@Article{Escalante2018Non,
  author     = {Escalante, C. and Morales de Luna, T. and Castro, M. J.},
  title      = {Non-hydrostatic pressure shallow flows: {GPU} implementation using finite volume and finite difference scheme},
  journal    = {Applied Mathematics and Computation},
  year       = {2018},
  volume     = {338},
  pages      = {631--659},
  month      = dec,
  issn       = {0096-3003},
  abstract   = {We consider the depth-integrated non-hydrostatic system derived by Yamazaki et al. An efficient formally second-order well-balanced hybrid finite volume finite difference numerical scheme is proposed. The scheme consists of a two-step algorithm based on a projection-correction type scheme initially introduced by Chorin–Temam [15]. First, the hyperbolic part of the system is discretized using a Polynomial Viscosity Matrix path-conservative finite volume method. Second, the dispersive terms are solved by means of compact finite differences. A new methodology is also presented to handle wave breaking over complex bathymetries. This adapts well to GPU-architectures and guidelines about its GPU implementation are introduced. The method has been applied to idealized and challenging experimental test cases, which shows the efficiency and accuracy of the method.},
  doi        = {10.1016/j.amc.2018.06.035},
  keywords   = {Finite difference, Finite volume, GPU, Non-hydrostatic Shallow-Water, Tsunami simulation, Wave breaking},
  shorttitle = {Non-hydrostatic pressure shallow flows},
  url        = {http://www.sciencedirect.com/science/article/pii/S0096300318305241},
  urldate    = {2018-10-25},
}

We consider the depth-integrated non-hydrostatic system derived by Yamazaki et al. An efficient formally second-order well-balanced hybrid finite volume finite difference numerical scheme is proposed. The scheme consists of a two-step algorithm based on a projection-correction type scheme initially introduced by Chorin–Temam [15]. First, the hyperbolic part of the system is discretized using a Polynomial Viscosity Matrix path-conservative finite volume method. Second, the dispersive terms are solved by means of compact finite differences. A new methodology is also presented to handle wave breaking over complex bathymetries. This adapts well to GPU-architectures and guidelines about its GPU implementation are introduced. The method has been applied to idealized and challenging experimental test cases, which shows the efficiency and accuracy of the method.

@Article{Escalante2018Efficient,
  author   = {Escalante, C. and Fernández-Nieto, E. D. and Morales de Luna, T. and Castro, M. J.},
  title    = {An {Efficient} {Two}-{Layer} {Non}-hydrostatic {Approach} for {Dispersive} {Water} {Waves}},
  journal  = {Journal of Scientific Computing},
  year     = {2018},
  month    = oct,
  issn     = {1573-7691},
  abstract = {In this paper, we propose a two-layer depth-integrated non-hydrostatic system with improved dispersion relations. This improvement is obtained through three free parameters: two of them related to the representation of the pressure at the interface and a third one that controls the relative position of the interface concerning the total height. These parameters are then optimized to improve the dispersive properties of the resulting system. The optimized model shows good linear wave characteristics up to kH{\textbackslash}approx 10𝑘𝐻≈10kH{\textbackslash}approx 10, that can be improved for long waves. The system is solved using an efficient formally second-order well-balanced and positive preserving hybrid finite volume/difference numerical scheme. The scheme consists of a two-step algorithm based on a projection-correction type scheme. First, the hyperbolic part of the system is discretized using a Polynomial Viscosity Matrix path-conservative finite-volume method. Second, the dispersive terms are solved using finite differences. The method has been applied to idealized and challenging physical situations that involve nearshore breaking. Agreement with laboratory data is excellent. This technique results in an accurate and efficient method.},
  doi      = {10.1007/s10915-018-0849-9},
  keywords = {Breaking waves, Dispersive waves, Finite-difference, Finite-volume, Non-hydrostatic, Shallow-water},
  language = {en},
  url      = {https://doi.org/10.1007/s10915-018-0849-9},
  urldate  = {2018-10-25},
}

In this paper, we propose a two-layer depth-integrated non-hydrostatic system with improved dispersion relations. This improvement is obtained through three free parameters: two of them related to the representation of the pressure at the interface and a third one that controls the relative position of the interface concerning the total height. These parameters are then optimized to improve the dispersive properties of the resulting system. The optimized model shows good linear wave characteristics up to kH\approx 10𝑘𝐻≈10kH\approx 10, that can be improved for long waves. The system is solved using an efficient formally second-order well-balanced and positive preserving hybrid finite volume/difference numerical scheme. The scheme consists of a two-step algorithm based on a projection-correction type scheme. First, the hyperbolic part of the system is discretized using a Polynomial Viscosity Matrix path-conservative finite-volume method. Second, the dispersive terms are solved using finite differences. The method has been applied to idealized and challenging physical situations that involve nearshore breaking. Agreement with laboratory data is excellent. This technique results in an accurate and efficient method.

