chapter   = {5},
  pages     = {237-244},
  title     = {{IFCP} {R}iemann solver: {A}pplication to tsunami modelling using {GPU}s},
  publisher = {CRC Press},
  year      = {2012},
  author    = {Castro D{\'i}az, Manuel J. and de la Asunci{\'o}n, Marc and Mac{\'i}as, Jorge and Par{\'e}s, Carlos and Fern{\'a}ndez Nieto, E. D. and Gonz{\'a}lez-Vida, J.-M. and Morales de Luna, Tom{\'a}s},
  editor    = {E.-V{\'a}zquez and A.-Hidalgo and P.-Garc{\'i}a and L.-Cea},
}

@Article{jaas_2012,
  author        = {Moros, J. and Serrano, J. and S{\'a}nchez-Linares, Carlos and Mac{\'i}as, Jorge and Laserna, J.-J.},
  title         = {{N}ew chemometrics in laser-induced breakdown spectroscopy for recognizing},
  journal       = {Journal of Analytical Atomic Spectrometry},
  year          = {2012},
  impact_factor = {4.372},
  url           = {http://pubs.rsc.org},
}

@Conference{VBMFSDGC_2012,
  author    = {V{\'a}zquez, J.-T. and B{\'a}rcenas, P. and Mac{\'i}as, Jorge and Fern{\'a}ndez-Salas, L.-M. and D{\'i}az-del-R{\'i}o, V. and Gonz{\'a}lez-Vida, J.-M. and Castro D{\'i}az, Manuel J.},
  title     = {{T}sunamis generados por deslizamientos submarinos en el {M}ar de {A}lbor{\'a}n: {S}imulaci{\'o}n de un posible tsunami a partir del sistema de movimientos en masa {A}l-{B}oran{\'i}},
  booktitle = {Res{\'u}menes extendidos del VIII Congreso Geol{\'o}gico de Espa{\~n}a},
  year      = {2012},
  editor    = {Fern{\'a}ndez, L.-P. and Fern{\'a}ndez, A. and Cuesta, A. and Bahamonte, J.-R.},
  volume    = {13},
  pages     = {1761-1764},
}

@Article{moralesdeluna2010Duality,
  author   = {Morales de Luna, Tom{\'a}s and Castro D{\'i}az, Manuel J. and Par{\'e}s, Carlos},
  title    = {{A} {D}uality {M}ethod for {S}ediment {T}ransport {B}ased on\ a\ {M}odified {M}eyer-{P}eter {\&} {M}{\"u}ller {M}od},
  journal  = {Journal of Scientific Computing},
  year     = {2011},
  volume   = {48},
  number   = {1},
  pages    = {258-273},
  abstract = {This article focuses on the simulation of the sediment transport by a fluid in contact with a sediment layer. This phenomena can be modelled by using a coupled model constituted by a hydrodynamical component, described by a shallow water system, and a morphodynamical one, which depends on a solid transport flux given by some empirical law. The solid transport discharge proposed by Meyer-Peter {\&} M{\"u}ller is one of the most popular but it has the inconvenient of not including pressure forces. Due to this, this formula produces numerical simulations that are not realistic in zones where gravity effects are relevant, e.g. advancing front of the sand layer. Moreover, the thickness of the sediment layer is not taken into account and, as a consequence, mass conservation of sediment may fail. Fowler et al. proposed a generalization that takes into account gravity effects as well as the thickness of the sediment layer which is in better agreement with the physics of the problem. We propose to solve this system by using a path-conservative scheme for the hydrodynamical part and a duality method based on Berm{\'u}dez-Moreno algorithm for the morphodynamical component.
},
  url      = {http://dx.doi.org/10.1007/s10915-010-9447-1},
}

This article focuses on the simulation of the sediment transport by a fluid in contact with a sediment layer. This phenomena can be modelled by using a coupled model constituted by a hydrodynamical component, described by a shallow water system, and a morphodynamical one, which depends on a solid transport flux given by some empirical law. The solid transport discharge proposed by Meyer-Peter & Müller is one of the most popular but it has the inconvenient of not including pressure forces. Due to this, this formula produces numerical simulations that are not realistic in zones where gravity effects are relevant, e.g. advancing front of the sand layer. Moreover, the thickness of the sediment layer is not taken into account and, as a consequence, mass conservation of sediment may fail. Fowler et al. proposed a generalization that takes into account gravity effects as well as the thickness of the sediment layer which is in better agreement with the physics of the problem. We propose to solve this system by using a path-conservative scheme for the hydrodynamical part and a duality method based on Bermúdez-Moreno algorithm for the morphodynamical component.

@Article{cordier2011Bedload,
  author   = {Cordier, St{\'e}phane and Morales de Luna, Tom{\'a}s and Le, M.},
  title    = {{B}edload transport in shallow water models: {W}hy splitting (may) fail, how hyperbolicity (can) help},
  journal  = {Advances in Water Resources},
  year     = {2011},
  volume   = {34},
  number   = {8},
  pages    = {980--989},
  abstract = {{{\textless}p{\textgreater}{\textless}br/{\textgreater}In} this paper, we are concerned with sediment transport models consisting of a shallow water system coupled with the so called Exner equation to describe the evolution of the topography. We show that, for some bedload transport models like the well-known {Meyer-Peter} and M{\"u}ller model, the system is hyperbolic and, thus, linearly stable, only under some constraint on the velocity. In practical situations, this condition is hopefully fulfilled. Numerical approximations of such system are often based on a splitting method, solving first shallow water equation on a time step and, updating afterwards the topography. It is shown that this strategy can create spurious/unphysical oscillations which are related to the study of hyperbolicity. Using an upper bound of the largest eigenvector may improve the results although the instabilities cannot be always avoided, e.g. in supercritical regions.{\textless}/p{\textgreater}},
  keywords = {Exner equation, Hyperbolicity, Sediment transport, Shallow water system, Splitting methods, Stability},
  url      = {http://www.sciencedirect.com/science/article/pii/S0309170811000935},
}

\textlessp\textgreater\textlessbr/\textgreaterIn this paper, we are concerned with sediment transport models consisting of a shallow water system coupled with the so called Exner equation to describe the evolution of the topography. We show that, for some bedload transport models like the well-known Meyer-Peter and Müller model, the system is hyperbolic and, thus, linearly stable, only under some constraint on the velocity. In practical situations, this condition is hopefully fulfilled. Numerical approximations of such system are often based on a splitting method, solving first shallow water equation on a time step and, updating afterwards the topography. It is shown that this strategy can create spurious/unphysical oscillations which are related to the study of hyperbolicity. Using an upper bound of the largest eigenvector may improve the results although the instabilities cannot be always avoided, e.g. in supercritical regions.\textless/p\textgreater

@Article{castro2011comptes,
  author   = {Castro D{\'i}az, Manuel J. and Ortega-Acosta, Sergio and de la Asunci{\'o}n, Marc and Mantas, Jos{\'e} Miguel and Gallardo, Jos{\'e}-M.},
  title    = {{GPU} computing for shallow water flow simulation based on finite volume schemes},
  journal  = {Comptes Rendus M{\'e}canique},
  year     = {2011},
  volume   = {339},
  number   = {2},
  pages    = {165-184},
  abstract = {This article is a review of the work that we are carrying out to efficiently simulate shallow water flows. In this paper, we focus on the efficient implementation of path-conservative Roe type high-order finite volume schemes to simulate shallow flows that are supposed to be governed by the one-layer or two-layer shallow water systems, formulated under the form of a conservation law with source terms. The implementation of the scheme is carried out on Graphics Processing Units (GPUs), thus achieving a substantial improvement of the speedup with respect to normal CPUs. Finally, some numerical experiments are presented. },
}

This article is a review of the work that we are carrying out to efficiently simulate shallow water flows. In this paper, we focus on the efficient implementation of path-conservative Roe type high-order finite volume schemes to simulate shallow flows that are supposed to be governed by the one-layer or two-layer shallow water systems, formulated under the form of a conservation law with source terms. The implementation of the scheme is carried out on Graphics Processing Units (GPUs), thus achieving a substantial improvement of the speedup with respect to normal CPUs. Finally, some numerical experiments are presented.

@Article{castro2011loss,
  author   = {Castro D{\'i}az, Manuel J. and Fern{\'a}ndez Nieto, E. D. and Gonz{\'a}lez Vida, Jos{\'e} M. and Par{\'e}s, Carlos},
  title    = {{N}umerical {T}reatment of the {L}oss of {H}yperbolicity of\ the\ {T}wo-{L}ayer {S}hallow-{W}ater {S}yst},
  journal  = {Journal of Scientific Computing},
  year     = {2011},
  volume   = {48},
  number   = {1},
  pages    = {16-40},
  abstract = {This article is devoted to the numerical solution of the inviscid two-layer shallow water system. This system may lose the hyperbolic character when the shear between the layer is big enough. This loss of hyperbolicity is related to the appearance of shear instabilities that leads, in real flows, to intense mixing of the two layers that the model is not able to simulate. The strategy here is to add some extra friction terms, which are supposed to parameterize the loss of mechanical energy due to mixing, to get rid of this difficulty. The main goal is to introduce a technique allowing one to add locally and automatically an ‘optimal’ amount of shear stress to make the flow to remain in the hyperbolicity region. To do this, first an easy criterium to check the hyperbolicity of the system for a given state is proposed and checked. Next, we introduce a predictor/corrector strategy. In the predictor stage, a numerical scheme is applied to the system without extra friction. In the second stage, a discrete semi-implicit linear friction law is applied at any cell in which the predicted states are not in the hyperbolicity region. The coefficient of this law is calculated so that the predicted states are driven to the boundary of the hyperbolicity region according to the proposed criterium. The numerical scheme to be used at the first stage has to be able to advance in time in presence of complex eigenvalues: we propose here a family of path-conservative numerical scheme having this property. Finally, some numerical tests have been performed to assess the efficiency of the proposed strategy.
},
  url      = {http://dx.doi.org/10.1007/s10915-010-9427-5},
}

This article is devoted to the numerical solution of the inviscid two-layer shallow water system. This system may lose the hyperbolic character when the shear between the layer is big enough. This loss of hyperbolicity is related to the appearance of shear instabilities that leads, in real flows, to intense mixing of the two layers that the model is not able to simulate. The strategy here is to add some extra friction terms, which are supposed to parameterize the loss of mechanical energy due to mixing, to get rid of this difficulty. The main goal is to introduce a technique allowing one to add locally and automatically an ‘optimal’ amount of shear stress to make the flow to remain in the hyperbolicity region. To do this, first an easy criterium to check the hyperbolicity of the system for a given state is proposed and checked. Next, we introduce a predictor/corrector strategy. In the predictor stage, a numerical scheme is applied to the system without extra friction. In the second stage, a discrete semi-implicit linear friction law is applied at any cell in which the predicted states are not in the hyperbolicity region. The coefficient of this law is calculated so that the predicted states are driven to the boundary of the hyperbolicity region according to the proposed criterium. The numerical scheme to be used at the first stage has to be able to advance in time in presence of complex eigenvalues: we propose here a family of path-conservative numerical scheme having this property. Finally, some numerical tests have been performed to assess the efficiency of the proposed strategy.

