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 hcong CDelta 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.
Highlights
The second-order scalar wave equation provides an important model for many hyperbolic wave problems in physics, engineering and life sciences
Many of these numerical discretizations follow the method of lines approach, where the wave equation is first discretized in space, after which the resulting system of ordinary differential equations is discretized with a suitable, often explicit, time integration method
This has resulted in many accurate and efficient numerical discretizations of the wave equation that can be found in nearly any text book on the numerical analysis of partial differential equations
Summary
The second-order scalar wave equation provides an important model for many hyperbolic wave problems in physics, engineering and life sciences. The space-time IP-DG discretization uses tensor product basis functions that are discontinuous in space and in time This provides a very natural way to construct an arbitrary higher-order accurate conservative discretization of the wave equation that allows for local mesh refinement and local time stepping. The key components in the error analysis are several stability bounds given by Theorems 1 and 2 and Corollary 1, which are used in a backward in time error-analysis as outlined in [28, Chapter 12] This allows us to prove in Theorem 3 stability, convergence and optimal order accuracy of the space-time IP-DG discretization in a DG-norm for general locally refined space-time meshes and arbitrary order tensor product basis functions in space and in time and without a CFL constraint for stability.
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