@article{morales_de_luna_relation_2014,
	title = {Relation between {PVM} schemes and simple Riemann solvers},
	volume = {30(4)},
	url = {http://onlinelibrary.wiley.com/doi/10.1002/num.21871/abstract},
	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},
	pages = {1315--1341},
	journaltitle = {Numerical Methods for Partial Differential Equations},
	author = {Morales de Luna, Tomás and Castro Díaz, Manuel J. and Parés, Carlos},
	date = {2014-03},
	keywords = {finite volume schemes, {PVM} schemes, Riemann solvers, Shallow water equations},
}

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{morales_de_luna_reliability_2013,
	title = {Reliability of first order numerical schemes for solving shallow water system over abrupt topography},
	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.},
	pages = {9012--9032},
	number = {17},
	journaltitle = {Applied Mathematics and Computation},
	author = {Morales de Luna, Tomás and Castro Díaz, Manuel J. and Parés, Carlos},
	date = {2013},
}

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{castro_diaz_hllc_2012,
	title = {A {HLLC} scheme for nonconservative hyperbolic problems. Application to turbidity currents with sediment transport},
	volume = {47},
	url = {http://www.esaim-m2an.org/action/displayAbstract?fromPage=online&aid=8654736},
	pages = {1--32},
	number = {1},
	journaltitle = {{ESAIM}: Mathematical Modelling and Numerical Analysis},
	author = {Castro Díaz, Manuel J. and Fernández Nieto, E. D. and Morales de Luna, Tomás and Narbona-Reina, Gladys and Parés, Carlos},
	date = {2012},
}

@book{macias_deslizamientos_2012,
	location = {Avda. del Brasil, 31. 28020 Madrid},
	title = {Deslizamientos Submarinos y Tsunamis en el Mar de Alborán. Un Ejemplo de Modelización.},
	volume = {6},
	url = {http://hdl.handle.net/10508/1073},
	series = {Temas de Oceanografía},
	publisher = {Instituto Español de Oceanografía},
	author = {Macías, Jorge and Fernández-Salas, L.-M. and González-Vida, J.-M. and Vázquez, J.-T. and Castro Díaz, Manuel J. and Bárcenas, P. and Díaz del Río, V. and Morales de Luna, Tomás and de la Asunción, Marc and Parés, Carlos},
	date = {2012},
	keywords = {Deslizamientos submarinos, Simulación de Tsunamis, Tsunami de Alborán},
}

@incollection{castro_diaz_ifcp_2012,
	title = {{IFCP} Riemann solver: Application to tsunami modelling using {GPUs}},
	pages = {237--244},
	publisher = {{CRC} Press},
	author = {Castro Díaz, Manuel J. and de la Asunción, Marc and Macías, Jorge and Parés, Carlos and Fernández Nieto, E. D. and González-Vida, J.-M. and Morales de Luna, Tomás},
	editor = {{E.-Vázquez} and {A.-Hidalgo} and {P.-García} and {L.-Cea}},
	date = {2012},
	note = {Section: 5},
}

@article{morales_de_luna_duality_2011,
	title = {A Duality Method for Sediment Transport Based on\ a\ Modified Meyer-Peter \& Müller Mod},
	volume = {48},
	url = {http://dx.doi.org/10.1007/s10915-010-9447-1},
	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 \& 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.},
	pages = {258--273},
	number = {1},
	journaltitle = {Journal of Scientific Computing},
	author = {Morales de Luna, Tomás and Castro Díaz, Manuel J. and Parés, Carlos},
	date = {2011},
}

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{castro_diaz_numerical_2011,
	title = {Numerical Treatment of the Loss of Hyperbolicity of\ the\ Two-Layer Shallow-Water Syst},
	volume = {48},
	url = {http://dx.doi.org/10.1007/s10915-010-9427-5},
	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.},
	pages = {16--40},
	number = {1},
	journaltitle = {Journal of Scientific Computing},
	author = {Castro Díaz, Manuel J. and Fernández Nieto, E. D. and González Vida, José M. and Parés, Carlos},
	date = {2011},
}