@Article{castro2011ifcp,
  author   = {Fern{\'a}ndez Nieto, E. D. and Castro D{\'i}az, Manuel J. and Par{\'e}s, Carlos},
  title    = {{O}n an {I}ntermediate {F}ield {C}apturing {R}iemann {S}olver {B}ased on a {P}arabolic {V}iscosity {M}atrix for the {T}wo-{L}ayer {S}hallow {W}ater {S}ystem},
  journal  = {Journal of Scientific Computing},
  year     = {2011},
  volume   = {48},
  number   = {1},
  pages    = {117-140},
  abstract = {The goal of this article is to design a new approximate Riemann solver for the two-layer shallow water system which is fast compared to Roe schemes and accurate compared to Lax-Friedrichs, FORCE, or GFORCE schemes (see Castro et al. in Math. Comput. 79:1427–1472, 2010 ). This Riemann solver is based on a suitable decomposition of a Roe matrix (see Toumi in J. Comput. Phys. 102(2):360–373, 1992 ) by means of a parabolic viscosity matrix (see Degond et al. in C. R. Acad. Sci. Paris 1 328:479–483, 1999 ) that captures some information concerning the intermediate characteristic fields. The corresponding first order numerical scheme, which is called IFCP (Intermediate Field Capturing Parabola) is linearly L ∞ -stable, well-balanced, and it doesn’t require an entropy-fix technique. Some numerical experiments are presented to compare the behavior of this new scheme with Roe and GFORCE methods.},
  url      = {http://dx.doi.org/10.1007/s10915-011-9465-7},
}

The goal of this article is to design a new approximate Riemann solver for the two-layer shallow water system which is fast compared to Roe schemes and accurate compared to Lax-Friedrichs, FORCE, or GFORCE schemes (see Castro et al. in Math. Comput. 79:1427–1472, 2010 ). This Riemann solver is based on a suitable decomposition of a Roe matrix (see Toumi in J. Comput. Phys. 102(2):360–373, 1992 ) by means of a parabolic viscosity matrix (see Degond et al. in C. R. Acad. Sci. Paris 1 328:479–483, 1999 ) that captures some information concerning the intermediate characteristic fields. The corresponding first order numerical scheme, which is called IFCP (Intermediate Field Capturing Parabola) is linearly L ∞ -stable, well-balanced, and it doesn’t require an entropy-fix technique. Some numerical experiments are presented to compare the behavior of this new scheme with Roe and GFORCE methods.

@Article{munozpares2011,
  author  = {Mu{\~n}oz Ruiz, Mar{\'i}a Luz and Par{\'e}s, Carlos},
  title   = {{O}n the convergence and well-balanced property of path-conservative numerical schemes for systems of balance laws},
  journal = {Journal of Scientific Computing},
  year    = {2011},
  volume  = {48},
  number  = {1-3},
  pages   = {274-295},
  url     = {http://www.springerlink.com/content/h84677m854r25780/},
}

@Article{asuncion2011cuda1L,
  author   = {de la Asunci{\'o}n, Marc and Mantas, Jos{\’e}-Miguel and Castro D{\'i}az, Manuel J.},
  title    = {{S}imulation of one-layer shallow water systems on\ multicore and {CUDA} architecture},
  journal  = {The Journal of Supercomputing},
  year     = {2011},
  volume   = {58},
  number   = {2},
  pages    = {206-214},
  abstract = {The numerical solution of shallow water systems is useful for several applications related to geophysical flows, but the big dimensions of the domains suggests the use of powerful accelerators to obtain numerical results in reasonable times. This paper addresses how to speed up the numerical solution of a first order well-balanced finite volume scheme for 2D one-layer shallow water systems by using modern Graphics Processing Units (GPUs) supporting the NVIDIA CUDA programming model. An algorithm which exploits the potential data parallelism of this method is presented and implemented using the CUDA model in single and double floating point precision. Numerical experiments show the high efficiency of this CUDA solver in comparison with a CPU parallel implementation of the solver and with respect to a previously existing GPU solver based on a shading language.
},
  url      = {http://dx.doi.org/10.1007/s11227-010-0406-2},
}

The numerical solution of shallow water systems is useful for several applications related to geophysical flows, but the big dimensions of the domains suggests the use of powerful accelerators to obtain numerical results in reasonable times. This paper addresses how to speed up the numerical solution of a first order well-balanced finite volume scheme for 2D one-layer shallow water systems by using modern Graphics Processing Units (GPUs) supporting the NVIDIA CUDA programming model. An algorithm which exploits the potential data parallelism of this method is presented and implemented using the CUDA model in single and double floating point precision. Numerical experiments show the high efficiency of this CUDA solver in comparison with a CPU parallel implementation of the solver and with respect to a previously existing GPU solver based on a shading language.

@Article{2011_Barcenas_et_al_JIG,
  author        = {B{\'a}rcenas, P. and Lobo, F.-J. and Mac{\'i}as, Jorge and Fern{\'a}ndez-Salas, L.-M. and D{\'i}az-del-R{\'i}o, V.},
  title         = {{S}patial variability of surficial sediments and bakscatter on the northern shelf of the {A}lboran {S}ea: {T}he effects of hydrodynamic forcing and local sediment supplies},
  journal       = {J. Iber. Geol.},
  year          = {2011},
  volume        = {37},
  number        = {2},
  pages         = {195:214},
  impact_factor = {0.973},
}

@Article{gallardo2011compact,
  author   = {Gallardo, Jos{\'e}-M. and Ortega-Acosta, Sergio and de la Asunci{\'o}n, Marc and Mantas-Ruiz, Jos{\'e}-Miguel},
  title    = {{T}wo-{D}imensional {C}ompact {T}hird-{O}rder {P}olynomial {R}econstructions. {S}olving {N}onconservative {H}yperbolic {S}ystems {U}sing {GPU}s},
  journal  = {Journal of Scientific Computing},
  year     = {2011},
  volume   = {48},
  number   = {1},
  pages    = {141-163},
  abstract = {We present a new kind of high-order reconstruction operator of polynomial type, which is used in combination with the scheme presented in Castro et al. (J. Sci. Comput. 39:67–114, 2009 ) for solving nonconservative hyperbolic systems. The implementation of the scheme is carried out on Graphics Processing Units (GPUs), thus achieving a substantial improvement of the speedup with respect to normal CPUs. As an application, the two-dimensional shallow water equations with geometrical source term due to the bottom slope is considered.
},
  url      = {http://dx.doi.org/10.1007/s10915-011-9470-x},
}

We present a new kind of high-order reconstruction operator of polynomial type, which is used in combination with the scheme presented in Castro et al. (J. Sci. Comput. 39:67–114, 2009 ) for solving nonconservative hyperbolic systems. The implementation of the scheme is carried out on Graphics Processing Units (GPUs), thus achieving a substantial improvement of the speedup with respect to normal CPUs. As an application, the two-dimensional shallow water equations with geometrical source term due to the bottom slope is considered.

@Article{bouchut2010subsonic,
  author   = {Bouchut, Fran{\c{c}}ois and Morales de Luna, Tom{\'a}s},
  title    = {{A} {S}ubsonic well-balanced reconstruction scheme for shallow water flows},
  journal  = {SIAM Journal on Numerical Analysis},
  year     = {2010},
  volume   = {48},
  number   = {5},
  pages    = {1733},
  abstract = {We consider the Saint-Venant system for shallow water flows with non-flat
	bottom. In the past years, efficient well-balanced methods have been
	proposed in order to well resolve solutions close to steady states
	at rest. Here we describe a strategy based on a local subsonic steady-state
	reconstruction that allows to derive a subsonic-well-balanced scheme,
	preserving exactly all the subsonic steady states. It generalizes
	the now wellknown hydrostatic solver, and as the latter it preserves
	nonnegativity of water height and satisfies a semi-discrete entropy
	inequality. An application to the Euler-Poisson system is proposed.},
  keywords = {shallow water, subsonic reconstruction, subsonic steady states, well-balanced scheme, semi-discrete entropy inequality},
  url      = {http://dx.doi.org/10.1137/090758416},
}

We consider the Saint-Venant system for shallow water flows with non-flat bottom. In the past years, efficient well-balanced methods have been proposed in order to well resolve solutions close to steady states at rest. Here we describe a strategy based on a local subsonic steady-state reconstruction that allows to derive a subsonic-well-balanced scheme, preserving exactly all the subsonic steady states. It generalizes the now wellknown hydrostatic solver, and as the latter it preserves nonnegativity of water height and satisfies a semi-discrete entropy inequality. An application to the Euler-Poisson system is proposed.

