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

doi:10.1007/s42967-021-00155-0

Communications on Applied Mathematics and Computation ›› 2022, Vol. 4 ›› Issue (3): 904-944.doi: 10.1007/s42967-021-00155-0

• ORIGINAL PAPER • 上一篇下一篇

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

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

Department of Applied Mathematics, University of Twente, P.O. Box 217, 7500 AE Enschede, The Netherlands

收稿日期:2020-12-20 修回日期:2021-06-11 出版日期:2022-09-20 发布日期:2022-07-04
通讯作者: J. J. W. van der Vegt,E-mail:j.j.w.vandervegt@utwente.nl;Poorvi Shukla,E-mail:p.shukla@utwente.nl E-mail:j.j.w.vandervegt@utwente.nl;p.shukla@utwente.nl
基金资助:
This research was funded by the Shell-NWO Computational Sciences for Energy Research Program.

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

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

Department of Applied Mathematics, University of Twente, P.O. Box 217, 7500 AE Enschede, The Netherlands

Received:2020-12-20 Revised:2021-06-11 Online:2022-09-20 Published:2022-07-04
Contact: J. J. W. van der Vegt,E-mail:j.j.w.vandervegt@utwente.nl;Poorvi Shukla,E-mail:p.shukla@utwente.nl E-mail:j.j.w.vandervegt@utwente.nl;p.shukla@utwente.nl
Supported by:
This research was funded by the Shell-NWO Computational Sciences for Energy Research Program.

摘要/Abstract

摘要： 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 $.

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

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

中图分类号:

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

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

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