MultiLevel Local Time-Stepping Methods of Runge--Kutta-type for Wave Equations

Abstract

Local mesh refinement significantly influences 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, multilevel local time-stepping (MLTS) methods have been proposed. Starting from the Runge--Kutta-based LTS methods derived by Grote, Mehlin, and Mitkova [SIAM J. Sci. Comput., 37 (2015), pp. A747--A775], 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 r...