@Article{bruno_m_gv_v_2010,
  author  = {Bruno, Miguel and Mac{\'i}as, Jorge and Gonz{\'a}lez-Vida, J.-M. and V{\'a}zquez, {\'A}gueda},
  title   = {{A}nalysing the tidal-related origin of subinertial flows through the {S}trait of {G}ibraltar},
  journal = {J. Geophys. Res.},
  year    = {2010},
  url     = {http://www.agu.org/journals/jc/papersinpress.shtml#id2010JC006499},
}

@Article{barcenas_fs_m_l_dr_2010,
  author        = {B{\'a}rcenas, P. and Fern{\'a}ndez-Salas, Lu{\'i}s-Miguel and Mac{\'i}as, Jorge and J. Lobo, Francisco and D{\'i}az, V{\'i}ctor},
  title         = {{E}studio morfom{\'e}trico comparativo entre las ondulaciones de los prodeltas de los r{\'i}os de {A}ndaluc{\'i}a {O}riental},
  journal       = {Rev. Soc. Geol. Espa{\~n}a},
  year          = {2010},
  volume        = {22},
  number        = {1-2},
  impact_factor = {0.973},
}

@Article{M.Dumbser2010,
  author   = {Dumbser, Michael and Hidalgo, A. and Castro D{\'i}az, Manuel J. and Par{\'e}s, Carlos and Toro, E.-F.},
  title    = {{FORCE} schemes on unstructured meshes {II}: {N}onconservative hyperbolic systems},
  journal  = {Comput. Methods Appl. Mech. Engrg.},
  year     = {2010},
  volume   = {199},
  number   = {9-12},
  pages    = {625–647},
  abstract = {In this paper we propose a new high order accurate centered path-conservative method on unstructured triangular and tetrahedral meshes for the solution of multi-dimensional non-conservative hyperbolic systems, as they typically arise in the context of compressible multi-phase flows. Our path-conservative centered scheme is an extension of the centered method recently proposed in [40] for conservation laws, to which it reduces if the system matrix is the Jacobian of a flux function. The main advantage in the proposed centered approach compared to upwind methods is that no information about the eigenstructure of the system or Roe averages are needed. The final fully discrete high order accurate formulation in space and time is obtained using the general framework of PNPM schemes proposed in [16], which unifies in one single general family of schemes classical finite volume and discontinuous Galerkin methods. These PNPM methods can also be called reconstructed discontinuous Galerkin schemes, due to the use of the PNPM least-squares reconstruction operator. We show applications of our high order accurate unstructured centered method to the two- and three-dimensional Baer–Nunziato equations of compressible multiphase flows as introduced in [4].},
  url      = {http://www.sciencedirect.com/science/article/pii/S0045782509003612},
}

In this paper we propose a new high order accurate centered path-conservative method on unstructured triangular and tetrahedral meshes for the solution of multi-dimensional non-conservative hyperbolic systems, as they typically arise in the context of compressible multi-phase flows. Our path-conservative centered scheme is an extension of the centered method recently proposed in [40] for conservation laws, to which it reduces if the system matrix is the Jacobian of a flux function. The main advantage in the proposed centered approach compared to upwind methods is that no information about the eigenstructure of the system or Roe averages are needed. The final fully discrete high order accurate formulation in space and time is obtained using the general framework of PNPM schemes proposed in [16], which unifies in one single general family of schemes classical finite volume and discontinuous Galerkin methods. These PNPM methods can also be called reconstructed discontinuous Galerkin schemes, due to the use of the PNPM least-squares reconstruction operator. We show applications of our high order accurate unstructured centered method to the two- and three-dimensional Baer–Nunziato equations of compressible multiphase flows as introduced in [4].

@Article{castro2010force,
  author   = {Castro D{\'i}az, Manuel J. and Pardo, Alberto and Par{\'e}s, Carlos and Toro, E.-F.},
  title    = {{O}n some fast well-balanced first order solvers for nonconservative systems.},
  journal  = {MATHEMATICS OF COMPUTATION},
  year     = {2010},
  volume   = {79},
  number   = {271},
  pages    = {1427–1472},
  abstract = {The goal of this article is to design robust and simple first order
explicit solvers for one-dimensional nonconservative hyperbolic systems. These
solvers are intended to be used as the basis for higher order methods for one or
multidimensional problems. The starting point for the development of these
solvers is the general definition of a Roe linearization introduced by Toumi in
1992 based on the use of a family of paths. Using this concept, Roe methods
can be extended to nonconservative systems. These methods have good wellbalanced
and robustness properties, but they have also some drawbacks: in
particular, their implementation requires the explicit knowledge of the eigenstructure
of the intermediate matrices. Our goal here is to design numerical
methods based on a Roe linearization which overcome this drawback. The
idea is to split the Roe matrices into two parts which are used to calculate
the contributions at the cells to the right and to the left, respectively. This
strategy is used to generate two different one-parameter families of schemes
which contain, as particular cases, some generalizations to nonconservative systems
of the well-known Lax-Friedrichs, Lax-Wendroff, FORCE, and GFORCE
schemes. Some numerical experiments are presented to compare the behaviors
of the schemes introduced here with Roe methods.},
}

The goal of this article is to design robust and simple first order explicit solvers for one-dimensional nonconservative hyperbolic systems. These solvers are intended to be used as the basis for higher order methods for one or multidimensional problems. The starting point for the development of these solvers is the general definition of a Roe linearization introduced by Toumi in 1992 based on the use of a family of paths. Using this concept, Roe methods can be extended to nonconservative systems. These methods have good wellbalanced and robustness properties, but they have also some drawbacks: in particular, their implementation requires the explicit knowledge of the eigenstructure of the intermediate matrices. Our goal here is to design numerical methods based on a Roe linearization which overcome this drawback. The idea is to split the Roe matrices into two parts which are used to calculate the contributions at the cells to the right and to the left, respectively. This strategy is used to generate two different one-parameter families of schemes which contain, as particular cases, some generalizations to nonconservative systems of the well-known Lax-Friedrichs, Lax-Wendroff, FORCE, and GFORCE schemes. Some numerical experiments are presented to compare the behaviors of the schemes introduced here with Roe methods.

@Article{castro2010SEMA,
  author  = {Castro D{\'i}az, Manuel J. and de la Asunci{\'o}n, Marc and Mantas, Jos{\'e} Miguel and Ortega-Acosta, Sergio},
  title   = {{O}n the benefits of using {GPU}s to simulate shallow flows with finite volume schemes},
  journal = {Bol. Soc. Esp. Mat. Apl.},
  year    = {2010},
  volume  = {50},
  pages   = {27-44},
}

@InBook{asuncion2010cuda2L,
  pages     = {353-364},
  title     = {{P}rogramming {CUDA}-{B}ased {GPU}s to {S}imulate {T}wo-{L}ayer {S}hallow {W}ater {F}lows},
  publisher = {Springer Berlin / Heidelberg},
  year      = {2010},
  author    = {de la Asunci{\'o}n, Marc and Mantas, Jos{\'e}-Miguel and Castro D{\'i}az, Manuel J.},
  editor    = {D'Ambra, P. and Guarracino, M. and Talia, D.},
  volume    = {6272},
  series    = {Lecture Notes in Computer Science},
  abstract  = {The two-layer shallow water system is used as the numerical model to simulate several phenomena related to geophysical flows such as the steady exchange of two different water flows, as occurs in the Strait of Gibraltar, or the tsunamis generated by underwater landslides. The numerical solution of this model for realistic domains imposes great demands of computing power and modern Graphics Processing Units (GPUs) have demonstrated to be a powerful accelerator for this kind of computationally intensive simulations. This work describes an accelerated implementation of a first order well-balanced finite volume scheme for 2D two-layer shallow water systems using GPUs supporting the CUDA (Compute Unified Device Architecture) programming model and double precision arithmetic. This implementation uses the CUDA framewok to exploit efficiently the potential fine-grain data parallelism of the numerical algorithm. Two versions of the GPU solver are implemented and studied: one using both single and double precision, and another using only double precision. Numerical experiments show the efficiency of this CUDA solver on several GPUs and a comparison with an efficient multicore CPU implementation of the solver is also reported.},
  url       = {http://dx.doi.org/10.1007/978-3-642-15291-7_32},
}

The two-layer shallow water system is used as the numerical model to simulate several phenomena related to geophysical flows such as the steady exchange of two different water flows, as occurs in the Strait of Gibraltar, or the tsunamis generated by underwater landslides. The numerical solution of this model for realistic domains imposes great demands of computing power and modern Graphics Processing Units (GPUs) have demonstrated to be a powerful accelerator for this kind of computationally intensive simulations. This work describes an accelerated implementation of a first order well-balanced finite volume scheme for 2D two-layer shallow water systems using GPUs supporting the CUDA (Compute Unified Device Architecture) programming model and double precision arithmetic. This implementation uses the CUDA framewok to exploit efficiently the potential fine-grain data parallelism of the numerical algorithm. Two versions of the GPU solver are implemented and studied: one using both single and double precision, and another using only double precision. Numerical experiments show the efficiency of this CUDA solver on several GPUs and a comparison with an efficient multicore CPU implementation of the solver is also reported.

@Article{Dumbser2009,
  author   = {Dumbser, Michael and Castro D{\'i}az, Manuel J. and Par{\'e}s, Carlos and F.-Toro, Eleuterio},
  title    = {{ADER} schemes on unstructured meshes for nonconservative hyperbolic systems: {A}pplications to geophysical flows},
  journal  = {Computers {\&} Fluids},
  year     = {2009},
  volume   = {38},
  number   = {9},
  pages    = {1731–1748},
  abstract = {We develop a new family of well-balanced path-conservative quadrature-free one-step ADER finite volume and discontinuous Galerkin finite element schemes on unstructured meshes for the solution of hyperbolic partial differential equations with non-conservative products and stiff source terms. The fully discrete formulation is derived using the recently developed framework of explicit one-step PNPM schemes of arbitrary high order of accuracy in space and time for conservative hyperbolic systems [Dumbser M, Balsara D, Toro EF, Munz CD. A unified framework for the construction of one-step finite-volume and discontinuous Galerkin schemes. J Comput Phys 2008;227:8209–53]. The two key ingredients of our high order approach are: first, the high order accurate PNPM reconstruction operator on unstructured meshes, using the WENO strategy presented in [Dumbser M, K{\"a}ser M, Titarev VA Toro EF. Quadrature-free non-oscillatory finite volume schemes on unstructured meshes for nonlinear hyperbolic systems. J Comput Phys 2007;226:204–43] to ensure monotonicity at discontinuities, and second, a local space–time Galerkin scheme to predict the evolution of the reconstructed polynomial data inside each element during one time step to obtain a high order accurate one-step time discretization. This approach is also able to deal with stiff source terms as shown in [Dumbser M, Enaux C, Toro EF. Finite volume schemes of very high order of accuracy for stiff hyperbolic balance laws. J Comput Phys 2008;227:3971–4001]. These two key ingredients are combined with the recently developed path-conservative methods of Par{\'e}s [Par{\'e}s C. Numerical methods for nonconservative hyperbolic systems: a theoretical framework. SIAM J Numer Anal 2006;44:300–21] and Castro et al. [Castro MJ, Gallardo JM, Par{\'e}s C. High-order finite volume schemes based on reconstruction of states for solving hyperbolic systems with nonconservative products. Applications to shallow-water systems. Math Comput 2006;75:1103–34] to treat the non-conservative products properly. We show applications of our method to the two-layer shallow water equations as well as applications to the recently published depth-averaged two-fluid flow model of Pitman and Le [Pitman EB, Le L. A two-fluid model for avalanche and debris flows. Philos Trans Roy Soc A 2005;363:1573–601].},
  url      = {http://www.sciencedirect.com/science/article/pii/S0045793009000498},
}

