Hostname: page-component-78c5997874-xbtfd Total loading time: 0 Render date: 2024-11-10T16:51:01.895Z Has data issue: false hasContentIssue false
  • English
  • Français

An entropy-correction free solver for non-homogeneous shallow water equations

Published online by Cambridge University Press:  15 November 2003

Tomás Chacón Rebollo ,
Antonio Domínguez Delgado  and
Enrique D. Fernández Nieto
Tomás Chacón Rebollo
Affiliation:
Departamento de Ecuaciones Diferenciales y Análisis Numérico, Universidad de Sevilla, C/ Tarfia, s/n, 41080 Sevilla, Spain. chacon@numer.us.es.
Antonio Domínguez Delgado
Affiliation:
Departamento de Matemática Aplicada I, Universidad de Sevilla, E.T.S. Arquitectura Avda, Reina Mercedes s/n, 41012 Sevilla, Spain. domdel@us.es.
Enrique D. Fernández Nieto
Affiliation:
Departamento de Matemática Aplicada I, Universidad de Sevilla, E.T.S. Arquitectura. Avda, Reina Mercedes, s/n, 41012 Sevilla, Spain. edofer@us.es.
Get access

Abstract

In this work we introduce an accurate solver for theShallow Water Equations with source terms. This scheme does not need any kind of entropy correction to avoid instabilities near critical points. The scheme also solves the non-homogeneous case, in such a way that all equilibria are computed at least with second order accuracy. We perform several tests for relevant flows showing the performance of our scheme.

Keywords

Finite Volume Methodupwindingshallow waterHarten Regularizationsource termsentropy-correction
Type
Research Article
Information
ESAIM: Mathematical Modelling and Numerical Analysis , Volume 37 , Issue 5 , September 2003 , pp. 755 - 772
Copyright
© EDP Sciences, SMAI, 2003

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Bermudez, A., Dervieux, A., Desideri, J.A. and Cendón, M.E.V., Upwind schemes for the two-dimensional shallow water equations with variable depth using unstructured meshes. Comput. Methods Appl. Mech. Engrg. 155 (1998) 49-72. CrossRef
A. Bermúdez and M.E.V. Cendón, Upwind Methods for Hyperbolic Conservation Laws with Source Terms. Comput. & Fluids 23 (1994) 1049-1071.
F. Bouchut, An introduction to finite volume methods for hyperbolic systems of conservation laws with source. Actas Ecole CEA-EDF-INRIA, Free surface geophysical flows, 7-10 Octobre, INRIA Rocquencourt, France (2002).
Dubois, F. and Mehlman, G., A non-parameterized entropy correction for Roe's approximate Riemann solver. Numer. Math. 73 (1996) 169-208. CrossRef
P. Brufau, Simulación bidimensional de flujos hidrodinámicos transitorios en gemotrías irregulares. Ph.D. thesis, Universidad de Zaragoza (2000).
Rebollo, T.C., Nieto, E.D.F. and Mármol, M.G., A flux-splitting solver for shallow watter equations with source terms. Int. J. Num. Methods Fluids 42 (2003) 23-55. CrossRef
Rebollo, T.C., Delgado, A.D. and Nieto, E.D.F., A family of stable numerical solvers for Shallow Water equations with source terms. Comput. Methods Appl. Mech. Engrg. 192 (2003) 203-225. CrossRef
T. Gallouët, J.-M. Hérard and N. Seguin, Some approximate Godunov schemes to compute shallow-water equations with topography. Comput. & Fluids 32 (2003) 479-513.
E. Godlewski and P.A. Raviart, Hyperbolic systems of conservation laws. Math. Appl. (1991).
E. Godlewski and P.A. Raviart, Numerical Approximation of Hyperbolic Systems of Conservation Laws. Springer-Verlag (1996).
Harten, A., Lax, P. and Van Leer, A., On upstream differencing and Godunov-type scheme for hyperbolic conservation laws. SIAM Rev. 25 (1983) 35. CrossRef
Jin, S., A steady-state capturing method for hyperbolic systems with geometrical source terms. ESAIM: M2AN 35 (2001) 631-645. CrossRef
Kurganov, A. and Levy, D., Central-upwind schemes for the saint-venant system. ESAIM: M2AN 36 (2002) 397-425. CrossRef
Kurganov, A. and Tadmor, E., New High-Resolution Central Schemes for Nonlinear Conservations Laws and Convection-Diffusion Equations. J. Comput. Phys. 160 (2000) 214-282.
Le Veque, R.J. and Yee, H.C., A study of numerical methods for hyperbolic conservation laws with stiff source terms. J. Comput. Phys. 86 (1990) 187-210. CrossRef
Le Veque, R.J., Balancing Source Terms and Flux Gradients in High-Resolution Godunov Methods: The Quasi-Steady Wave-Propagation Algorithm. J. Comp. Phys. 146 (1998) 346-365. CrossRef
Perthame, B. and Simeoni, C., A kinetic scheme for the Saint-Venant system with a source term. Calcolo 38 (2001) 201-231. CrossRef
P.L. Roe, Upwind differencing schemes for hyperbolic conservation laws with source terms, in Nonlinear Hyperbolic Problems, C. Carraso, P.A. Raviart and D. Serre, Eds., Springer-Verlag, Lecture Notes in Math. 1270 (1986) 41-51.
E.F. Toro, Riemann Solvers and Numerical Methods for Fluid Dynamics. Springer (1997).
M.E.V. Cendon, Estudio de esquemas descentrados para su aplicacion a las leyes de conservación hiperbólicas con términos fuente. Ph.D. thesis, Universidad de Santiago de Compostela (1994).
Cendón, M.E.V., Improved Treatment of Source Terms in Upwind Schemes for the Shallow Water Equations in Channels with Irregular Geometry. J. Comp. Phys. 148 (1999) 497-526. CrossRef
High-order, J.P. Vila schemes and entropy condition for nonlinear hyperbolic systems of conservations laws. Math. Comp. 50 (1988) 53-73.
Zhou, J.G., Causon, D.M., Mingham, C.G. and Ingram, D.M., The Surface Gradient Method for the Treatment of Source Terms in the Sallow-Water Equations. J. Comput. Phys. 168 (2001) 1-25. CrossRef