@Article{CastroDiaz2018Fully,
  author    = {Castro D{\'{\i}}az, Manuel J. and Chalons, Christophe and Morales de Luna, Tom{\'{a}}s},
  title     = {A Fully Well-Balanced Lagrange--Projection-Type Scheme for the Shallow-Water Equations},
  journal   = {{SIAM} Journal on Numerical Analysis},
  year      = {2018},
  volume    = {56},
  number    = {5},
  pages     = {3071--3098},
  month     = {jan},
  doi       = {10.1137/17m1156101},
  publisher = {Society for Industrial {\&} Applied Mathematics ({SIAM})},
}

@InCollection{castro2017well,
  author    = {Castro, M. J. and Morales de Luna, T. and Par{\'e}s, C.},
  title     = {Well-{Balanced} {Schemes} and {Path}-{Conservative} {Numerical} {Methods}},
  booktitle = {Handbook of {Numerical} {Analysis}},
  publisher = {Elsevier},
  year      = {2017},
  editor    = {Shu, R�mi Abgrall {and} Chi-Wang},
  volume    = {18},
  series    = {Handbook of {Numerical} {Methods} for {Hyperbolic} {ProblemsApplied} and {Modern} {Issues}},
  pages     = {131--175},
  note      = {DOI: 10.1016/bs.hna.2016.10.002},
  abstract  = {In this chapter we describe a general methodology for developing high-order well-balanced schemes for hyperbolic system with nonconservative products and/or source terms. We briefly recall the Dal Maso?LeFloch?Murat theory to define weak solutions of nonconservative systems and how it has been used to establish the notion of path-conservative schemes. We show that, under this framework, it is possible to extend to the nonconservative case many well-known numerical schemes that are commonly used for system of conservation laws. Moreover, their extension to high order can be done as well. Next the well-balanced property of the proposed methods is analyzed with an illustrative 1d example. Finally, we point out the difficulties related to the right definition of weak solution and the design of numerical schemes converging to them.},
  doi       = {10.1016/bs.hna.2016.10.002},
  file      = {:castro2017well.pdf:PDF},
  issn      = {1570-8659},
  keywords  = {35L60, 65M08, 65M20, 76L05, 76M12, Balance laws, Nonconservative hyperbolic systems, Path-conservative method, Well-balanced schemes},
  url       = {http://www.sciencedirect.com/science/article/pii/S1570865916300333},
  urldate   = {2017-07-24},
}

In this chapter we describe a general methodology for developing high-order well-balanced schemes for hyperbolic system with nonconservative products and/or source terms. We briefly recall the Dal Maso?LeFloch?Murat theory to define weak solutions of nonconservative systems and how it has been used to establish the notion of path-conservative schemes. We show that, under this framework, it is possible to extend to the nonconservative case many well-known numerical schemes that are commonly used for system of conservation laws. Moreover, their extension to high order can be done as well. Next the well-balanced property of the proposed methods is analyzed with an illustrative 1d example. Finally, we point out the difficulties related to the right definition of weak solution and the design of numerical schemes converging to them.

@Article{fernandez-nieto2017formal,
  Title                    = {Formal deduction of the {Saint}-{Venant}-{Exner} model including arbitrarily sloping sediment beds and associated energy},
  Author                   = {Fern\'andez-Nieto, Enrique D. and Morales de Luna, Tom\'as and Narbona-Reina, Gladys and Zabsonr\'e, Jean de Dieu},
  Journal                  = {ESAIM: Mathematical Modelling and Numerical Analysis},
  Year                     = {2017},

  Month                    = jan,
  Number                   = {1},
  Pages                    = {115--145},
  Volume                   = {51},