We develop a new family of well-balanced path-conservative quadrature-free one-step ADER finite volume and discontinuous Galerkin finite element schemes on unstructured meshes for the solution of hyperbolic partial differential equations with non-conservative products and stiff source terms. The fully discrete formulation is derived using the recently developed framework of explicit one-step PNPM schemes of arbitrary high order of accuracy in space and time for conservative hyperbolic systems [Dumbser M, Balsara D, Toro EF, Munz CD. A unified framework for the construction of one-step finite-volume and discontinuous Galerkin schemes. J Comput Phys 2008;227:8209–53]. The two key ingredients of our high order approach are: first, the high order accurate PNPM reconstruction operator on unstructured meshes, using the WENO strategy presented in [Dumbser M, Käser M, Titarev VA Toro EF. Quadrature-free non-oscillatory finite volume schemes on unstructured meshes for nonlinear hyperbolic systems. J Comput Phys 2007;226:204–43] to ensure monotonicity at discontinuities, and second, a local space–time Galerkin scheme to predict the evolution of the reconstructed polynomial data inside each element during one time step to obtain a high order accurate one-step time discretization. This approach is also able to deal with stiff source terms as shown in [Dumbser M, Enaux C, Toro EF. Finite volume schemes of very high order of accuracy for stiff hyperbolic balance laws. J Comput Phys 2008;227:3971–4001]. These two key ingredients are combined with the recently developed path-conservative methods of Parés [Parés C. Numerical methods for nonconservative hyperbolic systems: a theoretical framework. SIAM J Numer Anal 2006;44:300–21] and Castro et al. [Castro MJ, Gallardo JM, Parés C. High-order finite volume schemes based on reconstruction of states for solving hyperbolic systems with nonconservative products. Applications to shallow-water systems. Math Comput 2006;75:1103–34] to treat the non-conservative products properly. We show applications of our method to the two-layer shallow water equations as well as applications to the recently published depth-averaged two-fluid flow model of Pitman and Le [Pitman EB, Le L. A two-fluid model for avalanche and debris flows. Philos Trans Roy Soc A 2005;363:1573–601].

@Book{moralesdeluna09entropy,
  title     = {{E}ntropy satisfying schemes for shallow water systems},
  publisher = {VDM Verlag},
  year      = {2009},
  author    = {Morales de Luna, Tom{\'a}s},
  abstract  = {Entropy inequalities play an essential role in the election of a unique solution among all possible weak solutions for a system of conservation laws. Existence of an entropy will also allow to study the stability of numerical schemes for such equations and the main ideas can be easily extended to the more general case of quasi-linear systems. Nevertheless, developing numerical schemes that agree to this notion of entropy and at the same time grant some other #good# properties is not an easy task. Here we study the case of shallow water systems. In particular, the classical one-layer and two-layer shallow water systems and a generalized Savage-Hutter model are considered. Each particular case is studied and some schemes that agree to an entropy inequality are presented. We seek that these schemes will also satisfy properties like steady-states preservation and positiviy of water heights.},
  url       = {http://www.amazon.com/exec/obidos/ASIN/363917271X/ejelta5-20},
}

Entropy inequalities play an essential role in the election of a unique solution among all possible weak solutions for a system of conservation laws. Existence of an entropy will also allow to study the stability of numerical schemes for such equations and the main ideas can be easily extended to the more general case of quasi-linear systems. Nevertheless, developing numerical schemes that agree to this notion of entropy and at the same time grant some other #good# properties is not an easy task. Here we study the case of shallow water systems. In particular, the classical one-layer and two-layer shallow water systems and a generalized Savage-Hutter model are considered. Each particular case is studied and some schemes that agree to an entropy inequality are presented. We seek that these schemes will also satisfy properties like steady-states preservation and positiviy of water heights.

@InBook{barcenas_fs_l_m_dr_2009,
  pages  = {13-17},
  title  = {{E}studio morfom{\'e}trico de las ondulaciones de los r{\'i}os de {A}ndaluc{\'i}a oriental},
  year   = {2009},
  author = {B{\'a}rcenas, P. and Fern{\'a}ndez-Salas, Lu{\'i}s Miguel and Lobo, Francisco and Mac{\'i}as, Jorge and D{\'i}az, V{\'i}ctor},
  editor = {J.A. Morales, M. Cantano, A. Rodr{\'i}guez-Ram{\'i}rez and I. Delgado,},
}

@InBook{N_contribuciones_2009,
  pages   = {13-17},
  title   = {{E}studio morfom{\'e}trico entre las ondulaciones de los prodeltas de los r{\'i}os de {A}ndaluc{\'i}a {O}riental},
  year    = {2009},
  author  = {B{\'a}rcenas, P. and Fern{\'a}ndez-Salas, L.-M. and Mac{\'i}as, Jorge and Lobo, F.-J. and D{\'i}az-del-R{\'i}o, V.},
  editor  = {J.A. Morales, M. Cantano, A. Rodr{\'i}guez-Ram{\'i}rez and I. Delgado,},
  series  = {Nuevas contribuciones sobre la Geomorfolog{\'i}a Litoral},
  journal = {Rev. Soc. Geol. Espa{\~n}a},
}

@Article{MR2495820,
  author  = {Castro D{\'i}az, Manuel J. and Fern{\'a}ndez Nieto, E. D. and Ferreiro, A.-M. and Garc{\'i}a-Rodr{\'i}guez, J. A. and Par{\'e}s, Carlos},
  title   = {{H}igh order extensions of {R}oe schemes for two-dimensional nonconservative hyperbolic systems},
  journal = {J. Sci. Comput.},
  year    = {2009},
  volume  = {39},
  number  = {1},
  pages   = {67–114},
}

@Article{moralesdeluna09shallow,
  author   = {Morales de Luna, Tom{\'a}s and Castro D{\'i}az, Manuel J. and Par{\'e}s, Carlos and Fern{\'a}ndez Nieto, Enrique D.},
  title    = {{O}n a shallow water model for the simulation of turbidity currents},
  journal  = {Communications in Computational Physics},
  year     = {2009},
  abstract = {We present a model for hyperpycnal plumes or turbidity currents that
	takes into account the interaction between the turbidity current
	and the bottom, considering deposition and erosion effects as well
	as solid transport of particles at the bed load due to the current.
	Water entrainment from the ambient water in which the turbidity current
	plunges is also considered. Motion of ambient water is neglected
	and the rigid lid assumption is considered. The model is obtained
	as a depth-average system of equations under the shallow water hypothesis
	describing the balance of fluid mass, sediment mass and mean flow.
	The character of the system is analyzed and numerical simulations
	are carried out using finite volume schemes and path-conservative
	Roe schemes.},
  keywords = {Turbidity currents, hyperbolic systems, finite volume methods, path-conservative schemes, numerical modeling},
  url      = {http://www.global-sci.com/cgi-bin/fulltext/6/848/full},
}

We present a model for hyperpycnal plumes or turbidity currents that takes into account the interaction between the turbidity current and the bottom, considering deposition and erosion effects as well as solid transport of particles at the bed load due to the current. Water entrainment from the ambient water in which the turbidity current plunges is also considered. Motion of ambient water is neglected and the rigid lid assumption is considered. The model is obtained as a depth-average system of equations under the shallow water hypothesis describing the balance of fluid mass, sediment mass and mean flow. The character of the system is analyzed and numerical simulations are carried out using finite volume schemes and path-conservative Roe schemes.

@Article{c.pares09some,
  author  = {Par{\'e}s, Carlos and Mu{\~n}oz Ruiz, Mar{\'i}a Luz},
  title   = {{O}n some difficulties of the numerical approximation of nonconservative hyperbolic systems},
  journal = {Bol. Soc. Esp. Mat. Apl},
  year    = {2009},
}

@Article{M.J.Castro2009,
  author   = {Castro D{\'i}az, Manuel J. and Pardo, A. and Par{\'e}s, Carlos and Toro, E.-F.},
  title    = {{O}n some fast well-balanced first order solvers for nonconservative systems},
  journal  = {Math. Comp.},
  year     = {2009},
  volume   = {79},
  number   = {271},
  pages    = {1427–1472},
  abstract = {The goal of this article is to design robust and simple first-order explicit solvers for one-dimensional nonconservative hyperbolic systems. These solvers are intended to be used as the basis for higher-order methods for one- or multi-dimensional problems. The starting point for the development of these solvers is the general definition of a Roe linearization introduced by Toumi in 1992 based on the use of a family of paths. Using this concept, Roe methods can be extended to nonconservative systems. These methods have good well-balance and robustness properties, but they have also some drawbacks: in particular, their implementation requires the explicit knowledge of the eigenstructure of the intermediate matrices. Our goal here is to design numerical methods based on a Roe linearization which overcome this drawback. The idea is to split the Roe matrices into two parts which are used to calculate the contributions at the cells to the right and to the left, respectively. This strategy is used to generate two different one-parameter families of schemes which contain, as particular cases, some generalizations to nonconservative systems of the well-known Lax-Friedrichs, Lax-Wendroff, FORCE, and GFORCE schemes. Some numerical experiments are presented to compare the behaviors of the schemes introduced here with Roe methods},
}

The goal of this article is to design robust and simple first-order explicit solvers for one-dimensional nonconservative hyperbolic systems. These solvers are intended to be used as the basis for higher-order methods for one- or multi-dimensional problems. The starting point for the development of these solvers is the general definition of a Roe linearization introduced by Toumi in 1992 based on the use of a family of paths. Using this concept, Roe methods can be extended to nonconservative systems. These methods have good well-balance and robustness properties, but they have also some drawbacks: in particular, their implementation requires the explicit knowledge of the eigenstructure of the intermediate matrices. Our goal here is to design numerical methods based on a Roe linearization which overcome this drawback. The idea is to split the Roe matrices into two parts which are used to calculate the contributions at the cells to the right and to the left, respectively. This strategy is used to generate two different one-parameter families of schemes which contain, as particular cases, some generalizations to nonconservative systems of the well-known Lax-Friedrichs, Lax-Wendroff, FORCE, and GFORCE schemes. Some numerical experiments are presented to compare the behaviors of the schemes introduced here with Roe methods

@InBook{2009_CGMP_Realistic,
  pages  = {1429-1432},
  title  = {{R}ealistic application of a tidal 2{D} two-layer shallow water model to the {S}trait of {G}ibraltar},
  year   = {2009},
  author = {Castro D{\'i}az, Manuel J. and Gonz{\'a}lez-Vida, J.-M. and Mac{\'i}as, Jorge and Par{\'e}s, Carlos},
  series = {Numerical Analysis and Applied Mathematics},
}

@Article{bouchut09semi-discrete,
  author   = {Bouchut, Fran{\c{c}}ois and Morales de Luna, Tom{\'a}s},
  title    = {{S}emi-discrete {E}ntropy {S}atisfying {A}pproximate {R}iemann {S}olvers. {T}he {C}ase of the {S}uliciu {R}elaxation {A}pproximation},
  journal  = {Journal of Scientific Computing},
  year     = {2009},
  volume   = {41},
  number   = {3},
  pages    = {483-509},
  abstract = {Abstract In this work we establish conditions for an approximate simple
	Riemann solver to satisfy a semi-discrete entropy inequality. The
	semi-discrete approach is less restrictive than the fully-discrete
	case and allows to grant some other good properties for numerical
	schemes. First, conditions are established in an abstract framework
	for simple Riemann solvers to satisfy a semi-discrete entropy inequality
	and then the results are applied, as a particular case, to the Suliciu
	system. This will lead in particular to the definition of schemes
	for the isentropic gas dynamics and the full gas dynamics system
	that are stable and preserve the stationary shocks.},
  url      = {http://dx.doi.org/10.1007/s10915-009-9311-3},
}

Abstract In this work we establish conditions for an approximate simple Riemann solver to satisfy a semi-discrete entropy inequality. The semi-discrete approach is less restrictive than the fully-discrete case and allows to grant some other good properties for numerical schemes. First, conditions are established in an abstract framework for simple Riemann solvers to satisfy a semi-discrete entropy inequality and then the results are applied, as a particular case, to the Suliciu system. This will lead in particular to the definition of schemes for the isentropic gas dynamics and the full gas dynamics system that are stable and preserve the stationary shocks.