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{fernandez_nieto_intermediate_2011,
	title = {On an Intermediate Field Capturing Riemann Solver Based on a Parabolic Viscosity Matrix for the Two-Layer Shallow Water System},
	volume = {48},
	url = {http://dx.doi.org/10.1007/s10915-011-9465-7},
	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.},
	pages = {117--140},
	number = {1},
	journaltitle = {Journal of Scientific Computing},
	author = {Fernández Nieto, E. D. and Castro Díaz, Manuel J. and Parés, Carlos},
	date = {2011},
}

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{munoz_ruiz_convergence_2011,
	title = {On the convergence and well-balanced property of path-conservative numerical schemes for systems of balance laws},
	volume = {48},
	url = {http://www.springerlink.com/content/h84677m854r25780/},
	pages = {274--295},
	number = {1},
	journaltitle = {Journal of Scientific Computing},
	author = {Muñoz Ruiz, María Luz and Parés, Carlos},
	date = {2011},
}

@article{dumbser_force_2010,
	title = {{FORCE} schemes on unstructured meshes {II}: Nonconservative hyperbolic systems},
	volume = {199},
	url = {http://www.sciencedirect.com/science/article/pii/S0045782509003612},
	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].},
	pages = {625--647},
	number = {9},
	journaltitle = {Comput. Methods Appl. Mech. Engrg.},
	author = {Dumbser, Michael and Hidalgo, A. and Castro Díaz, Manuel J. and Parés, Carlos and Toro, E.-F.},
	date = {2010},
}

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{castro_diaz_fast_2010,
	title = {On some fast well-balanced first order solvers for nonconservative systems.},
	volume = {79},
	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.},
	pages = {1427--1472},
	number = {271},
	journaltitle = {{MATHEMATICS} {OF} {COMPUTATION}},
	author = {Castro Díaz, Manuel J. and Pardo, Alberto and Parés, Carlos and Toro, E.-F.},
	date = {2010},
}

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{dumbser_ader_2009,
	title = {{ADER} schemes on unstructured meshes for nonconservative hyperbolic systems: Applications to geophysical flows},
	volume = {38},
	url = {http://www.sciencedirect.com/science/article/pii/S0045793009000498},
	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, 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].},
	pages = {1731--1748},
	number = {9},
	journaltitle = {Computers \& Fluids},
	author = {Dumbser, Michael and Castro Díaz, Manuel J. and Parés, Carlos and F.-Toro, Eleuterio},
	date = {2009},
}

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].

@article{castro_diaz_high_2009,
	title = {High order extensions of Roe schemes for two-dimensional nonconservative hyperbolic systems},
	volume = {39},
	pages = {67--114},
	number = {1},
	journaltitle = {J. Sci. Comput.},
	author = {Castro Díaz, Manuel J. and Fernández Nieto, E. D. and Ferreiro, A.-M. and García-Rodríguez, J. A. and Parés, Carlos},
	date = {2009},
}

@article{morales_de_luna_shallow_2009,
	title = {On a shallow water model for the simulation of turbidity currents},
	url = {http://www.global-sci.com/cgi-bin/fulltext/6/848/full},
	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.},
	journaltitle = {Communications in Computational Physics},
	author = {Morales de Luna, Tomás and Castro Díaz, Manuel J. and Parés, Carlos and Fernández Nieto, Enrique D.},
	date = {2009},
	keywords = {finite volume methods, hyperbolic systems, numerical modeling, path-conservative schemes, Turbidity currents},
}

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{pares_difficulties_2009,
	title = {On some difficulties of the numerical approximation of nonconservative hyperbolic systems},
	journaltitle = {Bol. Soc. Esp. Mat. Apl},
	author = {Parés, Carlos and Muñoz Ruiz, María Luz},
	date = {2009},
}

