The Runge-Kutta local projection \(P^ 1\)-discontinuous-Galerkin finite element method for scalar conservation laws. (English) Zbl 0732.65094

Cockburn, Bernardo; Shu, Chi-Wang

RAIRO, Modélisation Math. Anal. Numér. 25, No. 3, 337-361 (1991).

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 \(L^{\infty}(L^{\infty}_{loc})\).

Reviewer: T.Petrila (Cluj-Napoca)

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

Keywords:

hyperbolic conservation laws; discontinuous Galerkin finite element; total variation diminishing Runge-Kutta time discretization; local projection; global stability; maximum principle; entropy solution; order of convergence

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