Finite volume simulation of the geostrophic adjustment in a rotating shallow-water system.
Díaz, Manuel J., C.; López, J.; and Parés, C.
*SIAM J. Sci. Comput.*, 31(1): 444–477. 2008.

Paper
link
bibtex
abstract
@Article{CastroDiaz2008,
author = {Castro D{\'i}az, Manuel J. and L{\'o}pez, Juan-Antonio and Par{\'e}s, Carlos},
journal = {SIAM J. Sci. Comput.},
title = {{F}inite volume simulation of the geostrophic adjustment in a rotating shallow-water system},
year = {2008},
number = {1},
pages = {444–477},
volume = {31},
abstract = {The goal of this article is to simulate rotating flows of shallow layers of fluid by means of finite volume numerical schemes. More precisely, we focus on the simulation of the geostrophic adjustment phenomenon. As spatial discretization, a first order Roe-type method and some higher-order extensions are developed. The time discretization is designed in order to provide suitable approximations of inertial oscillations, taking into account the Hamiltonian structure of the system for these solutions. The numerical dispersion laws and the wave amplifications of the schemes are studied, and their well-balanced properties are analyzed. Finally, some numerical experiments for one-dimensional (1d) and two-dimensional (2d) problems are shown.},
url = {http://epubs.siam.org/sisc/resource/1/sjoce3/v31/i1/p444_s1},
}

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

High-order finite volume schemes for shallow water equations with topography and dry areas.
Gallardo, J.; Díaz, Manuel J., C.; and Parés, C.
2008.

link
bibtex
@Conference{Gallardo2008,
author = {Gallardo, Jos{\'e}-M. and Castro D{\'i}az, Manuel J. and Par{\'e}s, Carlos},
booktitle = {Hyperbolic problems: Theory, numerics and applications},
title = {{H}igh-order finite volume schemes for shallow water equations with topography and dry areas},
year = {2008},
editor = {Eitan, Tadmor and Jian-Guo, Liu and Athanasios, Tzavaras},
pages = {585-594},
publisher = {American Mathematical Society},
}

Numerical solution of a 1-D elastohydrodynamic problem in magnetic storage devices.
Arregui, I.; Cendán, J. J.; Parés, C.; and Vázquez, C.
*M2AN Math. Model. Numer. Anal.*, 42(4): 645–665. 2008.

Paper
link
bibtex
abstract
@Article{Arregui2008,
author = {Arregui, I{\~n}igo and Cend{\'a}n, Jos{\'e} Jes{\'u}s and Par{\'e}s, Carlos and V{\'a}zquez, Carlos},
journal = {M2AN Math. Model. Numer. Anal.},
title = {{N}umerical solution of a 1-{D} elastohydrodynamic problem in magnetic storage devices},
year = {2008},
number = {4},
pages = {645–665},
volume = {42},
abstract = {In this work we present new numerical methods to simulate the mechanics of head-tape magnetic storage devices. The elastohydrodynamic problem is formulated in terms of a coupled system which is governed by a nonlinear compressible Reynolds equation for the air pressure over the head, and a rod model for the tape displacement. A fixed point algorithm between the solutions of the elastic and hydrodynamic problems is proposed. For the nonlinear Reynolds equation, a characteristics method and a duality algorithm are developed to cope with the convection dominating and nonlinear diffusion features, respectively. Furthermore, in the duality method the convergence and optimal choice of the parameters are analyzed. At each fixed point iteration, in the elastic model a complementarity formulation is required and appropriate numerical techniques are used. For the spatial discretization different finite element spaces are chosen. Finally, numerical test examples illustrate the theoretical results, as well as the good performance in the simulation of real devices},
url = {http://www.esaim-m2an.org/action/displayAbstract?fromPage=online{\&amp;}aid=8194733},
}

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

.
Díaz, Manuel J., C.; González-Vida, J.; Macías, Jorge; and Parés, C.
Simulation of tidal currents in the Strait of Gibraltar. Mofdi, E.; Mohamed, S.; and Naje, Y., editor(s). Universidad Rey Juan Carlos, 2008.

link
bibtex
@InBook{CastroDiaz2008b,
author = {Castro D{\'i}az, Manuel J. and Gonz{\'a}lez-Vida, J.-M. and Mac{\'i}as, Jorge and Par{\'e}s, Carlos},
editor = {Mofdi, El-Amrani and Mohamed, Sead and Naje, Yebari},
publisher = {Universidad Rey Juan Carlos},
title = {{S}imulation of tidal currents in the {S}trait of {G}ibraltar},
year = {2008},
}

