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

Poorvi Shukla, J.J.W. van der Vegt*

*Corresponding author for this work

Research output: Contribution to journalArticleAcademicpeer-review

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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 Δ t satisfy h≅ CΔ 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 and L2-norms is order p+ 1 for polynomial orders p= 1 and p= 3 and order p for polynomial order p= 2.

Original languageEnglish
Pages (from-to)904-944
Number of pages41
JournalCommunications on Applied Mathematics and Computation
Volume4
Early online date5 Jan 2022
DOIs
Publication statusPublished - Sep 2022

Keywords

  • A priori error analysis
  • Discontinuous Galerkin methods
  • Interior penalty method
  • Space-time methods
  • Wave equation
  • UT-Hybrid-D

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