Convergence analysis of energy conserving explicit local time-stepping methods for the wave equation

Local adaptivity and mesh refinement are key to the efficient simulation of wave phenomena in heterogeneous media or complex geometry. Locally refined meshes, however, dictate a small time-step everywhere with a crippling effect on any explicit time-marching method. In [18] a leap-frog (LF) based explicit local time-stepping (LTS) method was proposed, which overcomes the severe bottleneck due to a few small elements by taking small time-steps in the locally refined region and larger steps elsewhere. Here a rigorous convergence proof is presented for the fully-discrete LTS-LF method when combined with a standard conforming finite element method (FEM) in space. Numerical results further illustrate the usefulness of the LTS-LF Galerkin FEM in the presence of corner singularities

Multi-level local time-stepping methods of Runge-Kutta type forwave equations

Local mesh refinement significantly in uences the performance of explicit time-stepping methods for numerical wave propagation. Local time-stepping (LTS) methods improve the efficiency by using smaller time-steps precisely where the smallest mesh elements are located, thus permitting a larger time-step in the coarser regions of the mesh without violating the stability condition. However, when the mesh contains nested patches of refinement, any local time-step will be unnecessarily small in some regions. To allow for an appropriate time-step at each level of mesh refinement, multi-level local time-stepping (MLTS) methods have been proposed. Starting from the Runge{Kutta-based LTS methods derived by Grote et al. [17], we propose explicit MLTS methods of arbitrarily high accuracy. Numerical experiments with finite difference and continuous finite element spatial discretizations illustrate the usefulness of the novel MLTS methods and show that they retain the high accuracy and stability of the underlying Runge{Kutta methods

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

Runge-Kutta Based Explicit Local Time-Stepping Methods for Wave Propagation

Locally refined meshes severely impede the efficiency of explicit Runge-Kutta (RK) methods for the simulation of time-dependent wave phenomena. By taking smaller time-steps precisely where the smallest elements are located, local time-stepping (LTS) methods overcome the bottleneck caused by the stringent stability constraint of but a few small elements in the mesh. Starting from classical or low-storage explicit RK methods, explicit LTS methods of arbitrarily high accuracy are derived. When combined with an essentially diagonal finite element mass matrix, the resulting time-marching schemes retain the high accuracy, stability, and efficiency of the original RK methods while circumventing the geometry-induced stiffness. 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

High-Order Local Time-Stepping with Explicit Runge-Kutta Methods

We propose explicit local time-stepping (LTS) schemes of high accuracy based either on classical or low- storage Runge-Kutta schemes for time dependent Maxwell’s equations. By using smaller time steps precisely where smaller elements in the mesh are located, these methods overcome the bottleneck caused by local mesh refinement in explicit time integrators