@Article{M.Lastra2009,
  author  = {Lastra, M. and Mantas, J.-M. and Ure{\~n}a, C. and Castro D{\'i}az, Manuel J. and Garc{\'i}a, J.-A.},
  title   = {{S}imulation of shallow-water systems using graphics processing units},
  journal = {Submitted to Math. Comp. Simul.},
  year    = {2009},
}

@Article{MR2437587,
  author  = {Fern{\'a}ndez Nieto, E. D. and Bouchut, Fran{\c{c}}ois and Bresch, D. and Castro D{\'i}az, Manuel J. and Mangeney, A.},
  title   = {{A} new {{S}}avage-{{H}}utter type model for submarine avalanches and generated tsunami},
  journal = {J. Comput. Phys.},
  year    = {2008},
}

@Article{moralesdeluna08saint,
  author   = {Morales de Luna, Tom{\'a}s},
  title    = {{A} {S}aint {V}enant model for gravity driven shallow water flows with variable density and compressibility effects},
  journal  = {Mathematical and Computer Modelling},
  year     = {2008},
  abstract = {We introduce a new model for shallow water flows with non-flat bottom
	made of two layers of compressible{\textendash}incompressible fluids.
	The classical Savage{\textendash}Hutter model for gravity driven
	shallow water flows is derived from incompressible Euler equations.
	Here, we generalize this model by adding an upper compressible layer.
	We obtain a model of shallow water type, that admits an entropy dissipation
	inequality, preserves the steady state of a lake at rest and gives
	an approximation of the free surface compressible{\textendash}incompressible
	Euler equations. Keywords: Shallow water; Variable density; Compressibility;
	Entropy inequality; Savage{\textendash}Hutter model },
  keywords = {Compressibility,Entropy inequality,Savage Hutter model,Shallow water,Variable density},
  url      = {http://www.sciencedirect.com/science?\_ob=ArticleURL{\&}\_udi=B6V0V-4NTJH02-6{\&}\_user=10{\&}\_rdoc=1{\&}\_fmt={\&}\_orig=search{\&}\_sort=d{\&}view=c{\&}\_acct=C000050221{\&}\_version=1{\&}\_urlVersion=0{\&}\_userid=10{\&}md5=c714fddce3783429707813d440feb62a},
}

We introduce a new model for shallow water flows with non-flat bottom made of two layers of compressible–incompressible fluids. The classical Savage–Hutter model for gravity driven shallow water flows is derived from incompressible Euler equations. Here, we generalize this model by adding an upper compressible layer. We obtain a model of shallow water type, that admits an entropy dissipation inequality, preserves the steady state of a lake at rest and gives an approximation of the free surface compressible–incompressible Euler equations. Keywords: Shallow water; Variable density; Compressibility; Entropy inequality; Savage–Hutter model

@Article{bouchut08entropy,
  author   = {Bouchut, Fran{\c{c}}ois and Morales de Luna, Tom{\'a}s},
  title    = {{A}n entropy satisfying scheme for two-layer shallow water equations with uncoupled treatment},
  journal  = {ESAIM: Mathematical Modelling and Numerical Analysis (ESAIM: M2AN)},
  year     = {2008},
  abstract = {We consider the system of partial differential equations governing
	the one-dimensional flow of two superposed immiscible layers of shallow
	water. The difficulty in this system comes from the coupling terms
	involving some derivatives of the unknowns that make the system nonconservative,
	and eventually nonhyperbolic. Due to these terms, a numerical scheme
	obtained by performing an arbitrary scheme to each layer, and using
	time-splitting or other similar techniques leads to instabilities
	in general. Here we use entropy inequalities in order to control
	the stability. We introduce a stable well-balanced time-splitting
	scheme for the two-layer shallow water system that satisfies a fully
	discrete entropy inequality. In contrast with Roe type solvers, it
	does not need the computation of eigenvalues, which is not simple
	for the two-layer shallow water system. The solver has the property
	to keep the water heights nonnegative, and to be able to treat vanishing
	values.},
  keywords = {Two-layer shallow water, nonconservative system, complex eigenvalues, time-splitting, entropy inequality, well-balanced scheme, nonnegativity},
  url      = {http://dx.doi.org/10.1051/m2an:2008019},
}

We consider the system of partial differential equations governing the one-dimensional flow of two superposed immiscible layers of shallow water. The difficulty in this system comes from the coupling terms involving some derivatives of the unknowns that make the system nonconservative, and eventually nonhyperbolic. Due to these terms, a numerical scheme obtained by performing an arbitrary scheme to each layer, and using time-splitting or other similar techniques leads to instabilities in general. Here we use entropy inequalities in order to control the stability. We introduce a stable well-balanced time-splitting scheme for the two-layer shallow water system that satisfies a fully discrete entropy inequality. In contrast with Roe type solvers, it does not need the computation of eigenvalues, which is not simple for the two-layer shallow water system. The solver has the property to keep the water heights nonnegative, and to be able to treat vanishing values.

@Article{MR2460785,
  author   = {Castro D{\'i}az, Manuel J. and L{\'o}pez, Juan-Antonio and Par{\'e}s, Carlos},
  title    = {{F}inite volume simulation of the geostrophic adjustment in a rotating shallow-water system},
  journal  = {SIAM J. Sci. Comput.},
  year     = {2008},
  volume   = {31},
  number   = {1},
  pages    = {444–477},
  abstract = {The goal of this article is to simulate rotating flows of shallow layers of fluid by means of finite volume numerical schemes. More precisely, we focus on the simulation of the geostrophic adjustment phenomenon. As spatial discretization, a first order Roe-type method and some higher-order extensions are developed. The time discretization is designed in order to provide suitable approximations of inertial oscillations, taking into account the Hamiltonian structure of the system for these solutions. The numerical dispersion laws and the wave amplifications of the schemes are studied, and their well-balanced properties are analyzed. Finally, some numerical experiments for one-dimensional (1d) and two-dimensional (2d) problems are shown.},
  url      = {http://epubs.siam.org/sisc/resource/1/sjoce3/v31/i1/p444_s1},
}

The goal of this article is to simulate rotating flows of shallow layers of fluid by means of finite volume numerical schemes. More precisely, we focus on the simulation of the geostrophic adjustment phenomenon. As spatial discretization, a first order Roe-type method and some higher-order extensions are developed. The time discretization is designed in order to provide suitable approximations of inertial oscillations, taking into account the Hamiltonian structure of the system for these solutions. The numerical dispersion laws and the wave amplifications of the schemes are studied, and their well-balanced properties are analyzed. Finally, some numerical experiments for one-dimensional (1d) and two-dimensional (2d) problems are shown.

@Conference{HYP2008,
  author    = {Gallardo, Jos{\'e}-M. and Castro D{\'i}az, Manuel J. and Par{\'e}s, Carlos},
  title     = {{H}igh-order finite volume schemes for shallow water equations with topography and dry areas},
  booktitle = {Hyperbolic problems: Theory, numerics and applications},
  year      = {2008},
  editor    = {Eitan, Tadmor and Jian-Guo, Liu and Athanasios, Tzavaras},
  pages     = {585-594},
  publisher = {American Mathematical Society},
}

@Article{MR2437777,
  author   = {Arregui, I{\~n}igo and Cend{\'a}n, Jos{\'e} Jes{\'u}s and Par{\'e}s, Carlos and V{\'a}zquez, Carlos},
  title    = {{N}umerical solution of a 1-{D} elastohydrodynamic problem in magnetic storage devices},
  journal  = {M2AN Math. Model. Numer. Anal.},
  year     = {2008},
  volume   = {42},
  number   = {4},
  pages    = {645–665},
  abstract = {In this work we present new numerical methods to simulate the mechanics of head-tape magnetic storage devices. The elastohydrodynamic problem is formulated in terms of a coupled system which is governed by a nonlinear compressible Reynolds equation for the air pressure over the head, and a rod model for the tape displacement. A fixed point algorithm between the solutions of the elastic and hydrodynamic problems is proposed. For the nonlinear Reynolds equation, a characteristics method and a duality algorithm are developed to cope with the convection dominating and nonlinear diffusion features, respectively. Furthermore, in the duality method the convergence and optimal choice of the parameters are analyzed. At each fixed point iteration, in the elastic model a complementarity formulation is required and appropriate numerical techniques are used. For the spatial discretization different finite element spaces are chosen. Finally, numerical test examples illustrate the theoretical results, as well as the good performance in the simulation of real devices},
  url      = {http://www.esaim-m2an.org/action/displayAbstract?fromPage=online{\&}aid=8194733},
}

In this work we present new numerical methods to simulate the mechanics of head-tape magnetic storage devices. The elastohydrodynamic problem is formulated in terms of a coupled system which is governed by a nonlinear compressible Reynolds equation for the air pressure over the head, and a rod model for the tape displacement. A fixed point algorithm between the solutions of the elastic and hydrodynamic problems is proposed. For the nonlinear Reynolds equation, a characteristics method and a duality algorithm are developed to cope with the convection dominating and nonlinear diffusion features, respectively. Furthermore, in the duality method the convergence and optimal choice of the parameters are analyzed. At each fixed point iteration, in the elastic model a complementarity formulation is required and appropriate numerical techniques are used. For the spatial discretization different finite element spaces are chosen. Finally, numerical test examples illustrate the theoretical results, as well as the good performance in the simulation of real devices