@article{castro_diaz_fast_2009,
	title = {On some fast well-balanced first order solvers for nonconservative systems},
	volume = {79},
	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},
	pages = {1427--1472},
	number = {271},
	journaltitle = {Math. Comp.},
	author = {Castro Díaz, Manuel J. and Pardo, A. and Parés, Carlos and Toro, E.-F.},
	date = {2009},
}

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

@incollection{castro_diaz_realistic_2009,
	title = {Realistic application of a tidal 2D two-layer shallow water model to the Strait of Gibraltar},
	series = {Numerical Analysis and Applied Mathematics},
	pages = {1429--1432},
	author = {Castro Díaz, Manuel J. and González-Vida, J.-M. and Macías, Jorge and Parés, Carlos},
	date = {2009},
}

@article{castro_diaz_finite_2008,
	title = {Finite volume simulation of the geostrophic adjustment in a rotating shallow-water system},
	volume = {31},
	url = {http://epubs.siam.org/sisc/resource/1/sjoce3/v31/i1/p444_s1},
	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.},
	pages = {444--477},
	number = {1},
	journaltitle = {{SIAM} J. Sci. Comput.},
	author = {Castro Díaz, Manuel J. and López, Juan-Antonio and Parés, Carlos},
	date = {2008},
}

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.

@inproceedings{gallardo_high-order_2008,
	title = {High-order finite volume schemes for shallow water equations with topography and dry areas},
	pages = {585--594},
	booktitle = {Hyperbolic problems: Theory, numerics and applications},
	publisher = {American Mathematical Society},
	author = {Gallardo, José-M. and Castro Díaz, Manuel J. and Parés, Carlos},
	editor = {Eitan, Tadmor and Jian-Guo, Liu and Athanasios, Tzavaras},
	date = {2008},
}

@article{arregui_numerical_2008,
	title = {Numerical solution of a 1-D elastohydrodynamic problem in magnetic storage devices},
	volume = {42},
	url = {http://www.esaim-m2an.org/action/displayAbstract?fromPage=online&aid=8194733},
	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},
	pages = {645--665},
	number = {4},
	journaltitle = {M2AN Math. Model. Numer. Anal.},
	author = {Arregui, Iñigo and Cendán, José Jesús and Parés, Carlos and Vázquez, Carlos},
	date = {2008},
}

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

@incollection{castro_diaz_simulation_2008,
	title = {Simulation of tidal currents in the Strait of Gibraltar},
	publisher = {Universidad Rey Juan Carlos},
	author = {Castro Díaz, Manuel J. and González-Vida, J.-M. and Macías, Jorge and Parés, Carlos},
	editor = {Mofdi, El-Amrani and Mohamed, Sead and Naje, Yebari},
	date = {2008},
}

@article{castro_diaz_simulation_2008-1,
	title = {Simulation of tidal currents in the Strait of Gibraltar using two-dimensional two-layer shallow-water models},
	journaltitle = {Bol. Soc. Esp. Mat. Apl.},
	author = {Castro Díaz, Manuel J. and García-Rodríguez, J.-A. and González-Vida, J.-M. and Macías, Jorge and Parés, Carlos},
	date = {2008},
}

@article{castro_diaz_solving_2008,
	title = {Solving shallow-water systems in 2D domains using finite volume methods and multimedia {SSE} instructions},
	volume = {221},
	url = {http://www.sciencedirect.com/science/article/pii/S0377042707005201},
	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. 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.},
	pages = {16--32},
	number = {1},
	journaltitle = {J. Comput. Appl. Math.},
	author = {Castro Díaz, Manuel J. and García-Rodríguez, J.-A. and González-Vida, J.-M. and Parés, Carlos},
	date = {2008},
}

