Efficient explicit time integration for the simulation of acoustic and electromagnetic waves

The efficient and accurate numerical simulation of time-dependent wave phenomena is of fundamental importance in acoustic, electromagnetic or seismic wave propagation. Model problems describing wave propagation include the wave equation and Maxwell's equations, which we study in this work. Both models are partial differential equations in space and time. Following the method-of-lines approach we first discretize the two model problems in space using finite element methods (FEM) in their continuous or discontinuous form. FEM are increasingly popular in the presence of heterogeneous media or complex geometry due to their inherent flexibility: elements can be small precisely where small features are located, and larger elsewhere. Such a local mesh refinement, however, also imposes severe stability constraints on explicit time integration, as the maximal time-step is dictated by the smallest elements in the mesh. When mesh refinement is restricted to a small region, the use of implicit methods, or a very small time-step in the entire computational domain, are generally too high a price to pay. Local time-stepping (LTS) methods alleviate that geometry induced stability restriction by dividing the elements into two distinct regions: the "coarse region" which contains the larger elements and is integrated in time using an explicit method, and the "fine region" which contains the smaller elements and is integrated in time using either smaller time-steps or an implicit scheme. Here we first present LTS schemes based on explicit Runge-Kutta (RK) methods. Starting from classical or low-storage explicit RK methods, we derive explicit LTS methods of arbitrarily high accuracy. We prove that the LTS-RKs(p) methods yield the same rate of convergence as the underlying RKs scheme. Numerical experiments with continuous and discontinuous Galerkin finite element discretizations corroborate the expected rates of convergence and illustrate the usefulness of these LTS-RK methods. As a second method we propose local exponential Adams-Bashforth (LexpAB) schemes. Unlike LTS schemes, LexpAB methods overcome the severe stability restrictions caused by local mesh refinement not by integrating with a smaller time-step but by using the exact matrix exponential in the fine region. Thus, they present an interesting alternative to the LTS schemes. Numerical experiments in 1D and 2D confirm the expected order of convergence and demonstrate the versatility of the approach in cases of extreme refinement