This paper is concerned with the numerical approximations of Cauchy problems for one-dimensional nonconservative hyperbolic systems. The first goal is to introduce a general concept of well-balancing for numerical schemes solving this kind of systems. Once this concept stated, we investigate the well-balance properties of numerical schemes based on the generalized Roe linearizations introduced by [Toumi, J. Comp. Phys. 102 (1992) 360-373]. Next, this general theory is applied to obtain well-balanced schemes for solving coupled systems of conservation laws with source terms. Finally, we focus on applications to shallow water systems: the numerical schemes obtained and their properties are compared, in the case of one layer flows, with those introduced by [Bermúdez and Vázquez-Cendón, Comput. Fluids 23 (1994) 1049-1071]; in the case of two layer flows, they are compared with the numerical scheme presented by [Castro, Macías and Parés, ESAIM: M2AN 35 (2001) 107-127].
Mots clés : nonconservative hyperbolic systems, well-balanced schemes, Roe method, source terms, shallow-water systems
@article{M2AN_2004__38_5_821_0, author = {Par\'es, Carlos and Castro, Manuel}, title = {On the well-balance property of {Roe's} method for nonconservative hyperbolic systems. {Applications} to shallow-water systems}, journal = {ESAIM: Mod\'elisation math\'ematique et analyse num\'erique}, pages = {821--852}, publisher = {EDP-Sciences}, volume = {38}, number = {5}, year = {2004}, doi = {10.1051/m2an:2004041}, zbl = {1130.76325}, language = {en}, url = {http://www.numdam.org/articles/10.1051/m2an:2004041/} }
TY - JOUR AU - Parés, Carlos AU - Castro, Manuel TI - On the well-balance property of Roe's method for nonconservative hyperbolic systems. Applications to shallow-water systems JO - ESAIM: Modélisation mathématique et analyse numérique PY - 2004 SP - 821 EP - 852 VL - 38 IS - 5 PB - EDP-Sciences UR - http://www.numdam.org/articles/10.1051/m2an:2004041/ DO - 10.1051/m2an:2004041 LA - en ID - M2AN_2004__38_5_821_0 ER -
%0 Journal Article %A Parés, Carlos %A Castro, Manuel %T On the well-balance property of Roe's method for nonconservative hyperbolic systems. Applications to shallow-water systems %J ESAIM: Modélisation mathématique et analyse numérique %D 2004 %P 821-852 %V 38 %N 5 %I EDP-Sciences %U http://www.numdam.org/articles/10.1051/m2an:2004041/ %R 10.1051/m2an:2004041 %G en %F M2AN_2004__38_5_821_0
Parés, Carlos; Castro, Manuel. On the well-balance property of Roe's method for nonconservative hyperbolic systems. Applications to shallow-water systems. ESAIM: Modélisation mathématique et analyse numérique, Tome 38 (2004) no. 5, pp. 821-852. doi : 10.1051/m2an:2004041. http://www.numdam.org/articles/10.1051/m2an:2004041/
[1] On the solution to the Riemann problem for the compressible duct flow. SIAM J. Appl. Math. 64 (2004) 878-901. | Zbl
and ,[2] An introduction to finite volume methods for hyperbolic systems of conservation laws with source, in Free surface geophysical flows. Tutorial Notes. INRIA, Rocquencourt (2002).
,[3] Upwind methods for hyperbolic conservation laws with source terms. Comput. Fluids 23 (1994) 1049-1071. | Zbl
and ,[4] A -Scheme for a class of systems of coupled conservation laws with source term. Application to a two-layer 1-D shallow water system. ESAIM: M2AN 35 (2001) 107-127. | Numdam | Zbl
, and ,[5] Numerical simulation of two-layer Shallow Water flows through channels with irregular geometry. J. Comp. Phys. 195 (2004) 202-235. | Zbl
, , , , and ,[6] A family of stable numerical solvers for Shallow Water equations with source terms. Comp. Meth. Appl. Mech. Eng. 192 (2003) 203-225. | Zbl
, and ,[7] An entropy-correction free solver for non-homogeneous shallow water equations. ESAIM: M2AN 37 (2003) 755-772. | Numdam | Zbl
, and ,[8] A flux-splitting solver for shallow water equations with source terms. Int. Jour. Num. Meth. Fluids 42 (2003) 23-55. | Zbl
, and ,[9] Asymptotically balanced schemes for non-homogeneous hyperbolic systems - application to the Shallow Water equations. C.R. Acad. Sci. Paris, Ser. I 338 (2004) 85-90. | Zbl
, and ,[10] Microscopic profiles of shock waves and ambiguities in multiplications of distributions. SIAM J. Num. Anal. 26 (1989) 871-883. | Zbl
, , and ,[11] Definition and weak stability of nonconservative products. J. Math. Pures Appl. 74 (1995) 483-548. | Zbl
, and ,[12] Aproximación Numérica de Leyes de Conservación Hiperbólicas No Homogéneas. Aplicación a las Ecuaciones de Aguas Someras. Ph.D. Thesis, Universidad de Sevilla (2003).