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{castro_diaz_well-balanced_2008,
	title = {Well-balanced finite volume schemes for 2d non-homogeneous hyperboli systems. Application to the dam break of Aznalcóllar},
	volume = {197},
	url = {http://www.sciencedirect.com/science/article/pii/S0045782508001394},
	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: Aznalcóllar, an ecological disaster that happened in the province of Seville, Spain in 1998},
	pages = {3932--3950},
	number = {45},
	journaltitle = {Comput. Methods Appl. Mech. Engrg.},
	author = {Castro Díaz, Manuel J. and Chacón-Rebollo, T. and Fernández Nieto, E. D. and González-Vida, J.-M. and Parés, Carlos},
	date = {2008},
}

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{castro_diaz_well-balanced_2008-1,
	title = {Well-balanced high order extensions of Godunov's method for semilinear balance laws},
	volume = {46},
	url = {http://epubs.siam.org/sinum/resource/1/sjnaam/v46/i2/p1012_s1},
	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.},
	pages = {1012--1039},
	number = {2},
	journaltitle = {{SIAM} J. Numer. Anal.},
	author = {Castro Díaz, Manuel J. and Gallardo, José-M. and López-García, Juan-A. and Parés, Carlos},
	date = {2008},
}

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{castro_diaz_why_2008,
	title = {Why many theories of shock waves are necessary: convergence error in formally path-consistent schemes},
	volume = {227},
	url = {http://www.sciencedirect.com/science/article/pii/S0021999108002842},
	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.},
	pages = {8107--8129},
	number = {17},
	journaltitle = {J. Comput. Phys.},
	author = {Castro Díaz, Manuel J. and {LeFloch}, Philippe-G. and Muñoz Ruiz, María Luz and Parés, Carlos},
	date = {2008},
}

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{munoz_ruiz_godunov_2007,
	title = {Godunov method for nonconservative hyperbolic systems},
	volume = {41},
	pages = {169--185},
	number = {1},
	journaltitle = {M2AN Math. Model. Numer. Anal.},
	author = {Muñoz Ruiz, María Luz and Parés, Carlos},
	date = {2007},
}

@article{castro_diaz_improved_2007,
	title = {Improved {FVM} for two-layer shallow-water models: Application to the Strait of Gibraltar},
	journaltitle = {Advances in Engineering Software},
	author = {Castro Díaz, Manuel J. and García-Rodríguez, J.-A. and González-Vida, J.-M. and Macías, Jorge and Parés, Carlos},
	date = {2007},
}

@article{gallardo_well-balanced_2007,
	title = {On a well-balanced high-order finite volume scheme for shallow water equations with topography and dry areas},
	volume = {227},
	pages = {574--601},
	number = {1},
	journaltitle = {J. Comput. Phys.},
	author = {Gallardo, Jos\{{\textbackslash}textbackslash\}’e-M. and Parés, Carlos and Castro Díaz, Manuel J.},
	date = {2007},
}

@article{castro_diaz_well-balanced_2007,
	title = {On well-balanced finite volume methods for nonconservative nonhomogeneous hyperbolic systems},
	volume = {29},
	pages = {1093--1126},
	number = {3},
	journaltitle = {{SIAM} J. Sci. Comput.},
	author = {Castro Díaz, Manuel J. and Chacón, T. and Fernández-Nieto, E. D. and Parés, Carlos},
	date = {2007},
}

@article{castro_diaz_well-balanced_2007-1,
	title = {Well-balanced numerical schemes based on a generalized hydrostatic reconstruction technique},
	volume = {17},
	pages = {2055--2113},
	number = {12},
	journaltitle = {Math. Models Methods Appl. Sci.},
	author = {Castro Díaz, Manuel J. and Pardo, A. and Parés, Carlos},
	date = {2007},
}

@article{castro_diaz_parallel_2006,
	title = {A parallel 2d finite volume scheme for solving systems of balance laws with nonconservative products: application to shallow flows},
	volume = {195},
	pages = {2788--2815},
	number = {19},
	journaltitle = {Comput. Methods Appl. Mech. Engrg.},
	author = {Castro Díaz, Manuel J. and García-Rodríguez, J.-A. and González-Vida, J.-M. and Parés, Carlos},
	date = {2006},
}