  Doi                      = {10.1051/m2an/2016018},
  File                     = {:fernandez-nieto2017formal.pdf:PDF},
  ISSN                     = {0764-583X, 1290-3841},
  Language                 = {en},
  Url                      = {http://dx.doi.org/10.1051/m2an/2016018},
  Urldate                  = {2017-01-03}
}

@Article{moralesdeluna2017derivation,
  Title                    = {Derivation of a Multilayer Approach to Model Suspended Sediment Transport: Application to Hyperpycnal and Hypopycnal Plumes},
  Author                   = {Morales de Luna, Tom\'{a}s and Fern\'{a}ndez Nieto, Enrique D.. and Castro D\'{i}az, Manuel J.},
  Journal                  = {Communications in Computational Physics},
  Year                     = {2017},
  Number                   = {5},
  Pages                    = {1439-1485},
  Volume                   = {22},

  Category                 = {PHYSICS, MATHEMATICAL},
  Doi                      = {10.4208/cicp.OA-2016-0215},
  File                     = {:moralesdeluna2017derivation.pdf:PDF}
}

@Article{sanchez-linares2016hllc,
  Title                    = {A {HLLC} scheme for {Ripa} model},
  Author                   = {S\'anchez-Linares, C. and Morales de Luna, T. and Castro D\'iaz, M. J.},
  Journal                  = {Applied Mathematics and Computation},
  Year                     = {2016},

  Month                    = jan,
  Pages                    = {369--384},
  Volume                   = {272, Part 2},

  Abstract                 = {We consider the one-dimensional system of shallow-water equations with horizontal temperature gradients (the Ripa system). We derive a HLLC scheme for Ripa system which falls into the theory of path-conservative approximate Riemann solvers. The resulting scheme is robust, easy to implement, well-balanced, positivity preserving and entropy dissipative for the case of flat or continuous bottom.},
  Doi                      = {10.1016/j.amc.2015.05.137},
  File                     = {:sanchez-linares2016hllc.pdf:PDF},
  ISSN                     = {0096-3003},
  Keywords                 = {Finite volume schemes, HLLC, Path-conservative schemes, Ripa system},
  Series                   = {Recent {Advances} in {Numerical} {Methods} for {Hyperbolic} {Partial} {Differential} {Equations}},
  Url                      = {http://www.sciencedirect.com/science/article/pii/S0096300315007687},
  Urldate                  = {2017-01-03}
}

We consider the one-dimensional system of shallow-water equations with horizontal temperature gradients (the Ripa system). We derive a HLLC scheme for Ripa system which falls into the theory of path-conservative approximate Riemann solvers. The resulting scheme is robust, easy to implement, well-balanced, positivity preserving and entropy dissipative for the case of flat or continuous bottom.

@Article{berthon2015efficient,
  Title                    = {An efficient splitting technique for two-layer shallow-water model},
  Author                   = {Berthon, Christophe and Foucher, Fran\c{c}oise and Morales de Luna, Tom\'as},
  Journal                  = {Numerical Methods for Partial Differential Equations},
  Year                     = {2015},

  Month                    = sep,
  Number                   = {5},
  Pages                    = {1396--1423},
  Volume                   = {31},

  Doi                      = {10.1002/num.21949},
  File                     = {:berthon2015efficient.pdf:PDF},
  ISSN                     = {1098-2426},
  Keywords                 = {Finite volume schemes, non-negative preserving schemes, source term approximations, splitting schemes, two-layer shallow-water model, Well-balanced schemes},
  Language                 = {en},
  Url                      = {http://onlinelibrary.wiley.com/doi/10.1002/num.21949/abstract},
  Urldate                  = {2017-01-03}
}

@Article{fernandez-nieto2014influence,
  Title                    = {On the influence of the thickness of the sediment moving layer in the definition of the bedload transport formula in Exner systems},
  Author                   = {Fern{\'a}ndez-Nieto, E. D. and Lucas, C. and Morales de Luna, T. and Cordier, S.},
  Journal                  = {Computers \& Fluids},
  Year                     = {2014},

  Month                    = mar,
  Pages                    = {87--106},
  Volume                   = {91},

  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.},
  Doi                      = {10.1016/j.compfluid.2013.11.031},
  File                     = {:fernandez-nieto2014influence.pdf:PDF},
  ISSN                     = {0045-7930},
  Keywords                 = {Exner system, Finite volume method, Sediment transport, Shallow water},
  Url                      = {http://www.sciencedirect.com/science/article/pii/S0045793013004866},
  Urldate                  = {2014-01-27}
}

