A Space-Time Interior Penalty Discontinuous Galerkin Method for the Wave Equation

doi:10.1007/s42967-021-00155-0

Abstract

Abstract: A new higher-order accurate space-time discontinuous Galerkin (DG) method using the interior penalty flux and discontinuous basis functions, both in space and in time, is presented and fully analyzed for the second-order scalar wave equation. Special attention is given to the definition of the numerical fluxes since they are crucial for the stability and accuracy of the space-time DG method. The theoretical analysis shows that the DG discretization is stable and converges in a DG-norm on general unstructured and locally refined meshes, including local refinement in time. The space-time interior penalty DG discretization does not have a CFL-type restriction for stability. Optimal order of accuracy is obtained in the DG-norm if the mesh size h and the time step $ \Delta t $ satisfy $ h\cong C\Delta t $, with C a positive constant. The optimal order of accuracy of the space-time DG discretization in the DG-norm is confirmed by calculations on several model problems. These calculations also show that for pth-order tensor product basis functions the convergence rate in the $ L^\infty $ and $ L^2 $-norms is order $ p+1 $ for polynomial orders $ p=1 $ and $ p=3 $ and order p for polynomial order $ p=2 $.

Key words: Wave equation, Space-time methods, Discontinuous Galerkin methods, Interior penalty method, A priori error analysis

CLC Number:

Poorvi Shukla, J. J. W. van der Vegt. A Space-Time Interior Penalty Discontinuous Galerkin Method for the Wave Equation[J]. Communications on Applied Mathematics and Computation, 2022, 4(3): 904-944.

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