@article{castro_diaz_high_2006,
	title = {High order finite volume schemes based on reconstruction of states for solving hyperbolic systems with nonconservative products. Applications to shallow-water systems},
	volume = {75},
	pages = {1103--1134},
	number = {255},
	journaltitle = {Math. Comp.},
	author = {Castro Díaz, Manuel J. and Gallardo, José-M. and Parés, Carlos},
	date = {2006},
}

@article{pares_numerical_2006,
	title = {Numerical methods for nonconservative hyperbolic systems: a theoretical framework},
	volume = {44},
	pages = {300--321},
	number = {1},
	journaltitle = {{SIAM} J. Numer. Anal.},
	author = {Parés, Carlos},
	date = {2006},
}

@article{castro_diaz_numerical_2006,
	title = {Numerical treatment of wet/dry fronts in shallow flows with a modified Roe scheme},
	volume = {16},
	pages = {897--931},
	number = {6},
	journaltitle = {Math. Models Methods Appl. Sci.},
	author = {Castro Díaz, Manuel J. and González Vida, José M. and Parés, Carlos},
	date = {2006},
}

@inproceedings{gallardo_well-balanced_2006,
	title = {On a well-balanced high-order finite volume scheme for the shallow water equations with bottom topography and dry areas},
	pages = {259--270},
	booktitle = {Hyperbolic problems: Theory, numerics and applications},
	publisher = {Springer},
	author = {Gallardo, Jos\{{\textbackslash}textbackslash\}’e-M. and Castro Díaz, Manuel J. and Parés, Carlos and González-Vida, J.-M.},
	editor = {Sylvie Benzoni-Gavage, Denis Serre},
	date = {2006},
}

@article{gallardo_generalized_2005,
	title = {A generalized duality method for solving variational inequalities. Applications to some nonlinear Dirichlet problems},
	volume = {100},
	pages = {259--291},
	number = {2},
	journaltitle = {Numer. Math.},
	author = {Gallardo, José-M. and Castro Díaz, Manuel J. and Parés, Carlos},
	date = {2005},
}

@article{pares_mathematical_2005,
	title = {Mathematical models for the simulation of environmental flows: from the Strait of Gibraltar to the Aznalcollar disaster},
	journaltitle = {{ERCIM} News},
	author = {Parés, Carlos and Macías, Jorge and Castro Díaz, Manuel J.},
	date = {2005},
}

@article{castro_diaz_numerical_2005,
	title = {The numerical treatment of wet/dry fronts in shallow flows: application to one-layer and two-layer systems},
	volume = {42},
	pages = {419--439},
	number = {3},
	journaltitle = {Math. Comput. Modelling},
	author = {Castro Díaz, Manuel J. and Ferreiro, A.-M. and García-Rodríguez, J.-A. and González-Vida, J.-M. and Macías, Jorge and Parés, Carlos and Vázquez-Cendón, M. Elena},
	date = {2005},
}

@article{castro_diaz_two-layer_2004,
	title = {A two-layer finite volume model for flows through channels with irregular geometry: Computation of maximal exchange solutions Application to the Strait of Gibraltar},
	volume = {9},
	pages = {241--249},
	number = {2},
	journaltitle = {Communications in Nonlinear Science and Numerical Simulation},
	author = {Castro Díaz, Manuel J. and Garcia-Rodriguez, J.-A. and Macías, Jorge and Parés, Carlos and Vázquez-Cendón, E.},
	date = {2004},
}

@article{castro_diaz_two-layer_2004-1,
	title = {A two-layer finite volume model for flows through channels with irregular geometry: computation of maximal exchange solutions. Application to the Strait of Gibraltar},
	journaltitle = {Commun. Nonlinear Sci. Numer. Simul.},
	author = {Castro Díaz, Manuel J. and Macías, Jorge and Parés, Carlos and García-Rodríguez, Jose A. and Vázquez-Cendón, Elena},
	date = {2004},
}