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,
  Title                    = {Relation between {PVM} schemes and simple Riemann solvers},
  Author                   = {Morales de Luna, Tom\'as and Castro D\'iaz, Manuel J. and Par\'es, Carlos},
  Journal                  = {Numerical Methods for Partial Differential Equations},
  Year                     = {2014},

  Month                    = mar,
  Number                   = {4},
  Pages                    = {1315--1341},
  Volume                   = {30},

  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},
  Copyright                = {Copyright 2014 Wiley Periodicals, Inc.},
  Doi                      = {10.1002/num.21871},
  File                     = {:moralesdeluna2014relation.pdf:PDF},
  ISSN                     = {1098-2426},
  Keywords                 = {finite volume schemes, {PVM} schemes, Riemann solvers, Shallow water equations},
  Language                 = {en},
  Url                      = {http://onlinelibrary.wiley.com/doi/10.1002/num.21871/abstract},
  Urldate                  = {2014-03-18}
}

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{castrodiaz2013hllc,
  Title                    = {A {HLLC} scheme for nonconservative hyperbolic problems. Application to turbidity currents with sediment transport},
  Author                   = {Castro D\'iaz, Manuel Jes\'us and Fern\'andez-Nieto, Enrique Domingo and Morales de Luna, Tom\'as and Narbona-Reina, Gladys and Par\'es Madro\ nal, Carlos},
  Journal                  = {{ESAIM:} Mathematical Modelling and Numerical Analysis},
  Year                     = {2013},

  Month                    = jul,
  Number                   = {1},
  Pages                    = {1--32},
  Volume                   = {47},

  Comment                  = {�ndice de impacto: 1.22, Posici�n: 52, Revista dentro del 25%: S�, Categor�a: MATHEMATICS, APPLIED, Num. revistas en cat.: 245, Fuente de impacto: WOS (JCR)},
  Doi                      = {10.1051/m2an/2012017},
  File                     = {:castrodiaz2012HLLC.pdf:PDF},
  ISSN                     = {0764-{583X}, 1290-3841},
  Url                      = {http://www.esaim-m2an.org/action/displayAbstract?fromPage=online&aid=8654736}
}

@Article{fernandez-nieto2013multilayer,
  Title                    = {A multilayer shallow water system for polydisperse sedimentation},
  Author                   = {E.D. Fern{\'a}ndez-Nieto and E.H. Kon{\'e} and Morales de Luna, Tom{\'a}s and R. B{\"u}rger},
  Journal                  = {Journal of Computational Physics},
  Year                     = {2013},

  Month                    = apr,
  Pages                    = {281--314},
  Volume                   = {238},

  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.},
  Comment                  = {�ndice de impacto: 2.31, Posici�n: 5, Revista dentro del 25%: S�, Num. revistas en cat.: 55, Fuente de impacto: WOS (JCR), Categor�a: PHYSICS, MATHEMATICAL},
  Doi                      = {10.1016/j.jcp.2012.12.008},
  File                     = {:fernandez-nieto13multilayer.pdf:PDF},
  ISSN                     = {0021-9991},
  Keywords                 = {Finite volume, Multilayer, Polydisperse, Sediment, Shallow water},
  Url                      = {http://www.sciencedirect.com/science/article/pii/S0021999112007395},
  Urldate                  = {2013-03-14}
}

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.

@InBook{moralesdeluna2013sedimentNova,
  Title                    = {Sediment transport: monitoring, modeling, and management},
  Author                   = {Morales de Luna, T.},
  Chapter                  = {Mathematical Models for Simulation of Bedload and Suspension Sediment Transport},
  Editor                   = {Khan, Abdul A and Wu, Weiming},
  Pages                    = {239--261},
  Publisher                = {Nova Publishers},
  Year                     = {2013},

  File                     = {:moralesdeluna2013sedimentNova_chapter.pdf:PDF;:moralesdeluna2013sedimentNova.pdf:PDF},
  ISBN                     = {978-1626186835},
  Language                 = {English},
  Shorttitle               = {Sediment transport},
  Url                      = {https://www.novapublishers.com/catalog/product_info.php?products_id=41654}
}