Simulation of tidal currents in the Strait of Gibraltar using two-dimensional two-layer shallow-water models.
Díaz, Manuel J., C.; García-Rodríguez, J.-A.; González-Vida, J.; Macías, Jorge; and Parés, C.
*Bol. Soc. Esp. Mat. Apl.*. 2008.

link
bibtex
@Article{CastroDiaz2008c,
author = {Castro D{\'i}az, Manuel J. and Garc{\'i}a-Rodr{\'i}guez, J.-A. and Gonz{\'a}lez-Vida, J.-M. and Mac{\'i}as, Jorge and Par{\'e}s, Carlos},
journal = {Bol. Soc. Esp. Mat. Apl.},
title = {{S}imulation of tidal currents in the {S}trait of {G}ibraltar using two-dimensional two-layer shallow-water models},
year = {2008},
}

Solving shallow-water systems in 2D domains using finite volume methods and multimedia SSE instructions.
Díaz, Manuel J., C.; García-Rodríguez, J.-A.; González-Vida, J.; and Parés, C.
*J. Comput. Appl. Math.*, 221(1): 16–32. 2008.

Paper
link
bibtex
abstract
@Article{CastroDiaz2008d,
author = {Castro D{\'i}az, Manuel J. and Garc{\'i}a-Rodr{\'i}guez, J.-A. and Gonz{\'a}lez-Vida, J.-M. and Par{\'e}s, Carlos},
journal = {J. Comput. Appl. Math.},
title = {{S}olving shallow-water systems in 2{D} domains using finite volume methods and multimedia {SSE} instructions},
year = {2008},
number = {1},
pages = {16–32},
volume = {221},
abstract = {The goal of this paper is to construct efficient parallel solvers for 2D hyperbolic systems of conservation laws with source terms and nonconservative products. The method of lines is applied: at every intercell a projected Riemann problem along the normal direction is considered which is discretized by means of well-balanced Roe methods. The resulting 2D numerical scheme is explicit and first-order accurate. In [M.J. Castro, J.A. Garc{\'i}a, J.M. Gonz{\'a}lez, C. Pares, A parallel 2D Finite Volume scheme for solving systems of balance laws with nonconservative products: Application to shallow flows, Comput. Methods Appl. Mech. Engrg. 196 (2006) 2788–2815] a domain decomposition method was used to parallelize the resulting numerical scheme, which was implemented in a PC cluster by means of MPI techniques.
In this paper, in order to optimize the computations, a new parallelization of SIMD type is performed at each MPI thread, by means of SSE (“Streaming SIMD Extensions”), which are present in common processors. More specifically, as the most costly part of the calculations performed at each processor consists of a huge number of small matrix and vector computations, we use the Intel© Integrated Performance Primitives small matrix library. To make easy the use of this library, which is implemented using assembler and SSE instructions, we have developed a C++ wrapper of this library in an efficient way. Some numerical tests were carried out to validate the performance of the C++ small matrix wrapper. The specific application of the scheme to one-layer Shallow-Water systems has been implemented on a PC’s cluster. The correct behavior of the one-layer model is assessed using laboratory data.},
url = {http://www.sciencedirect.com/science/article/pii/S0377042707005201},
}

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

Well-balanced finite volume schemes for 2d non-homogeneous hyperboli systems. Application to the dam break of Aznalcóllar.
Díaz, Manuel J., C.; Chacón-Rebollo, T.; Fernández Nieto, E. D.; González-Vida, J.; and Parés, C.
*Comput. Methods Appl. Mech. Engrg.*, 197(45-48): 3932–3950. 2008.

Paper
link
bibtex
abstract
@Article{CastroDiaz2008e,
author = {Castro D{\'i}az, Manuel J. and Chac{\'o}n-Rebollo, T. and Fern{\'a}ndez Nieto, E. D. and Gonz{\'a}lez-Vida, J.-M. and Par{\'e}s, Carlos},
journal = {Comput. Methods Appl. Mech. Engrg.},
title = {{W}ell-balanced finite volume schemes for 2d non-homogeneous hyperboli systems. {A}pplication to the dam break of {A}znalc{\'o}llar},
year = {2008},
number = {45-48},
pages = {3932–3950},
volume = {197},
abstract = {In this paper, we introduce a class of well-balanced finite volume schemes for 2D non-homogeneous hyperbolic systems. We extend the derivation of standard finite volume solvers for homogeneous systems to non-homogeneous ones using the method of lines. We study conservation and some well-balanced properties of the numerical scheme. We apply our solvers to shallow water equations: we prove that these exactly compute the water at rest solutions. We also perform some numerical tests by comparing with 1D solutions, simulating the formation of a hydraulic drop and a hydraulic jump, and studying a real dam break: Aznalc{\'o}llar, an ecological disaster that happened in the province of Seville, Spain in 1998},
url = {http://www.sciencedirect.com/science/article/pii/S0045782508001394},
}

