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The Runge-Kutta local projection \(P^ 1\)-discontinuous-Galerkin finite element method for scalar conservation laws. (English) Zbl 0732.65094
The construction of a general scheme for hyperbolic conservation laws is envisaged. This construction is based on a discontinuous Galerkin finite element with a high-order accurate total variation diminishing Runge- Kutta time discretization and a local projection which enforces the global stability of the scheme.
The resulting scheme which verifies a maximum principle, is total variation bounded in the means, linearly stable for CF\(\in [0,1/3]\) and formally uniformly second order accurate in time and space. It proves numerically that the scheme does converge to the entropy solution, and that the order of convergence is equal to 2 in the norm of \(L^{\infty}(L^{\infty}_{loc})\).

MSC:
65M60 Finite element, Rayleigh-Ritz and Galerkin methods for initial value and initial-boundary value problems involving PDEs
65M12 Stability and convergence of numerical methods for initial value and initial-boundary value problems involving PDEs
35L65 Hyperbolic conservation laws
65M20 Method of lines for initial value and initial-boundary value problems involving PDEs
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References:
[1] Y. BRENIER and S. OSHER, Approximate Riemman Solvers, and Numerical Flux Functions ICASE-NASA Langley Center report n^\circ 84-63, Hampton, VA, 1984. Zbl0597.65071 · Zbl 0597.65071
[2] G. CHAVENT and B. COCKBURN, Consistance et Stabilité des Schémas LRG pour les Lois de Conservation Scalaires, INRIA report # 370 (1987).
[3] G. CHAVENT and B. COCKBURN, The Local Projection Discontinuons Galerkin Finite Element Method for Scalar Conservation Laws, M2AN, 23 (1989), pp. 565-592. Zbl0715.65079 MR1025072 · Zbl 0715.65079
[4] G. CHAVENT and G. SALZANO, A Finite Element Method for the 1D Water Flooding Problem with Gravity, J. Comput. Phys., 45 (1982) pp. 307-344. Zbl0489.76106 MR666166 · Zbl 0489.76106
[5] P. G. CIARLET, The Finite Element Method for Elliptic Problems, North Holland, 1975. Zbl0383.65058 MR520174 · Zbl 0383.65058
[6] T. GEVECI, The Significance of the Stability of Difference Schemes in Different lp-spaces, SIAM Review, 24 (1982),pp. 413-426. Zbl0495.65043 MR678560 · Zbl 0495.65043
[7] B. A. FRYXEL, P. R. WOODWARD, P. COLLELA and K. H. WINKLER, An Implicit-Explicit Hybrid Method for Lagrangian Hydrodynamics, J. Comput Phys., 63 (1986), pp. 283-310. Zbl0596.76078 MR835820 · Zbl 0596.76078
[8] P. D. LAX and B. WENDROFF, Systems of Conservation Laws, Comm. Pure and Appl. Math., 13 (1960), pp. 217-237. Zbl0152.44802 MR120774 · Zbl 0152.44802
[9] 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
[10] B. VAN LEER, Towards the Ultimate Conservative Scheme, VI. A NewApproach to Numerical Convection J. Comput.Phys., 23 (1977), pp. 276-299. Zbl0339.76056 · Zbl 0339.76056
[11] C. W. SHU, TVB uniformly high-order schemes for conservation laws, Math.Comp., 49 (1987), pp. 105-121. Zbl0628.65075 MR890256 · Zbl 0628.65075
[12] C. W. SHU and S. OSHER, Efficient Implementation of Essentially Non-Oscillatory Shock-Capturing Schemes, J. Comput. Phys., 77 (1988), pp. 439-471. Zbl0653.65072 MR954915 · Zbl 0653.65072
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