@Article{moralesdeluna2013reliability,
  Title                    = {Reliability of first order numerical schemes for solving shallow water system over abrupt topography},
  Author                   = {Morales de Luna, T. and Castro D{\'i}az, {M.J.} and Par{\'e}s, C.},
  Journal                  = {Applied Mathematics and Computation},
  Year                     = {2013},

  Month                    = may,
  Number                   = {17},
  Pages                    = {9012--9032},
  Volume                   = {219},

  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.},
  Doi                      = {10.1016/j.amc.2013.03.033},
  File                     = {:moralesdeluna13Reliability.pdf:PDF},
  ISSN                     = {0096-3003},
  Keywords                 = {Finite volume schemes, {SHALLOW} {WATER} {EQUATIONS}, Well-balanced schemes},
  Url                      = {http://www.sciencedirect.com/science/article/pii/S0096300313002865},
  Urldate                  = {2013-04-22}
}

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.

@Article{cordier2011bedload,
  Title                    = {Bedload transport in shallow water models: Why splitting (may) fail, how hyperbolicity (can) help},
  Author                   = {Cordier, S. and Le, {M.H.} and Morales de Luna, T.},
  Journal                  = {Advances in Water Resources},
  Year                     = {2011},

  Month                    = aug,
  Number                   = {8},
  Pages                    = {980--989},
  Volume                   = {34},

  Abstract                 = {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�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.},
  Category                 = {WATER RESOURCES},
  Comment                  = {�ndice de impacto: 2.477, Revista dentro del 25%: S�, Num. revistas en cat.: 76, Categor�a: WATER RESOURCES , Posici�n: 4, Fuente de impacto: WOS (JCR)},
  Doi                      = {16/j.advwatres.2011.05.002},
  File                     = {:cordier2011Bedload.pdf:PDF},
  Impactfactor             = {2.477 (2010)},
  ISSN                     = {0309-1708},
  Keywords                 = {Exner equation, Hyperbolicity, Sediment transport, Shallow water system, Splitting methods, Stability},
  Position                 = {4},
  Revincat                 = {76},
  Shorttitle               = {Bedload transport in shallow water models},
  Url                      = {http://www.sciencedirect.com/science/article/pii/S0309170811000935}
}

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�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.

@Article{moralesdeluna2011duality,
  Title                    = {A Duality Method for Sediment Transport Based on a Modified {Meyer-Peter} \& {M}\"uller Model},
  Author                   = {Morales de Luna, T. and Castro D\'iaz, M. J. and Par\'es Madro\~nal, C.},
  Journal                  = {Journal of Scientific Computing},
  Year                     = {2011},

  Month                    = dec,
  Pages                    = {258-273},
  Volume                   = {48},

  Category                 = {MATHEMATICS, APPLIED},
  Comment                  = {�ndice de impacto: 1.252, Num. revistas en cat.: 236, Revista dentro del 25%: S�, Categor�a: MATHEMATICS, APPLIED, Posici�n: 53, Fuente de impacto: WOS (JCR)},
  Doi                      = {10.1007/s10915-010-9447-1},
  File                     = {:moralesdeluna2010Duality.pdf:PDF},
  Impactfactor             = {1.252 (2010)},
  ISSN                     = {0885-7474},
  Position                 = {53},
  Revincat                 = {236},
  Url                      = {http://www.springerlink.com/content/7401364w29464259/}
}

@Article{bouchut2010subsonic,
  Title                    = {A {Subsonic-Well-Balanced} Reconstruction Scheme for Shallow Water Flows},
  Author                   = {Bouchut, Francois and Morales de Luna, Tomas},
  Journal                  = {{SIAM} Journal on Numerical Analysis},
  Year                     = {2010},
  Number                   = {5},
  Pages                    = {1733--1758},
  Volume                   = {48},

  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.},
  Category                 = {MATHEMATICS, APPLIED},
  Comment                  = {�ndice de impacto: 1.664, Num. revistas en cat.: 236, Revista dentro del 25%: S�, Categor�a: MATHEMATICS, APPLIED, Fuente de impacto: WOS (JCR), Posici�n: 24},
  Doi                      = {10.1137/090758416},
  File                     = {:bouchut2010subsonic-well-balanced.pdf:PDF},
  Impactfactor             = {1.664},
  ISSN                     = {00361429},
  Keywords                 = {shallow water, subsonic reconstruction, subsonic steady states, well-balanced scheme, semi-discrete entropy inequality},
  Position                 = {24},
  Revincat                 = {236},
  Url                      = {http://link.aip.org/link/SJNAAM/v48/i5/p1733/s1&Agg=doi}
}

