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Arbitrary high order discontinuous Galerkin schemes. (English) Zbl 1210.65165
Cordier, Stéphane (ed.) et al., Numerical methods for hyperbolic and kinetic problems. CEMRACS 2003, summer research center in mathematics and advances in scientific computing, July 21 – August 29, 2003, CIRM, Marseille, France. Zürich: European Mathematical Society (EMS) (ISBN 3-03719-012-4/pbk). IRMA Lectures in Mathematics and Theoretical Physics 7, 295-333 (2005).
Summary: In this paper we apply the ADER one step time discretization to the discontinuous Galerkin framework for hyperbolic conservation laws. In the case of linear hyperbolic systems we obtain a quadrature-free explicit single-step scheme of arbitrary order of accuracy in space and time on Cartesian and triangular meshes. The ADER-DG scheme does not need more memory than a first order explicit Euler time-stepping scheme. This becomes possible because of an extensive use of the governing equations inside the numerical scheme. In the nonlinear case, quadrature of the ADER-DG scheme in space and time is performed with Gaussian quadrature formulae of suitable order of accuracy. We show numerical convergence results for the linearized Euler equations up to 10th order of accuracy in space and time on Cartesian and triangular meshes. Numerical results for the nonlinear Euler equations up to 6th order of accuracy in space and time are provided as well. In this paper we also show the possibility of applying a linear reconstruction operator of the order $$3N+2$$ to the degrees of freedom of the DG method resulting in a numerical scheme of the order $$3N+3$$ on Cartesian grids where $$N$$ is the order of the original basis functions before reconstruction.
For the entire collection see [Zbl 1062.76002].

##### MSC:
 65M60 Finite element, Rayleigh-Ritz and Galerkin methods for initial value and initial-boundary value problems involving PDEs
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