Enriched discrete spaces for time domain wave equations

Abstract

The second order linear wave equation is simple in representation but its numerical approximation is challenging, especially when the system contains waves of high frequencies. While 10 grid points per wavelength is regarded as the rule of thumb to achieve tolerable approximation with the standard numerical approach, high resolution or high grid density is often required at high frequency which is often computationally demanding. As a contribution to tackling this problem, we consider in this thesis the discretization of the problem in the framework of the space-time discontinuous Galerkin (DG) method while investigating the solution in a finite dimensional space whose building blocks are waves themselves. The motivation for this approach is to reduce the number of degrees of freedom per wavelength as well as to introduce some analytical features of the problem into its numerical approximation. The developed space-time DG method is able to accommodate any polynomial bases. However, the Trefftz based space-time method proves to be efficient even for a system operating at high frequency. Comparison with polynomial spaces of total degree shows that equivalent orders of convergence are obtainable with fewer degrees of freedom. Moreover, the implementation of the Trefftz based method is cheaper as integration is restricted to the space-time mesh skeleton. We also extend our technique to a more complicated wave problem called the telegraph equation or the damped wave equation. The construction of the Trefftz space for this problem is not trivial. However, the exibility of the DG method enables us to use a special technique of propagating polynomial initial data using a wave-like solution (analytical) formula which gives us the required wave-like local solutions for the construction of the space. This thesis contains important a priori analysis as well as the convergence analysis for the developed space-time method, and extensive numerical experiments.