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

Well-balanced high order extensions of Godunov's method for semilinear balance laws.
Díaz, Manuel J., C.; Gallardo, J.; López-García, Juan-A.; and Parés, C.
*SIAM J. Numer. Anal.*, 46(2): 1012–1039. 2008.

Paper
link
bibtex
abstract
@Article{CastroDiaz2008f,
author = {Castro D{\'i}az, Manuel J. and Gallardo, Jos{\'e}-M. and L{\'o}pez-Garc{\'i}a, Juan-A. and Par{\'e}s, Carlos},
journal = {SIAM J. Numer. Anal.},
title = {{W}ell-balanced high order extensions of {G}odunov's method for semilinear balance laws},
year = {2008},
number = {2},
pages = {1012–1039},
volume = {46},
abstract = {This paper is concerned with the development of well-balanced high order numerical schemes for systems of balance laws with a linear flux function, whose coefficients may be variable. First, well-balanced first order numerical schemes are obtained based on the use of exact solvers of Riemann problems that include both the flux and the source terms. Godunov's methods so obtained are extended to higher order schemes by using a technique of reconstruction of states. The main contribution of this paper is to introduce a reconstruction technique that preserves the well-balanced property of Godunov's methods. Some numerical experiments are presented to verify in practice the properties of the developed numerical schemes.},
url = {http://epubs.siam.org/sinum/resource/1/sjnaam/v46/i2/p1012_s1},
}

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

Why many theories of shock waves are necessary: convergence error in formally path-consistent schemes.
Díaz, Manuel J., C.; LeFloch, P.; Muñoz Ruiz, M.; and Parés, C.
*J. Comput. Phys.*, 227(17): 8107–8129. 2008.

Paper
link
bibtex
abstract
@Article{CastroDiaz2008g,
author = {Castro D{\'i}az, Manuel J. and LeFloch, Philippe-G. and Mu{\~n}oz Ruiz, Mar{\'i}a Luz and Par{\'e}s, Carlos},
journal = {J. Comput. Phys.},
title = {{W}hy many theories of shock waves are necessary: convergence error in formally path-consistent schemes},
year = {2008},
number = {17},
pages = {8107–8129},
volume = {227},
abstract = {We are interested in nonlinear hyperbolic systems in nonconservative form arising in fluid dynamics, and, for solutions containing shock waves, we investigate the convergence of finite difference schemes applied to such systems. According to Dal Maso, LeFloch, and Murat’s theory, a shock wave theory for a given nonconservative system requires prescribing a priori a family of paths in the phase space. In the present paper, we consider schemes that are formally consistent with a given family of paths, and we investigate their limiting behavior as the mesh is refined. we first generalize to systems a property established earlier by Hou and LeFloch for scalar conservation laws, and we prove that nonconservative schemes generate, at the level of the limiting hyperbolic system, an convergence error source-term which, provided the total variation of the approximations remains uniformly bounded, is a locally bounded measure. This convergence error measure is supported on the shock trajectories and, as we demonstrate here, is usually “small”. In the special case that the scheme converges in the sense of graphs – a rather strong convergence property often violated in practice – then this measure source-term vanishes. We also discuss the role of the equivalent equation associated with a difference scheme; here, the distinction between scalar equations and systems appears most clearly since, for systems, the equivalent equation of a scheme that is formally path-consistent depends upon the prescribed family of paths. The core of this paper is devoted to investigate numerically the approximation of several (simplified or full) hyperbolic models arising in fluid dynamics. This leads us to the conclusion that for systems having nonconservative products associated with linearly degenerate characteristic fields, the convergence error vanishes. For more general models, this measure is evaluated very accurately, especially by plotting the shock curves associated with each scheme under consideration; as we demonstrate, plotting the shock curves provide a convenient approach for evaluating the range of validity of a given scheme.},
url = {http://www.sciencedirect.com/science/article/pii/S0021999108002842},
}

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