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

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 $p=2$.

Funding

Shell-NWO Computational Sciences for Energy Research Program