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{moralesdeluna2010sediment,
  Title                    = {On a sediment transport model in shallow water equations with gravity effects},
  Author                   = {Morales de Luna, Tom\'{a}s and Castro D\'{i}az, Manuel J. and Par\'{e}s Madro\~{n}al, Carlos},
  Journal                  = {Numerical Mathematics and Advanced Applications 2009},
  Year                     = {2010},
  Pages                    = {655--662},

  Doi                      = {10.1007/978-3-642-11795-4},
  File                     = {:enumath2009-libro.pdf:PDF},
  Url                      = {http://www.springer.com/mathematics/computational+science+%26+engineering/book/978-3-642-11794-7}
}

@Article{bouchut2009semi,
  Title                    = {Semi-discrete Entropy Satisfying Approximate Riemann Solvers. The Case of the Suliciu Relaxation Approximation},
  Author                   = {Bouchut, Fran\c{c}ois and Morales de Luna, Tom\'{a}s},
  Journal                  = {Journal of Scientific Computing},
  Year                     = {2009},
  Pages                    = {483--509},
  Volume                   = {41},

  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.},
  Category                 = {MATHEMATICS, APPLIED},
  Comment                  = {�ndice de impacto: 1.707, Revista dentro del 25%: S�, Categor�a: MATHEMATICS, APPLIED, Num. revistas en cat.: 204, Fuente de impacto: WOS (JCR), Posici�n: 23},
  Doi                      = {10.1007/s10915-009-9311-3},
  File                     = {:bouchut09semi-discrete.pdf:PDF},
  Impactfactor             = {1.707},
  Position                 = {23},
  Revincat                 = {204},
  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.

@Book{moralesdeluna2009entropy,
  Title                    = {Entropy satisfying schemes for shallow water systems},
  Author                   = {Morales de Luna, Tom\'as},
  Publisher                = {{VDM} Verlag},
  Year                     = {2009},
  Month                    = sep,

  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.},
  ISBN                     = {{363917271X}},
  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.

@Article{moralesdeluna2009shallow,
  Title                    = {On a shallow water model for the simulation of turbidity currents},
  Author                   = {Morales de Luna, Tom\'{a}s and Castro D\'{i}az, Manuel J. and Par\'{e}s Madro\~{n}al, Carlos and Fern\'{a}ndez Nieto, Enrique D.},
  Journal                  = {Communications in Computational Physics},
  Year                     = {2009},
  Number                   = {4},
  Pages                    = {848--882},
  Volume                   = {6},

  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.},
  Category                 = {PHYSICS, MATHEMATICAL},
  Comment                  = {�ndice de impacto: 2.077, Revista dentro del 25%: S�, Posici�n: 8, Fuente de impacto: WOS (JCR), Num. revistas en cat.: 47, Categor�a: PHYSICS, MATHEMATICAL},
  Doi                      = {10.4208/cicp.2009.v6.p848},
  File                     = {:moralesdeluna09shallow.pdf:PDF},
  Impactfactor             = {2.077},
  Keywords                 = {Turbidity currents, hyperbolic systems, finite volume methods, path-conservative schemes, numerical modeling},
  Position                 = {8},
  Revincat                 = {47},
  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.

@InProceedings{moralesdeluna2009river,
  Title                    = {River Sediment Transport and Deposition Modeling},
  Author                   = {Morales de Luna, T. and Castro Diaz, M. J. and Pares Madronal, C.},
  Booktitle                = {NUMERICAL ANALYSIS AND APPLIED MATHEMATICS},
  Year                     = {2009},

  Address                  = {2 HUNTINGTON QUADRANGLE, STE 1NO1, MELVILLE, NY 11747-4501 USA},
  Editor                   = {Simos, TE and Psihoyios, G and Tsitouras, C},
  Note                     = {International Conference on Numerical Analysis and Applied Mathematics, Rethymno, GREECE, SEP 18-22, 2009},
  Organization             = {Greek Minist Educ \& Religious Affairs; European Soc Computat Methods Sci \& Engn},
  Pages                    = {1437-1440},
  Publisher                = {AMER INST PHYSICS},
  Series                   = {AIP Conference Proceedings},
  Volume                   = {1168},

