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On the well-balance property of Roe’s method for nonconservative hyperbolic systems. Applications to shallow-water systems. (English) Zbl 1130.76325
Summary: 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 I. Toumi [J. Comput. Phys. 102, No. 2, 360–373 (1992; Zbl 0783.65068)]. 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 [A. Bermúdez and M. E. Vázquez-Cendón, Comput. Fluids 23, No. 8, 1049–1071 (1994; Zbl 0816.76052)]; in the case of two layer flows, they are compared with the numerical scheme presented by M. Castro and C. Parés [ M2AN, Math. Model. Numer. Anal. 35, No. 1, 107–127 (2001; Zbl 1094.76046)].

##### MSC:
 76B55 Internal waves for incompressible inviscid fluids 76M20 Finite difference methods applied to problems in fluid mechanics 65M06 Finite difference methods for initial value and initial-boundary value problems involving PDEs 35L45 Initial value problems for first-order hyperbolic systems 35L60 First-order nonlinear hyperbolic equations
HE-E1GODF
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##### References:
 [1] N. Andronov and G. Warnecke , On the solution to the Riemann problem for the compressible duct flow . SIAM J. Appl. Math. 64 ( 2004 ) 878 - 901 . Zbl 1065.35191 · Zbl 1065.35191 [2] F. Bouchut , 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] A. Bermúdez and M.E. Vázquez , Upwind methods for hyperbolic conservation laws with source terms . Comput. Fluids 23 ( 1994 ) 1049 - 1071 . Zbl 0816.76052 · Zbl 0816.76052 [4] M.J. Castro , J. Macías and C. Parés , 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. ESAIM: M2AN 35 ( 2001 ) 107 - 127 . Numdam | Zbl 1094.76046 · Zbl 1094.76046 [5] M.J. Castro , J.A. García-Rodríguez , J.M. González-Vida , J. Macías , C. Parés and M.E. Vázquez-Cendón , Numerical simulation of two-layer Shallow Water flows through channels with irregular geometry . J. Comp. Phys. 195 ( 2004 ) 202 - 235 . Zbl 1087.76077 · Zbl 1087.76077 [6] T. Chacón , A. Domínguez and E.D. Fernández , A family of stable numerical solvers for Shallow Water equations with source terms . Comp. Meth. Appl. Mech. Eng. 192 ( 2003 ) 203 - 225 . Zbl 1083.76557 · Zbl 1083.76557 [7] T. Chacón , A. Domínguez and E.D. Fernández , An entropy-correction free solver for non-homogeneous shallow water equations . ESAIM: M2AN 37 ( 2003 ) 755 - 772 . Numdam | Zbl 1033.76032 · Zbl 1033.76032 [8] T. Chacón , E.D. Fernández and M. Gómez Mármol , A flux-splitting solver for shallow water equations with source terms . Int. Jour. Num. Meth. Fluids 42 ( 2003 ) 23 - 55 . Zbl 1033.76033 · Zbl 1033.76033 [9] T. Chacón , A. Domínguez and E.D. Fernández , 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 1038.65073 · Zbl 1038.65073 [10] J.F. Colombeau , A.Y. Le Roux , A. Noussair and B. Perrot , Microscopic profiles of shock waves and ambiguities in multiplications of distributions . SIAM J. Num. Anal. 26 ( 1989 ) 871 - 883 . Zbl 0674.76049 · Zbl 0674.76049 [11] G. Dal Masso , P.G. LeFloch and F. Murat , Definition and weak stability of nonconservative products . J. Math. Pures Appl. 74 ( 1995 ) 483 - 548 . Zbl 0853.35068 · Zbl 0853.35068 [12] E.D. Fernández Nieto , Aproximación Numérica de Leyes de Conservación Hiperbólicas No Homogéneas . Aplicación a las Ecuaciones de Aguas Someras. Ph.D. Thesis, Universidad de Sevilla ( 2003 ). [13] A.C. Fowler , Mathematical Model in the Applied Sciences . Cambridge ( 1997 ). Zbl 0997.00535 · Zbl 0997.00535 [14] P. García-Navarro and M.E. Vázquez-Cendón , On numerical treatment of the source terms in the shallow water equations . Comput. Fluids 29 ( 2000 ) 17 - 45 . Zbl 0986.76051 · Zbl 0986.76051 [15] P. Goatin