zbMATH — the first resource for mathematics

Convergence of an upstream finite volume scheme for a nonlinear hyperbolic equation on a triangular mesh. (English) Zbl 0801.65089
The following nonlinear hyperbolic problem in two-dimensional space is considered (1) \(u_ t (x,t) + \text{div} (v(x,t) f(u(x,t))) = 0\), \(x \in \mathbb{R}^ 2\), \(t \in\mathbb{R}_ +\), \(\text{div} v(x,t) = 0\), \(\sup_{x,t} | v(x,t) | < \infty\). An explicit Euler scheme is used for the time discretization of (1) and a triangular mesh for the spatial discretization. Under a usual stability condition,the convergence of the solution obtained by an upstream finite volume scheme towards the unique entropy weak solution of (1) is proved.
Reviewer: K.Zlateva (Russe)

65M12 Stability and convergence of numerical methods for initial value and initial-boundary value problems involving PDEs
65M60 Finite element, Rayleigh-Ritz and Galerkin methods for initial value and initial-boundary value problems involving PDEs
65M06 Finite difference methods for initial value and initial-boundary value problems involving PDEs
35L60 First-order nonlinear hyperbolic equations
Full Text: DOI EuDML
[1] Champier, S., Gallouët, T. (1992): Convergence d’un schéma décentré amont sur un maillage triangulaire pour un problème hyperbolique linéaire. MZAN,26(7), 835-853 · Zbl 0772.65065
[2] Cockburn, B., Coquel, F., Le Floch, P., Shu, C.W., Convergence of Finite Volume Methods for Scalar Conservation Laws (submitted)
[3] Crandall, M.G., Majda, A. (1980): Monotone difference approximations for scalar conservation laws. Math. Comput.,34, (149) 1-21 · Zbl 0423.65052 · doi:10.1090/S0025-5718-1980-0551288-3
[4] DiPerna R. (1985): Measure-valued solutions to conservation laws. Arch. Rat. Mech. Anal.88, 223-270 · Zbl 0616.35055 · doi:10.1007/BF00752112
[5] Eymard, R., Gallouët T.: Convergence d’un Schéma de type Eléments Finis?Volumes Finis pour un Système Couplé Elliptique?Hyperbolique (submitted)
[6] Godunov, S. (1976): Résolution numérique des problèmes multidimensionnels de la dynamique des gaz. Editions de Moscou. Moscou
[7] Gallouët, T., Herbin, R., A Uniqueness Result for Measure Valued Solutions of a Nonlinear Hyperbolic Equations. J. Differ. Equations (to appear) · Zbl 0806.35114
[8] Harten, A. (1983): On a class of high resolution total-variation-stable finite-difference schemes. J. Comput. Phys.49, 357 · Zbl 0565.65050 · doi:10.1016/0021-9991(83)90136-5
[9] Osher, S. (1984): Riemann solvers, the entropy condition, and difference approximations, SIAM J. Numer. Anal.21 2 · Zbl 0592.65069
[10] Sanders, R. (1983): On the convergence of monotone finite difference schemes with variable spatial differencing. Math. Comput.40 (161), 91-106 · Zbl 0533.65061 · doi:10.1090/S0025-5718-1983-0679435-6
[11] Szepessy, A. (1989): An existence result for scalar conservation laws using measure valued solutions. Comm. P.D.E.14(10), 1329-1350 · Zbl 0704.35022 · doi:10.1080/03605308908820657
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.