@incollection{macias_numerical_2004,
	title = {Numerical simulation in Oceanography. Application to the Alboran Sea and the Strait of Gibraltar},
	author = {Macías, Jorge and Parés, Carlos and Castro Díaz, Manuel J.},
	date = {2004},
}

@article{castro_diaz_numerical_2004,
	title = {Numerical simulation of two-layer shallow water flows through channels with irregular geometry},
	journaltitle = {J. Comput. Phys.},
	author = {Castro Díaz, Manuel J. and García-Rodríguez, José-A. and González-Vida, J.-M. and Macías, Jorge and Parés, Carlos and Vázquez-Cendón, M. Elena},
	date = {2004},
}

@article{pares_well-balance_2004,
	title = {On the well-balance property of Roe's method for nonconservative hyperbolic systems. Applications to shallow-water systems},
	journaltitle = {Mathematical Modelling and Numerical Analysis},
	author = {Parés, Carlos and Castro Díaz, Manuel J.},
	date = {2004},
}

@article{pares_well-balance_2004-1,
	title = {On the well-balance property of Roe's method for nonconservative hyperbolic systems. Applications to shallow-water systems},
	journaltitle = {M2AN Math. Model. Numer. Anal.},
	author = {Parés, Carlos and Castro Díaz, Manuel J.},
	date = {2004},
}

@article{munoz_ruiz_one-dimensional_2003,
	title = {On an one-dimensional bi-layer shallow-water problem},
	journaltitle = {Nonlinear Anal.},
	author = {Muñoz Ruiz, María Luz and Castro Díaz, Manuel J. and Parés, Carlos},
	date = {2003},
}

@article{gonzalez_vida_lagrangian_2002,
	title = {A Lagrangian finite element algorithm for the numerical solution of one-dimensional shallow water equations},
	journaltitle = {Rev. Internac. Métod. Numér. Cálc. Diseñ. Ingr.},
	author = {González Vida, José M. and Parés, Carlos},
	date = {2002},
}

@article{castro_diaz_numerical_2002,
	title = {Numerical simulation of internal tides in the Strait of Gibraltar},
	journaltitle = {{RACSAM} Rev. R. Acad. Cienc. Exactas F{\textbackslash}'ı s. Nat. Ser. A Mat.},
	author = {Castro Díaz, Manuel J. and González-Vida, J.-M. and Macías, Jorge and Muñoz Ruiz, María Luz and Parés, Carlos and García-Rodríguez, J. and Vázquez-Cendón, E.},
	date = {2002},
}

@article{pares_convergence_2002,
	title = {On the convergence of the Bermúdez-Moreno algorithm with constant parameters},
	journaltitle = {Numer. Math.},
	author = {Parés, Carlos and Castro Díaz, Manuel J. and Macías, Jorge},
	date = {2002},
}

@article{castro_diaz_q-scheme_2001,
	title = {A Q-scheme for a class of systems of coupled conservation laws with source term. Application to a two-layer 1-D shallow water system},
	volume = {35},
	pages = {107--127},
	number = {1},
	journaltitle = {M2AN Math. Model. Numer. Anal.},
	author = {Castro Díaz, Manuel J. and Macías, Jorge and Parés, Carlos},
	date = {2001},
}

@article{castro_diaz_incomplete_2001,
	title = {An incomplete {LU}-based family of preconditioners for numerical resolution of a shallow water system using a duality method. Applications},
	journaltitle = {Appl. Math. Lett.},
	author = {Castro Díaz, Manuel J. and Macías, Jorge and Parés, Carlos},
	date = {2001},
}

@article{pares_duality_2001,
	title = {Duality methods with an automatic choice of parameters. Application to shallow water equations in conservative form},
	journaltitle = {Numer. Math.},
	author = {Parés, Carlos and Macías, Jorge and Castro Díaz, Manuel J.},
	date = {2001},
}