@Article{castro2008sediment,
  author  = {Castro D{\'i}az, Manuel J. and Fern{\'a}ndez Nieto, E. D. and Ferreiro, A.-M.},
  title   = {{S}ediment transport models in {S}hallow {W}ater equations and numerical approach by high order finite volume methods},
  journal = {Comput Fluids},
  year    = {2008},
}

@InBook{CGMP_2008,
  title     = {{S}imulation of tidal currents in the {S}trait of {G}ibraltar},
  publisher = {Universidad Rey Juan Carlos},
  year      = {2008},
  author    = {Castro D{\'i}az, Manuel J. and Gonz{\'a}lez-Vida, J.-M. and Mac{\'i}as, Jorge and Par{\'e}s, Carlos},
  editor    = {Mofdi, El-Amrani and Mohamed, Sead and Naje, Yebari},
}

@Article{MR2401061,
  author  = {Castro D{\'i}az, Manuel J. and Garc{\'i}a-Rodr{\'i}guez, J.-A. and Gonz{\'a}lez-Vida, J.-M. and Mac{\'i}as, Jorge and Par{\'e}s, Carlos},
  title   = {{S}imulation of tidal currents in the {S}trait of {G}ibraltar using two-dimensional two-layer shallow-water models},
  journal = {Bol. Soc. Esp. Mat. Apl.},
  year    = {2008},
}

@Article{MR2458748,
  author   = {Castro D{\'i}az, Manuel J. and Garc{\'i}a-Rodr{\'i}guez, J.-A. and Gonz{\'a}lez-Vida, J.-M. and Par{\'e}s, Carlos},
  title    = {{S}olving shallow-water systems in 2{D} domains using finite volume methods and multimedia {SSE} instructions},
  journal  = {J. Comput. Appl. Math.},
  year     = {2008},
  volume   = {221},
  number   = {1},
  pages    = {16–32},
  abstract = {The goal of this paper is to construct efficient parallel solvers for 2D hyperbolic systems of conservation laws with source terms and nonconservative products. The method of lines is applied: at every intercell a projected Riemann problem along the normal direction is considered which is discretized by means of well-balanced Roe methods. The resulting 2D numerical scheme is explicit and first-order accurate. In [M.J. Castro, J.A. Garc{\'i}a, J.M. Gonz{\'a}lez, C. Pares, A parallel 2D Finite Volume scheme for solving systems of balance laws with nonconservative products: Application to shallow flows, Comput. Methods Appl. Mech. Engrg. 196 (2006) 2788–2815] a domain decomposition method was used to parallelize the resulting numerical scheme, which was implemented in a PC cluster by means of MPI techniques.

In this paper, in order to optimize the computations, a new parallelization of SIMD type is performed at each MPI thread, by means of SSE (“Streaming SIMD Extensions”), which are present in common processors. More specifically, as the most costly part of the calculations performed at each processor consists of a huge number of small matrix and vector computations, we use the Intel© Integrated Performance Primitives small matrix library. To make easy the use of this library, which is implemented using assembler and SSE instructions, we have developed a C++ wrapper of this library in an efficient way. Some numerical tests were carried out to validate the performance of the C++ small matrix wrapper. The specific application of the scheme to one-layer Shallow-Water systems has been implemented on a PC’s cluster. The correct behavior of the one-layer model is assessed using laboratory data.},
  url      = {http://www.sciencedirect.com/science/article/pii/S0377042707005201},
}

The goal of this paper is to construct efficient parallel solvers for 2D hyperbolic systems of conservation laws with source terms and nonconservative products. The method of lines is applied: at every intercell a projected Riemann problem along the normal direction is considered which is discretized by means of well-balanced Roe methods. The resulting 2D numerical scheme is explicit and first-order accurate. In [M.J. Castro, J.A. García, J.M. González, C. Pares, A parallel 2D Finite Volume scheme for solving systems of balance laws with nonconservative products: Application to shallow flows, Comput. Methods Appl. Mech. Engrg. 196 (2006) 2788–2815] a domain decomposition method was used to parallelize the resulting numerical scheme, which was implemented in a PC cluster by means of MPI techniques. In this paper, in order to optimize the computations, a new parallelization of SIMD type is performed at each MPI thread, by means of SSE (“Streaming SIMD Extensions”), which are present in common processors. More specifically, as the most costly part of the calculations performed at each processor consists of a huge number of small matrix and vector computations, we use the Intel© Integrated Performance Primitives small matrix library. To make easy the use of this library, which is implemented using assembler and SSE instructions, we have developed a C++ wrapper of this library in an efficient way. Some numerical tests were carried out to validate the performance of the C++ small matrix wrapper. The specific application of the scheme to one-layer Shallow-Water systems has been implemented on a PC’s cluster. The correct behavior of the one-layer model is assessed using laboratory data.

@Article{MR2458121,
  author   = {Castro D{\'i}az, Manuel J. and Chac{\'o}n-Rebollo, T. and Fern{\'a}ndez Nieto, E. D. and Gonz{\'a}lez-Vida, J.-M. and Par{\'e}s, Carlos},
  title    = {{W}ell-balanced finite volume schemes for 2d non-homogeneous hyperboli systems. {A}pplication to the dam break of {A}znalc{\'o}llar},
  journal  = {Comput. Methods Appl. Mech. Engrg.},
  year     = {2008},
  volume   = {197},
  number   = {45-48},
  pages    = {3932–3950},
  abstract = {In this paper, we introduce a class of well-balanced finite volume schemes for 2D non-homogeneous hyperbolic systems. We extend the derivation of standard finite volume solvers for homogeneous systems to non-homogeneous ones using the method of lines. We study conservation and some well-balanced properties of the numerical scheme. We apply our solvers to shallow water equations: we prove that these exactly compute the water at rest solutions. We also perform some numerical tests by comparing with 1D solutions, simulating the formation of a hydraulic drop and a hydraulic jump, and studying a real dam break: Aznalc{\'o}llar, an ecological disaster that happened in the province of Seville, Spain in 1998},
  url      = {http://www.sciencedirect.com/science/article/pii/S0045782508001394},
}

In this paper, we introduce a class of well-balanced finite volume schemes for 2D non-homogeneous hyperbolic systems. We extend the derivation of standard finite volume solvers for homogeneous systems to non-homogeneous ones using the method of lines. We study conservation and some well-balanced properties of the numerical scheme. We apply our solvers to shallow water equations: we prove that these exactly compute the water at rest solutions. We also perform some numerical tests by comparing with 1D solutions, simulating the formation of a hydraulic drop and a hydraulic jump, and studying a real dam break: Aznalcóllar, an ecological disaster that happened in the province of Seville, Spain in 1998

@Article{MR2383221,
  author   = {Castro D{\'i}az, Manuel J. and Gallardo, Jos{\'e}-M. and L{\'o}pez-Garc{\'i}a, Juan-A. and Par{\'e}s, Carlos},
  title    = {{W}ell-balanced high order extensions of {G}odunov's method for semilinear balance laws},
  journal  = {SIAM J. Numer. Anal.},
  year     = {2008},
  volume   = {46},
  number   = {2},
  pages    = {1012–1039},
  abstract = {This paper is concerned with the development of well-balanced high order numerical schemes for systems of balance laws with a linear flux function, whose coefficients may be variable. First, well-balanced first order numerical schemes are obtained based on the use of exact solvers of Riemann problems that include both the flux and the source terms. Godunov's methods so obtained are extended to higher order schemes by using a technique of reconstruction of states. The main contribution of this paper is to introduce a reconstruction technique that preserves the well-balanced property of Godunov's methods. Some numerical experiments are presented to verify in practice the properties of the developed numerical schemes.},
  url      = {http://epubs.siam.org/sinum/resource/1/sjnaam/v46/i2/p1012_s1},
}

This paper is concerned with the development of well-balanced high order numerical schemes for systems of balance laws with a linear flux function, whose coefficients may be variable. First, well-balanced first order numerical schemes are obtained based on the use of exact solvers of Riemann problems that include both the flux and the source terms. Godunov's methods so obtained are extended to higher order schemes by using a technique of reconstruction of states. The main contribution of this paper is to introduce a reconstruction technique that preserves the well-balanced property of Godunov's methods. Some numerical experiments are presented to verify in practice the properties of the developed numerical schemes.

@Article{MR2442446,
  author   = {Castro D{\'i}az, Manuel J. and LeFloch, Philippe-G. and Mu{\~n}oz Ruiz, Mar{\'i}a Luz and Par{\'e}s, Carlos},
  title    = {{W}hy many theories of shock waves are necessary: convergence error in formally path-consistent schemes},
  journal  = {J. Comput. Phys.},
  year     = {2008},
  volume   = {227},
  number   = {17},
  pages    = {8107–8129},
  abstract = {We are interested in nonlinear hyperbolic systems in nonconservative form arising in fluid dynamics, and, for solutions containing shock waves, we investigate the convergence of finite difference schemes applied to such systems. According to Dal Maso, LeFloch, and Murat’s theory, a shock wave theory for a given nonconservative system requires prescribing a priori a family of paths in the phase space. In the present paper, we consider schemes that are formally consistent with a given family of paths, and we investigate their limiting behavior as the mesh is refined. we first generalize to systems a property established earlier by Hou and LeFloch for scalar conservation laws, and we prove that nonconservative schemes generate, at the level of the limiting hyperbolic system, an convergence error source-term which, provided the total variation of the approximations remains uniformly bounded, is a locally bounded measure. This convergence error measure is supported on the shock trajectories and, as we demonstrate here, is usually “small”. In the special case that the scheme converges in the sense of graphs – a rather strong convergence property often violated in practice – then this measure source-term vanishes. We also discuss the role of the equivalent equation associated with a difference scheme; here, the distinction between scalar equations and systems appears most clearly since, for systems, the equivalent equation of a scheme that is formally path-consistent depends upon the prescribed family of paths. The core of this paper is devoted to investigate numerically the approximation of several (simplified or full) hyperbolic models arising in fluid dynamics. This leads us to the conclusion that for systems having nonconservative products associated with linearly degenerate characteristic fields, the convergence error vanishes. For more general models, this measure is evaluated very accurately, especially by plotting the shock curves associated with each scheme under consideration; as we demonstrate, plotting the shock curves provide a convenient approach for evaluating the range of validity of a given scheme.},
  url      = {http://www.sciencedirect.com/science/article/pii/S0021999108002842},
}

