×

zbMATH — the first resource for mathematics

The local projection \(P^ 0-P^ 1\)-discontinuous-Galerkin finite element method for scalar conservation laws. (English) Zbl 0715.65079
The local projection \(P^ 0-P^ 1\)-discontinuous Galerkin finite element method \((\Lambda \Pi P^ 0P^ 1\)-scheme) is used for solving numerically scalar conservation laws. The method is obtained modifying an explicit discontinuous Galerkin method introduced via a local projection. The resulting scheme is an extension of the Godunov scheme that verifies a local maximum principle. Numerical evidence is given.
Reviewer: P.Y.Yalamov

MSC:
65M60 Finite element, Rayleigh-Ritz and Galerkin methods for initial value and initial-boundary value problems involving PDEs
35L65 Hyperbolic conservation laws
PDF BibTeX XML Cite
Full Text: DOI EuDML
References:
[1] Y. BRENIER and S. OSHER, Approximate Riemman Solvers and Numerical Flux Functions, ICASE report n^\circ 84-63 (1984). Zbl0597.65071 · Zbl 0597.65071 · doi:10.1137/0723018
[2] G. CHAVENT and B. COCKBURN, Convergence et Stabilité des Schémas LRG, INRIA report.
[3] G. CHAVENT and G. SALZANO, A finite Element Method for the 1D Water Flooding Problem with Gravity, J. Comp. Phys., 45 (1982), pp. 307-344. Zbl0489.76106 MR666166 · Zbl 0489.76106 · doi:10.1016/0021-9991(82)90107-3
[4] B. COCKBURN, Le Schéma G-k/2 pour les Lois de Conservation Scalaires, Congrès National d’Analyse Numérique (1984), pp. 53-56.
[5] B. COCKBURN, The Quasi-Monotone schemes for Scalar Conservation Laws, IMA Preprint Séries n^\circ 263, 268 and 277. To appear in SIAM J. Numer. Anal. MR1025091
[6] A. HARTEN, On a class of high-resolution total-variation-stable finite-differene schemes, SIAM J. Numer. AnaL, 21 (1984), pp. 1-23. Zbl0547.65062 MR731210 · Zbl 0547.65062 · doi:10.1137/0721001
[7] C. JOHNSON and J. PITKARANTA, An Analysis of the Discontinuous Galerkin Method for a Scalar Hyperbolic Equation, Math, of Comp., 46 (1986), pp.1-26. Zbl0618.65105 MR815828 · Zbl 0618.65105 · doi:10.2307/2008211
[8] A. Y. LEROUX, A Numerical Conception of Entropy for Quasi-Linear Equations, Math. of Comp., 31 (1977), pp. 848-872. Zbl0378.65053 MR478651 · Zbl 0378.65053 · doi:10.2307/2006117
[9] P. LESAINT and P. A. RAVIART, On a Finite Element Method for Solving the Neutron Transport Equation, Mathematical Aspects of Finite Element in Partial Differential Equations, Academic Press, Ed. Carl de Boor, pp. 89-145. Zbl0341.65076 · Zbl 0341.65076
[10] S. OSHER, Convergence of Generalized MUSCL Schemes, SIAM J. Numer. Anal., 22 (1984), pp. 947-961. Zbl0627.35061 MR799122 · Zbl 0627.35061 · doi:10.1137/0722057
[11] S. OSHER, Riemman Solvers, the Entropy Condition and Difference Approximations, SIAM J. Numer. Anal., 21 (1984), pp. 217-235. Zbl0592.65069 MR736327 · Zbl 0592.65069 · doi:10.1137/0721016
[12] E. TADMOR, Numerical Viscosity and the Entropy Condition for Conservative Difference Schemes, Math. Comp., 43 (1984), pp. 369-381. Zbl0587.65058 MR758189 · Zbl 0587.65058 · doi:10.2307/2008282
[13] B. VAN LEER, Towards the Ultimate Conservative Scheme, II Monotonicity and Conservation Combined in a Second Order Scheme, J. Comput. Phys., 14 (1974), pp. 361-370. Zbl0276.65055 · Zbl 0276.65055 · doi:10.1016/0021-9991(74)90019-9
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.