,[13] Mathematical Model in the Applied Sciences. Cambridge (1997). | Zbl
,[14] On numerical treatment of the source terms in the shallow water equations. Comput. Fluids 29 (2000) 17-45. | Zbl
and ,[15] The Riemann problem for a class of resonant hyperbolic systems of balance laws, preprint (2003). | Numdam | MR
and ,[16] Numerical Approximation of Hyperbolic Systems of Conservation Laws. Springer-Verlag, New York (1996). | MR | Zbl
and ,[17] A well-balanced flux-vector splitting scheme designed for hyperbolic systems of conservation laws with source terms. Comp. Math. Appl. 39 (2000) 135-159. | Zbl
,[18] A well-balanced scheme using non-conservative products designed for hyperbolic system of conservation laws with source terms. Mat. Mod. Meth. Appl. Sc. 11 (2001) 339-365. | Zbl
,[19] A well balanced scheme for the numerical processing of source terms in hyperbolic equations. SIAM J. Numer. Anal. 33 (1996) 1-16. | Zbl
and ,[20] Analysis and approximation of conservation laws with source terms. SIAM J. Numer. Anal. 34 (1997) 1980-2007. | Zbl
, , and ,[21] Self-adjusting grid methods for one-dimensional hyperbolic conservation laws. J. Comp. Phys. 50 (1983) 235-269. | Zbl
and ,[22] Propagating phase boundaries; formulation of the problem and existence via Glimm scheme. Arch. Rat. Mech. Anal. 123 (1993) 153-197. | Zbl
,[23] Numerical Methods for Conservation Laws. Birkhäuser (1990). | MR | Zbl
,[24] Balancing source terms and flux gradients in high-resolution Godunov methods: the quasi-steady wave-propagation algorithm. J. Comp. Phys. 146 (1998) 346-365. | Zbl
,[25] Finite Volume Methods for Hyperbolic Problems. Cambridge University Press (2002). | MR | Zbl
,[26] A kinetic scheme for the Saint-Venant system with a source term. Calcolo 38 (2001) 201-231. | Zbl
and ,[27] Convergence of the upwind interface source method for hyperbolic conservation laws, in Proc. of Hyp 2002, Thou and Tadmor Eds., Springer (2003). | MR | Zbl
and ,[28] A nonconservative hyperbolic system modeling spray dynamics. I. Solution of the Riemann problem. Math. Mod. Meth. Appl. Sci. 5 (1995) 297-333. | Zbl
and ,[29] Approximate Riemann solvers, parameter vectors and difference schemes. J. Comp. Phys. 43 (1981) 357-371. | Zbl
,[30] Upwinding difference schemes for hyperbolic conservation laws with source terms, in Proc. of the Conference on Hyperbolic Problems, Carasso, Raviart and Serre Eds., Springer (1986) 41-51. | Zbl
,[31] Water Waves. Interscience, New York (1957). | Zbl
,[32] Riemann Solvers and Numerical Methods for Fluid Dynamics. A Practical Introduction. Springer-Verlag (1997). | MR | Zbl
,[33] Shock-Capturing Methods for Free-Surface Shallow Flows. Wiley (2001). | Zbl
,[34] Model hyperbolic systems with source terms: exact and numerical solutions, in Proc. of Godunov methods: Theory and Applications (2000). | MR | Zbl
and ,[35] A weak formulation of Roe's approximate Riemann Solver. J. Comp. Phys. 102 (1992) 360-373. | Zbl
,[36] Estudio de Esquemas Descentrados para su Aplicación a las Leyes de Conservación Hiperbólicas con Términos Fuente. Ph.D. Thesis, Universidad de Santiago de Compostela (1994).
,[37] 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. | Zbl
,[38] The space BV and quasilinear equations. Math. USSR Sbornik 73 (1967) 225-267. | Zbl
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