and P.G. LeFloch , The Riemann problem for a class of resonant hyperbolic systems of balance laws , preprint ( 2003 ). MR 2097035 · Zbl 1086.35069 [16] E. Godlewski and P.A. Raviart , Numerical Approximation of Hyperbolic Systems of Conservation Laws . Springer-Verlag, New York ( 1996 ). MR 1410987 | Zbl 0860.65075 · Zbl 0860.65075 [17] L. Gosse , 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 0963.65090 · Zbl 0963.65090 [18] L. Gosse , 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 1018.65108 · Zbl 1018.65108 [19] J.M. Greenberg and A.Y. LeRoux , A well balanced scheme for the numerical processing of source terms in hyperbolic equations . SIAM J. Numer. Anal. 33 ( 1996 ) 1 - 16 . Zbl 0876.65064 · Zbl 0876.65064 [20] J.M. Greenberg , A.Y. LeRoux , R. Baraille and A. Noussair , Analysis and approximation of conservation laws with source terms . SIAM J. Numer. Anal. 34 ( 1997 ) 1980 - 2007 . Zbl 0888.65100 · Zbl 0888.65100 [21] A. Harten and J.M. Hyman , Self-adjusting grid methods for one-dimensional hyperbolic conservation laws . J. Comp. Phys. 50 ( 1983 ) 235 - 269 . Zbl 0565.65049 · Zbl 0565.65049 [22] P.G. LeFloch , Propagating phase boundaries; formulation of the problem and existence via Glimm scheme . Arch. Rat. Mech. Anal. 123 ( 1993 ) 153 - 197 . Zbl 0784.73010 · Zbl 0784.73010 [23] R. LeVeque , Numerical Methods for Conservation Laws . Birkhäuser ( 1990 ). MR 1077828 | Zbl 0723.65067 · Zbl 0723.65067 [24] R. LeVeque , 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 0931.76059 · Zbl 0931.76059 [25] R. LeVeque , Finite Volume Methods for Hyperbolic Problems . Cambridge University Press ( 2002 ). MR 1925043 | Zbl 1010.65040 · Zbl 1010.65040 [26] B. Perthame and C. Simeoni , A kinetic scheme for the Saint-Venant system with a source term . Calcolo 38 ( 2001 ) 201 - 231 . Zbl 1008.65066 · Zbl 1008.65066 [27] B. Perthame and C. Simeoni , Convergence of the upwind interface source method for hyperbolic conservation laws , in Proc. of Hyp 2002, Thou and Tadmor Eds., Springer ( 2003 ). MR 2053160 | Zbl 1064.65098 · Zbl 1064.65098 [28] P.A. Raviart and L. Sainsaulieu , A nonconservative hyperbolic system modeling spray dynamics . I. Solution of the Riemann problem. Math. Mod. Meth. Appl. Sci. 5 ( 1995 ) 297 - 333 . Zbl 0837.76089 · Zbl 0837.76089 [29] P.L. Roe , Approximate Riemann solvers, parameter vectors and difference schemes . J. Comp. Phys. 43 ( 1981 ) 357 - 371 . Zbl 0474.65066 · Zbl 0474.65066 [30] P.L. Roe , 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 0626.65086 · Zbl 0626.65086 [31] J.J. Stoker , Water Waves . Interscience, New York ( 1957 ). Zbl 0078.40805 · Zbl 0078.40805 [32] E.F. Toro , Riemann Solvers and Numerical Methods for Fluid Dynamics . A Practical Introduction. Springer-Verlag ( 1997 ). MR 1474503 | Zbl 0801.76062 · Zbl 0801.76062 [33] E.F. Toro , Shock-Capturing Methods for Free-Surface Shallow Flows . Wiley ( 2001 ). Zbl 0996.76003 · Zbl 0996.76003 [34] E.F. Toro and M.E. Vázquez-Cendón , Model hyperbolic systems with source terms: exact and numerical solutions , in Proc. of Godunov methods: Theory and Applications ( 2000 ). MR 1963646 | Zbl 0989.65095 · Zbl 0989.65095 [35] I. Toumi , A weak formulation of Roe’s approximate Riemann Solver . J. Comp. Phys. 102 ( 1992 ) 360 - 373 . Zbl 0783.65068 · Zbl 0783.65068 [36] M.E. Vázquez-Cendón , 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] M.E. Vázquez-Cendón , 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 0931.76055 · Zbl 0931.76055 [38] A.I. Volpert , The space BV and quasilinear equations . Math. USSR Sbornik 73 ( 1967 ) 225 - 267 . Zbl 0168.07402 · Zbl 0168.07402
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