We are interested in nonlinear hyperbolic systems in nonconservative form arising in fluid dynamics, and, for solutions containing shock waves, we investigate the convergence of finite difference schemes applied to such systems. According to Dal Maso, LeFloch, and Murat’s theory, a shock wave theory for a given nonconservative system requires prescribing a priori a family of paths in the phase space. In the present paper, we consider schemes that are formally consistent with a given family of paths, and we investigate their limiting behavior as the mesh is refined. we first generalize to systems a property established earlier by Hou and LeFloch for scalar conservation laws, and we prove that nonconservative schemes generate, at the level of the limiting hyperbolic system, an convergence error source-term which, provided the total variation of the approximations remains uniformly bounded, is a locally bounded measure. This convergence error measure is supported on the shock trajectories and, as we demonstrate here, is usually “small”. In the special case that the scheme converges in the sense of graphs – a rather strong convergence property often violated in practice – then this measure source-term vanishes. We also discuss the role of the equivalent equation associated with a difference scheme; here, the distinction between scalar equations and systems appears most clearly since, for systems, the equivalent equation of a scheme that is formally path-consistent depends upon the prescribed family of paths. The core of this paper is devoted to investigate numerically the approximation of several (simplified or full) hyperbolic models arising in fluid dynamics. This leads us to the conclusion that for systems having nonconservative products associated with linearly degenerate characteristic fields, the convergence error vanishes. For more general models, this measure is evaluated very accurately, especially by plotting the shock curves associated with each scheme under consideration; as we demonstrate, plotting the shock curves provide a convenient approach for evaluating the range of validity of a given scheme.

@Article{MR2323696,
  author  = {Mu{\~n}oz Ruiz, Mar{\'i}a Luz and Par{\'e}s, Carlos},
  title   = {{G}odunov method for nonconservative hyperbolic systems},
  journal = {M2AN Math. Model. Numer. Anal.},
  year    = {2007},
  volume  = {41},
  number  = {1},
  pages   = {169–185},
}

@Article{castro2007improved,
  author  = {Castro D{\'i}az, Manuel J. and Garc{\'i}a-Rodr{\'i}guez, J.-A. and Gonz{\'a}lez-Vida, J.-M. and Mac{\'i}as, Jorge and Par{\'e}s, Carlos},
  title   = {{I}mproved {FVM} for two-layer shallow-water models: {A}pplication to the {S}trait of {G}ibraltar},
  journal = {Advances in Engineering Software},
  year    = {2007},
}

@Article{MR2361537,
  author  = {Gallardo, Jos{\’e}-M. and Par{\'e}s, Carlos and Castro D{\'i}az, Manuel J.},
  title   = {{O}n a well-balanced high-order finite volume scheme for shallow water equations with topography and dry areas},
  journal = {J. Comput. Phys.},
  year    = {2007},
  volume  = {227},
  number  = {1},
  pages   = {574–601},
}

@Article{MR2318699,
  author  = {Castro D{\'i}az, Manuel J. and Chac{\'o}n, T. and Fern{\'a}ndez-Nieto, E. D. and Par{\'e}s, Carlos},
  title   = {{O}n well-balanced finite volume methods for nonconservative nonhomogeneous hyperbolic systems},
  journal = {SIAM J. Sci. Comput.},
  year    = {2007},
  volume  = {29},
  number  = {3},
  pages   = {1093–1126},
}

@Article{MR2371563,
  author  = {Castro D{\'i}az, Manuel J. and Pardo, A. and Par{\'e}s, Carlos},
  title   = {{W}ell-balanced numerical schemes based on a generalized hydrostatic reconstruction technique},
  journal = {Math. Models Methods Appl. Sci.},
  year    = {2007},
  volume  = {17},
  number  = {12},
  pages   = {2055–2113},
}

@Article{MR2207488,
  author  = {Castro D{\'i}az, Manuel J. and Garc{\'i}a-Rodr{\'i}guez, J.-A. and Gonz{\'a}lez-Vida, J.-M. and Par{\'e}s, Carlos},
  title   = {{A} parallel 2d finite volume scheme for solving systems of balance laws with nonconservative products: application to shallow flows},
  journal = {Comput. Methods Appl. Mech. Engrg.},
  year    = {2006},
  volume  = {195},
  number  = {19-22},
  pages   = {2788–2815},
}

@Article{MR2219021,
  author  = {Castro D{\'i}az, Manuel J. and Gallardo, Jos{\'e}-M. and Par{\'e}s, Carlos},
  title   = {{H}igh order finite volume schemes based on reconstruction of states for solving hyperbolic systems with nonconservative products. {A}pplications to shallow-water systems},
  journal = {Math. Comp.},
  year    = {2006},
  volume  = {75},
  number  = {255},
  pages   = {1103–1134},
}

@Article{MR2217384,
  author  = {Par{\'e}s, Carlos},
  title   = {{N}umerical methods for nonconservative hyperbolic systems: a theoretical framework},
  journal = {SIAM J. Numer. Anal.},
  year    = {2006},
  volume  = {44},
  number  = {1},
  pages   = {300–321},
}

@Article{MR2232076,
  author  = {Castro D{\'i}az, Manuel J. and Gonz{\'a}lez Vida, Jos{\'e} M. and Par{\'e}s, Carlos},
  title   = {{N}umerical treatment of wet/dry fronts in shallow flows with a modified {{R}}oe scheme},
  journal = {Math. Models Methods Appl. Sci.},
  year    = {2006},
  volume  = {16},
  number  = {6},
  pages   = {897–931},
}

@Conference{HYP2006,
  author    = {Gallardo, Jos{\’e}-M. and Castro D{\'i}az, Manuel J. and Par{\'e}s, Carlos and Gonz{\'a}lez-Vida, J.-M.},
  title     = {{O}n a well-balanced high-order finite volume scheme for the shallow water equations with bottom topography and dry areas},
  booktitle = {Hyperbolic problems: Theory, numerics and applications},
  year      = {2006},
  editor    = {Sylvie Benzoni-Gavage, Denis Serre},
  pages     = {259--270},
  publisher = {Springer},
}

@Article{MR2135784,
  author  = {Gallardo, Jos{\'e}-M. and Castro D{\'i}az, Manuel J. and Par{\'e}s, Carlos},
  title   = {{A} generalized duality method for solving variational inequalities. {A}pplications to some nonlinear {D}irichlet problems},
  journal = {Numer. Math.},
  year    = {2005},
  volume  = {100},
  number  = {2},
  pages   = {259–291},
}

@Article{pares_m_c_2005,
  author  = {Par{\'e}s, Carlos and Mac{\'i}as, Jorge and Castro D{\'i}az, Manuel J.},
  title   = {{M}athematical models for the simulation of environmental flows: from the {S}trait of {G}ibraltar to the {A}znalcollar disaster},
  journal = {ERCIM News},
  year    = {2005},
}

@Article{MR2163780,
  author  = {Castro D{\'i}az, Manuel J. and Ferreiro, A.-M. and Garc{\'i}a-Rodr{\'i}guez, J.-A. and Gonz{\'a}lez-Vida, J.-M. and Mac{\'i}as, Jorge and Par{\'e}s, Carlos and V{\'a}zquez-Cend{\'o}n, M. Elena},
  title   = {{T}he numerical treatment of wet/dry fronts in shallow flows: application to one-layer and two-layer systems},
  journal = {Math. Comput. Modelling},
  year    = {2005},
  volume  = {42},
  number  = {3-4},
  pages   = {419–439},
}

@Article{macias_2004,
  author  = {Mac{\'i}as, Jorge},
  title   = {L} # os #{N} # i #{\~n} # os #de #los #{T} # r #{\'o} # picos #{: qu{\'e} son, sus modelos y su predicci{\'o}n},
  journal = {Bolet{\'i}n de la Sociedad Espa{\~n}ola de Matem{\'a}tica Aplicada},
  year    = {2004},
}

@Article{castro2004two,
  author  = {Castro D{\'i}az, Manuel J. and Garcia-Rodriguez, J.-A. and Mac{\'i}as, Jorge and Par{\'e}s, Carlos and V{\'a}zquez-Cend{\'o}n, E.},
  title   = {{A} two-layer finite volume model for flows through channels with irregular geometry: {C}omputation of maximal exchange solutions {A}pplication to the {S}trait of {G}ibraltar},
  journal = {Communications in Nonlinear Science and Numerical Simulation},
  year    = {2004},
  volume  = {9},
  number  = {2},
  pages   = {241–249},
}

@Article{MR2044136,
  author  = {Castro D{\'i}az, Manuel J. and Mac{\'i}as, Jorge and Par{\'e}s, Carlos and Garc{\'i}a-Rodr{\'i}guez, Jose A. and V{\'a}zquez-Cend{\'o}n, Elena},
  title   = {{A} two-layer finite volume model for flows through channels with irregular geometry: computation of maximal exchange solutions. {A}pplication to the {S}trait of {G}ibraltar},
  journal = {Commun. Nonlinear Sci. Numer. Simul.},
  year    = {2004},
  note    = {Recent advances in computational and mathematical methods for science and engineering},
}

@InBook{macias_p_c_2004,
  title  = {{N}umerical simulation in {O}ceanography. {A}pplication to the {A}lboran {S}ea and the {S}trait of {G}ibraltar},
  year   = {2004},
  author = {Mac{\'i}as, Jorge and Par{\'e}s, Carlos and Castro D{\'i}az, Manuel J.},
}

@Article{MR2043135,
  author  = {Castro D{\'i}az, Manuel J. and Garc{\'i}a-Rodr{\'i}guez, Jos{\'e}-A. and Gonz{\'a}lez-Vida, J.-M. and Mac{\'i}as, Jorge and Par{\'e}s, Carlos and V{\'a}zquez-Cend{\'o}n, M. Elena},
  title   = {{N}umerical simulation of two-layer shallow water flows through channels with irregular geometry},
  journal = {J. Comput. Phys.},
  year    = {2004},
}

@Article{Munoz2004,
  author  = {Mu{\~n}oz Ruiz, Mar{\'i}a Luz},
  title   = {{O}n a non-homogeneous bi-layer shallow-water problem: an existence theorem},
  journal = {Diff. Int. Eq.},
  year    = {2004},
}

@Article{pares2004well,
  author  = {Par{\'e}s, Carlos and Castro D{\'i}az, Manuel J.},
  title   = {{O}n the well-balance property of {R}oe's method for nonconservative hyperbolic systems. {A}pplications to shallow-water systems},
  journal = {Mathematical Modelling and Numerical Analysis},
  year    = {2004},
}

