×

zbMATH — the first resource for mathematics

Godunov method for nonconservative hyperbolic systems. (English) Zbl 1124.65077
The paper is concerned with the numerical approximation of Cauchy problems for one-dimensional nonconservative hyperbolic systems. For the definition of the weak solutions of the system, the theory: an interpretation of the nonconservative products as Borel measures is given, based on the choice of a family of paths drawn in the phase space, is used. Even if the family of paths can be chosen arbitrarily, it is natural to require this family to satisfy some hypotheses concerning the relation of the paths with the integral curves of the characteristic fields.
The investigation of the implication of three basic hypotheses of this nature is the main purpose of this work. It is also shown that when the family of paths satisfies these hypotheses, Godunov methods can be written in a natural form that generalizes their classical expression for systems of conservation laws. The well-balance properties of these methods are also studied. The consistency of the numerical scheme with the definition of weak solutions is finally proved.

MSC:
65M06 Finite difference methods for initial value and initial-boundary value problems involving PDEs
35L60 First-order nonlinear hyperbolic equations
65M12 Stability and convergence of numerical methods for initial value and initial-boundary value problems involving PDEs
Software:
HLLE
PDF BibTeX Cite
Full Text: DOI Numdam EuDML
References:
[1] F. Bouchut , Nonlinear Stability of Finite Volume Methods for Hyperbolic Conservation Laws and Well-Balanced Schemes for Sources . Birkhäuser ( 2004 ). MR 2128209 | Zbl 1086.65091 · Zbl 1086.65091
[2] A. Bressan , H.K. Jenssen and P. Baiti , An instability of the Godunov Scheme . arXiv:math.AP/0502125 v 2 ( 2005 ). arXiv | MR 2254446 | Zbl 1122.35074 · Zbl 1122.35074
[3] 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. Math. Mod. Num. Anal. 35 ( 2001 ) 107 - 127 . Numdam | Zbl 1094.76046 · Zbl 1094.76046
[4] F. Coquel , D. Diehl , C. Merkle and C. Rohde , Sharp and diffuse interface methods for phase transition problems in liquid-vapour flows , in Numerical Methods for Hyperbolic and Kinetic Problems, IRMA Lectures in Mathematics and Theoretical Physics, Proceedings of CEMRACS 2003. MR 2186374 · Zbl 1210.80016
[5] G. Dal Maso , 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
[6] F. De Vuyst , Schémas non-conservatifs et schémas cinétiques pour la simulation numérique d’écoulements hypersoniques non visqueux en déséquilibre thermochimique . Ph.D. thesis, University of Paris VI, France ( 1994 ).
[7] E. Godlewski and P.A. Raviart , Numerical Approximation of Hyperbolic Systems of Conservation Laws . Springer ( 1996 ). MR 1410987 | Zbl 0860.65075 · Zbl 0860.65075
[8] L. Gosse , A well-balanced flux-vector splitting scheme designed for hyperbolic systems of conservation laws with source terms . Comput. Math. Appl. 39 ( 2000 ) 135 - 159 . Zbl 0963.65090 · Zbl 0963.65090
[9] L. Gosse , A well-balanced scheme using non-conservative products designed for hyperbolic systems of conservation laws with source terms . Math. Models Methods Appl. Sci. 11 ( 2001 ) 339 - 365 . Zbl 1018.65108 · Zbl 1018.65108
[10] 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
[11] 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
[12] A. Harten , P.D. Lax and B. Van Leer , On upstream differencing and Godunov-type schemes for hyperbolic conservation laws . SIAM Rev. 25 ( 1983 ) 35 - 61 . Zbl 0565.65051 · Zbl 0565.65051
[13] T. Hou and P.G. LeFloch , Why nonconservative schemes converge to wrong solutions: error analysis . Math. Comp. 62 ( 1994 ) 497 - 530 . Zbl 0809.65102 · Zbl 0809.65102
[14] E. Isaacson and B. Temple , Convergence of the \(2\times 2\) Godunov method for a general resonant nonlinear balance law . SIAM J. Appl. Math. 55 ( 1995 ) 625 - 640 . Zbl 0838.35075 · Zbl 0838.35075
[15] P.D. Lax and B. Wendroff , Systems of conservation laws . Comm. Pure Appl. Math. 13 ( 1960 ) 217 - 237 . Zbl 0152.44802 · Zbl 0152.44802
[16] P.G. LeFloch , Shock waves for nonlinear hyperbolic systems in nonconservative form . Institute Math. Appl., Minneapolis, Preprint 593 ( 1989 ).
[17] P.G. LeFloch , Graph solutions of nonlinear hyperbolic systems . J. Hyper. Differ. Equa. 2 ( 2004 ) 643 - 689 . Zbl 1071.35078 · Zbl 1071.35078
[18] P.G. LeFloch and A.E. Tzavaras , Representation of weak limits and definition of nonconservative products . SIAM J. Math. Anal. 30 ( 1999 ) 1309 - 1342 . Zbl 0939.35115 · Zbl 0939.35115
[19] R.J. 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
[20] C. Parés , Numerical methods for nonconservative hyperbolic systems: a theoretical framework . SIAM J. Numer. Anal. 44 ( 2006 ) 300 - 321 . Zbl 1130.65089 · Zbl 1130.65089
[21] C. Parés and M. Castro , On the well-balance property of Roe’s method for nonconservative hyperbolic systems . Applications to shallow water systems. Math. Mod. Num. Anal. 38 ( 2004 ) 821 - 852 . Numdam | Zbl 1130.76325 · Zbl 1130.76325
[22] A.I. Volpert , The space BV and quasilinear equations . Math. USSR Sbornik 73 ( 1967 ) 225 - 267 . Zbl 0168.07402 · Zbl 0168.07402
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.