@article{macias_improvement_1999,
	title = {Improvement and generalization of a finite element shallow-water solver to multi-layer systems},
	journaltitle = {Internat. J. Numer. Methods Fluids},
	author = {Macías, Jorge and Parés, Carlos and Castro Díaz, Manuel J.},
	date = {1999},
}

@incollection{valle_numerical_1998,
	title = {Numerical resolution of a shallow-water system using a duality method. Application to the Alboran Sea},
	author = {Valle, Antonio and Parés, Carlos and Macías, Jorge and Castro Díaz, Manuel J.},
	date = {1998},
}

@incollection{castro_diaz_multilayer_1996,
	title = {A multilayer shallow water model. Application to the modelling of the Alboran Sea and the Strait of Gibraltar},
	author = {Castro Díaz, Manuel J. and Macías, Jorge and Parés, Carlos},
	date = {1996},
}

@article{conca_navier-stokes_1995,
	title = {Navier-Stokes equations with imposed pressure and velocity fluxes},
	journaltitle = {Internat. J. Numer. Methods Fluids},
	author = {Conca, C. and Parés, Carlos and Pironneau, O. and Thiriet, M.},
	date = {1995},
}

@article{pares_approximation_1994,
	title = {Approximation de la solution des équations d'un modèle de turbulence par une méthode de Lagrange-Galerkin},
	journaltitle = {Rev. Mat. Apl.},
	author = {Parés, Carlos},
	date = {1994},
}

@article{macias_numerical_1994,
	title = {Numerical modelling of water mass exchanges through the Strait of Gibraltar},
	journaltitle = {Revista {GAIA}},
	author = {Macías, Jorge and Castro Díaz, Manuel J. and Parés, Carlos},
	date = {1994},
}

@article{mj_castro_well-balanced_2020,
	title = {Well-Balanced High-Order Finite Volume Methods for Systems of Balance Laws},
	volume = {82},
	pages = {48},
	journaltitle = {J Sci Comput},
	author = {M.J. Castro, C. Parés},
	date = {2020},
}

@article{n_aissiouene_two-dimensional_2020,
	title = {A two-dimensional method for a family of dispersive shallow water models},
	volume = {6},
	pages = {187--226},
	journaltitle = {{SMAI} Journal of Computational Mathematics},
	author = {N. Aïssiouene, M.-O. Bristeau, E. Godlewski, A. Mangeney, C. Parés, J. Sainte-Marie},
	date = {2020},
}

@article{gomez_high-order_2021,
	title = {High-order well-balanced methods for systems of balance laws: a control-based approach},
	volume = {394},
	pages = {125820},
	journaltitle = {Applied Mathematics and Computation},
	author = {Gómez, I. and Castro, M.J. and Parés, C.},
	date = {2021},
}

@article{e_pimentel-garcia_efficient_2021,
	title = {On the efficient implementation of {PVM} methods and simple Riemann solvers. Application to the Roe method for large hyperbolic systems},
	volume = {388},
	pages = {125544},
	journaltitle = {Applied Mathematics and Computation},
	author = {E. Pimentel-García, C. Parés, M. J. Castro, J. Koellermeier},
	date = {2021},
}

@article{h_carrillo_lax-wendroff_2021,
	title = {Lax-Wendroff Approximate Taylor Methods with Fast and Optimized Weighted Essentially Non-oscillatory Reconstructions},
	volume = {86},
	pages = {15},
	journaltitle = {Journal of Scientific Computing},
	author = {H. Carrillo, C. Parés, D. Zorío},
	date = {2021},
}

@article{c_pares_well-balanced_2021,
	title = {Well-balanced high-order finite difference methods for systems of balance laws},
	volume = {425},
	pages = {109880},
	journaltitle = {Journal of Computational Physics},
	author = {C. Parés, C. Parés-Pulido},
	date = {2021},
}