  Abstract                 = {Sediment can be transported in several ways by the action of a river. During low transport stages, particles move by sliding and rolling over the surface of the bed. With the increase of the velocity, the sediment is entrained into suspension and travels significant distances before being deposed again. One can observe a continuous exchange between sediment at the river bed and sediment in suspension. Moreover, when the concentration of suspended sediment is elevated, the river can plunge into the ocean creating an hyperpycnal plume. All this phenomena may be modeled by means of a coupled model constituted by a hydrodynamical component, described by a Shallow water system and transport equations for sediment in suspension with erosion and deposition source terms, and a morphodynamical component, which depends on a bedload transport flux. The mathematical model proposed allows to model the phenomena previously described as well as pure bedload or suspension transport and hyperpycnal plumes. The equations are solved using path-conservative schemes described by Pares et al.},
  File                     = {:moralesdeluna2009River.pdf:PDF},
  ISBN                     = {978-0-7354-0709-1},
  ISSN                     = {0094-243X},
  Keywords                 = {Sediment transport; hyperbolic systems; finite volume methods; path-conservative schemes; numerical modeling},
  Keywords-plus            = {NONCONSERVATIVE HYPERBOLIC SYSTEMS; TURBIDITY CURRENTS; SIMULATION},
  Language                 = {English},
  Research-areas           = {Mathematics; Physics},
  Type                     = {Proceedings Paper},
  Web-of-science-categories = {Mathematics, Applied; Physics, Mathematical}
}

Sediment can be transported in several ways by the action of a river. During low transport stages, particles move by sliding and rolling over the surface of the bed. With the increase of the velocity, the sediment is entrained into suspension and travels significant distances before being deposed again. One can observe a continuous exchange between sediment at the river bed and sediment in suspension. Moreover, when the concentration of suspended sediment is elevated, the river can plunge into the ocean creating an hyperpycnal plume. All this phenomena may be modeled by means of a coupled model constituted by a hydrodynamical component, described by a Shallow water system and transport equations for sediment in suspension with erosion and deposition source terms, and a morphodynamical component, which depends on a bedload transport flux. The mathematical model proposed allows to model the phenomena previously described as well as pure bedload or suspension transport and hyperpycnal plumes. The equations are solved using path-conservative schemes described by Pares et al.

@Article{bouchut2008entropy,
  Title                    = {An entropy satisfying scheme for two-layer shallow water equations with uncoupled treatment},
  Author                   = {Bouchut, Fran\c{c}ois and Morales de Luna, Tom\'{a}s},
  Journal                  = {ESAIM: Mathematical Modelling and Numerical Analysis (ESAIM: M2AN)},
  Year                     = {2008},

  Month                    = {jun},
  Pages                    = {683--698},
  Volume                   = {42},

  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.},
  Category                 = {MATHEMATICS, APPLIED},
  Comment                  = {�ndice de impacto: 0.978, Posici�n: 57, Categor�a: MATHEMATICS, APPLIED, Fuente de impacto: WOS (JCR), Num. revistas en cat.: 175},
  Doi                      = {10.1051/m2an:2008019},
  File                     = {:bouchut08entropy.pdf:PDF},
  Impactfactor             = {0.978},
  Keywords                 = {Two-layer shallow water, nonconservative system, complex eigenvalues, time-splitting, entropy inequality, well-balanced scheme, nonnegativity},
  Position                 = {57},
  Revincat                 = {175},
  Url                      = {http://www.esaim-m2an.org/index.php?option=article\&access=standard\&Itemid=129\&url=/articles/m2an/abs/first/m2an0716/m2an0716.html}
}

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{moralesdeluna2008saint,
  Title                    = {A Saint Venant model for gravity driven shallow water flows with variable density and compressibility effects},
  Author                   = {Morales de Luna, Tom\'{a}s},
  Journal                  = {Mathematical and Computer Modelling},
  Year                     = {2008},

  Month                    = feb,
  Number                   = {3-4},
  Pages                    = {436--444},
  Volume                   = {47},

