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The Runge-Kutta discontinuous Galerkin method for conservation laws. I: Multidimensional systems. (English) Zbl 0920.65059
The paper is devoted to the construction and investigation of the so-called Runge-Kutta discontinuous Galerkin method for the numerical solution of hyperbolic conservation laws. Thereby, the method presented in previous publications is extended to systems of multidimensional nonlinear conservation laws. The algorithms are described and discussed, including algorithm formulation and practical implementation such as the numerical fluxes, quadrature rules, degrees of freedom, and the slope limiters, both in the triangular and the rectangular element cases. Numerical experiments for two-dimensional Euler equations of gas dynamics are presented that show the effect of the (formal) order of accuracy and the use of triangles or rectangles on the quality of the approximation.

##### MSC:
 65M20 Method of lines 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 65L06 Multistep, Runge-Kutta and extrapolation methods for ordinary differential equations 76M25 Other numerical methods (fluid mechanics) (MSC2010) 76N15 Gas dynamics, general 35L65 Hyperbolic conservation laws
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