@Article{MR2104431,
  author  = {Par{\'e}s, Carlos and Castro D{\'i}az, Manuel J.},
  title   = {{O}n the well-balance property of {{R}}oe's method for nonconservative hyperbolic systems. {{A}}pplications to shallow-water systems},
  journal = {M2AN Math. Model. Numer. Anal.},
  year    = {2004},
}

@Article{macias_2003,
  author  = {Mac{\'i}as, Jorge},
  title   = {{E}l fen{\'o}meno E} # l #{N} # i #{\~n} # o #{: rese{\~n}a hist{\'o}rica, modelos y predicciones},
  journal = {Bolet{\'i}n de la Sociedad Malague{\~n}a de Ciencias},
  year    = {2003},
}

@Article{ODE5,
  author  = {Gallardo, Jos{\'e}-M.},
  title   = {{G}eneration of analytic semigroups by differential operators with mixed boundary conditions},
  journal = {Rocky Mountain Journal of Mathematics},
  year    = {2003},
  volume  = {33},
  number  = {3},
  pages   = {831-863},
}

@Article{chatelon2003bi,
  author  = {Mu{\~n}oz Ruiz, Mar{\'i}a Luz and F.-J., Chatelon and Orenga, P.},
  title   = {{O}n a bi-layer shallow-water problem},
  journal = {Nonlinear Anal},
  year    = {2003},
}

@Article{MR1959560,
  author  = {Mu{\~n}oz Ruiz, Mar{\'i}a Luz and Castro D{\'i}az, Manuel J. and Par{\'e}s, Carlos},
  title   = {{O}n an one-dimensional bi-layer shallow-water problem},
  journal = {Nonlinear Anal.},
  year    = {2003},
}

@Article{MR1933457,
  author  = {Gonz{\'a}lez Vida, Jos{\'e} M. and Par{\'e}s, Carlos},
  title   = {{A} {{L}}agrangian finite element algorithm for the numerical solution of one-dimensional shallow water equations},
  journal = {Rev. Internac. M{\'e}tod. Num{\'e}r. C{\'a}lc. Dise{\~n}. Ingr.},
  year    = {2002},
}

@Article{MR1985739,
  author  = {Castro D{\'i}az, Manuel J. and Gonz{\'a}lez-Vida, J.-M. and Mac{\'i}as, Jorge and Mu{\~n}oz Ruiz, Mar{\'i}a Luz and Par{\'e}s, Carlos and Garc{\'i}a-Rodr{\'i}guez, J. and V{\'a}zquez-Cend{\'o}n, E.},
  title   = {{N}umerical simulation of internal tides in the {S}trait of {G}ibraltar},
  journal = {RACSAM Rev. R. Acad. Cienc. Exactas F\'\i s. Nat. Ser. A Mat.},
  year    = {2002},
  note    = {Mathematics and environment (Spanish) (Paris, 2002)},
}

@Article{MR1917367,
  author  = {Par{\'e}s, Carlos and Castro D{\'i}az, Manuel J. and Mac{\'i}as, Jorge},
  title   = {{O}n the convergence of the {B}erm{\'u}dez-{M}oreno algorithm with constant parameters},
  journal = {Numer. Math.},
  year    = {2002},
}

@Article{MR1811983,
  author  = {Castro D{\'i}az, Manuel J. and Mac{\'i}as, Jorge and Par{\'e}s, Carlos},
  title   = {{A} {Q}-scheme for a class of systems of coupled conservation laws with source term. {A}pplication to a two-layer 1-{D} shallow water system},
  journal = {M2AN Math. Model. Numer. Anal.},
  year    = {2001},
  volume  = {35},
  number  = {1},
  pages   = {107–127},
}

@Article{MR1832678,
  author  = {Castro D{\'i}az, Manuel J. and Mac{\'i}as, Jorge and Par{\'e}s, Carlos},
  title   = {{A}n incomplete {LU}-based family of preconditioners for numerical resolution of a shallow water system using a duality method. {A}pplications},
  journal = {Appl. Math. Lett.},
  year    = {2001},
}

@Article{ODE4,
  author  = {Gallardo, Jos{\'e}-M.},
  title   = {{D}ifferential operators with mixed boundary conditions: generation of analytic semigroups},
  journal = {Nonlinear Analysis: Theory, Methods and Applications},
  year    = {2001},
  volume  = {47},
  pages   = {1333-1344},
}

@Article{MR1846768,
  author  = {Par{\'e}s, Carlos and Mac{\'i}as, Jorge and Castro D{\'i}az, Manuel J.},
  title   = {{D}uality methods with an automatic choice of parameters. {{A}}pplication to shallow water equations in conservative form},
  journal = {Numer. Math.},
  year    = {2001},
}

@Article{macias_2001,
  author  = {Mac{\'i}as, Jorge},
  title   = {{E}studio del fen{\'o}meno E} # l #{N} # i #{\~n} # o #{ mediante un modelo acoplado h{\'i}brido de oc{\'e}ano-atm{\'o}sfera},
  journal = {Rev. Geof{\'i}s.},
  year    = {2001},
}

@Article{MR1834562,
  author  = {Castro D{\'i}az, Manuel J. and Garcia-Navarro, P.},
  title   = {{T}he application of a conservative grid adaptation technique to 1{{D}} shallow water equations},
  journal = {Math. Comput. Modelling},
  year    = {2001},
}

@Article{fvrier_multivariate_2000,
  author  = {F{\'e}vrier, Sabine and Frankignoul, Claude and Sirven, J{\'e}r{\^o}me and Mike-K., Davey and Pascale, Delecluse and Ineson, Sarah and Mac{\'i}as, Jorge and Senn{\'e}chael, Nathalie and Stephenson, David-B.},
  title   = {{A} multivariate intercomparison between three oceanic {GCM}s using observed current and thermocline depth anomalies in the tropical {P}acific during 1985-1992},
  journal = {Journal of Marine Systems},
  year    = {2000},
  url     = {http://www.sciencedirect.com/science?_ob=ArticleURL{\&}_udi=B6VF5-3YYTF6S-5{\&}_user=10{\&}_rdoc=1{\&}_fmt={\&}_orig=search{\&}_sort=d{\&}_docanchor={\&}view=c{\&}_acct=C000050221{\&}_version=1{\&}_urlVersion=0{\&}_userid=10{\&}md5=a96d467650e39d3fc1d6961f20ded545},
}

@Article{ODE2,
  author  = {Gallardo, Jos{\'e}-M.},
  title   = {{G}eneration of analytic semigroups by second-order differential operators with non-separated boundary conditions},
  journal = {Rocky Mountain Journal of Mathematics},
  year    = {2000},
  volume  = {30},
  number  = {3},
  pages   = {869-899},
}

@Article{ODE3,
  author  = {Gallardo, Jos{\'e}-M.},
  title   = {{S}econd-order differential operators with integral boundary conditions and generation of analytic semigroups},
  journal = {Rocky Mountain Journal of Mathematics},
  year    = {2000},
  volume  = {30},
  number  = {4},
  pages   = {1265-1291},
}

@Article{macias_k_s_1998,
  author  = {Mac{\'i}as, Jorge and Stephenson, David and Kearsley, Anthony},
  title   = {{A} basic reference state suitable for anomaly coupled ocean-atmosphere climate models},
  journal = {Appl. Math. Lett.},
  year    = {1999},
}

@Article{MR1726910,
  author  = {Mac{\'i}as, Jorge and Par{\'e}s, Carlos and Castro D{\'i}az, Manuel J.},
  title   = {{I}mprovement and generalization of a finite element shallow-water solver to multi-layer systems},
  journal = {Internat. J. Numer. Methods Fluids},
  year    = {1999},
}

@InBook{valle_p_m_c_1998,
  title  = {{N}umerical resolution of a shallow-water system using a duality method. {A}pplication to the {A}lboran {S}ea},
  year   = {1998},
  author = {Valle, Antonio and Par{\'e}s, Carlos and Mac{\'i}as, Jorge and Castro D{\'i}az, Manuel J.},
}

@Article{MR1463582,
  author  = {Castro D{\'i}az, Manuel J. and Hecht, F. and Mohammadi, B. and Pironneau, O.},
  title   = {{A}nisotropic unstructured mesh adaption for flow simulations},
  journal = {Internat. J. Numer. Methods Fluids},
  year    = {1997},
}

@Article{macias_s_1997,
  author  = {Mac{\'i}as, Jorge and Stephenson, David},
  title   = {{S}easonal and interannual {ENSO} variability in a {H}ybrid {C}oupled {M}odel of the tropical {P}acific},
  journal = {Comptes Rendus de l'Acad{\'e}mie des Sciencies de Paris, s{\'e}rie II a},
  year    = {1997},
}

@Article{ODE1,
  author  = {Gallardo, Jos{\'e}-M.},
  title   = {{S}econd-order {ODE}'s with non-separated boundary conditions and generation of analytic semigroups},
  journal = {Nonlinear Analysis: Theory, Methods and Applications},
  year    = {1997},
  volume  = {30},
  pages   = {4991-4994},
}

@InBook{castro_m_p_1996,
  title  = {{A} multilayer shallow water model. {A}pplication to the modelling of the {A}lboran {S}ea and the {S}trait of {G}ibraltar},
  year   = {1996},
  author = {Castro D{\'i}az, Manuel J. and Mac{\'i}as, Jorge and Par{\'e}s, Carlos},
}

@InBook{macias_s_t_b_1996,
  title  = {{I}nterannual variability simulated in the tropical {P}acific},
  year   = {1996},
  author = {Mac{\'i}as, Jorge and Stephenson, David and Laurent, Terray and Sophie, Belamari},
}

@Article{MR1316046,
  author  = {Conca, C. and Par{\'e}s, Carlos and Pironneau, O. and Thiriet, M.},
  title   = {{N}avier-{{S}}tokes equations with imposed pressure and velocity fluxes},
  journal = {Internat. J. Numer. Methods Fluids},
  year    = {1995},
}

@Article{MR1311657,
  author  = {Par{\'e}s, Carlos},
  title   = {{A}pproximation de la solution des {\'e}quations d'un mod{\`e}le de turbulence par une m{\'e}thode de {{L}}agrange-{{G}}alerkin},
  journal = {Rev. Mat. Apl.},
  year    = {1994},
}

@Article{macias_c_p_1994,
  author  = {Mac{\'i}as, Jorge and Castro D{\'i}az, Manuel J. and Par{\'e}s, Carlos},
  title   = {{N}umerical modelling of water mass exchanges through the {S}trait of {G}ibraltar},
  journal = {Revista GAIA},
  year    = {1994},
}