  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},
  Category                 = {MATHEMATICS, APPLIED},
  Comment                  = {�ndice de impacto: 1.032, Categor�a: MATHEMATICS, APPLIED, Fuente de impacto: WOS (JCR), Num. revistas en cat.: 175, Posici�n: 51},
  Doi                      = {10.1016/j.mcm.2007.04.016},
  File                     = {:moralesdeluna08saint.pdf:PDF},
  Impactfactor             = {1.032},
  Keywords                 = {Compressibility,Entropy inequality,Savage{\textendash}Hutter model,Shallow water,Variable density},
  Position                 = {51},
  Revincat                 = {175},
  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

@InProceedings{moralesdeluna2008semidiscrete,
  Title                    = {Semidiscrete Entropy Satisfying Approximate Riemann Solvers and Application to the Suliciu Relaxation Approximation},
  Author                   = {Morales de Luna, Tom\'{a}s and Bouchut, Fran\c{c}ois},
  Booktitle                = {Hyperbolic Problems: Theory, Numerics, Applications},
  Year                     = {2008},
  Pages                    = {739--746},
  Publisher                = {Springer},

  Abstract                 = {We establish conditions for an approximate simple Riemann solver to satisfy a semi-discrete entropy inequality. Classically, a discrete entropy inequality allows to analyze the stability of a numerical scheme for a conservative system. A semi-discrete entropy inequality gives a simpler and less restrictive approach than a fully discrete entropy inequality and leads to the definition of less restrictive conditions for numerical schemes to satisfy. 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. The Suliciu relaxation system is attached to the resolution of the isentropic gas dynamics system and can also handle full gas dynamics. It allows to define a simple approximate Riemann solver for gas dynamics. Conditions have already been established for the scheme to be entropy satisfying. Our approach allows to relax the conditions established in the fully discrete case and leads to the definition of a numerical scheme for gas dynamics which satisfies a semi-discrete entropy inequality while allowing exact resolution of shocks.},
  File                     = {:moralesdeluna08semidiscrete.pdf:PDF},
  ISBN                     = {978-3-540-75711-5},
  Url                      = {http://www.springer.com/east/home?SGWID=5-102-22-173765220-0\&changeHeader=true}
}

We establish conditions for an approximate simple Riemann solver to satisfy a semi-discrete entropy inequality. Classically, a discrete entropy inequality allows to analyze the stability of a numerical scheme for a conservative system. A semi-discrete entropy inequality gives a simpler and less restrictive approach than a fully discrete entropy inequality and leads to the definition of less restrictive conditions for numerical schemes to satisfy. 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. The Suliciu relaxation system is attached to the resolution of the isentropic gas dynamics system and can also handle full gas dynamics. It allows to define a simple approximate Riemann solver for gas dynamics. Conditions have already been established for the scheme to be entropy satisfying. Our approach allows to relax the conditions established in the fully discrete case and leads to the definition of a numerical scheme for gas dynamics which satisfies a semi-discrete entropy inequality while allowing exact resolution of shocks.

@InProceedings{moralesdeluna2008modeling,
  Title                    = {Modeling and Simulation of Turbidity Currents},
  Author                   = {Morales de Luna, Tom\'{a}s and Castro D\'{i}az, Manuel J. and Par\'{e}s Madro\~{n}al, Carlos},
  Booktitle                = {Finite Volumes for Complex Applications V},
  Year                     = {2008},
  Month                    = may,
  Note                     = {ISBN 978-1-84821-035-6},
  Pages                    = {593--600},
  Publisher                = {Wiley},

  Abstract                 = {We present a new model for hyperpycnal plumes or turbidity currents. This model takes into account deposition and erosion effects as well as solid transport of particles at the bed load due to the current. The model is obtained as depth-average equations under the shallow water hypothesis describing the balance of fluid mass, sediment mass and mean flow. The final system is hyperbolic and numerical simulations are carried out using finite volume schemes and path-conservative schemes.},
  File                     = {:moralesdeluna08modeling.pdf:PDF},
  ISBN                     = {9781848210356},
  Url                      = {http://www.iste.co.uk/index.php?p=a&ACTION=View&id=220}
}

We present a new model for hyperpycnal plumes or turbidity currents. This model takes into account deposition and erosion effects as well as solid transport of particles at the bed load due to the current. The model is obtained as depth-average equations under the shallow water hypothesis describing the balance of fluid mass, sediment mass and mean flow. The final system is hyperbolic and numerical simulations are carried out using finite volume schemes and path-conservative schemes.