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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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