Multi-Level Explicit Local Time-Stepping Methods for Second-Order Wave Equations

Local mesh refinement severly impedes the efficiency of explicit time-stepping methods for numerical wave propagation. Local time-stepping (LTS) methods overcome the bottleneck due to a few small elements by allowing smaller time-steps precisely where those elements are located. Yet when the region of local mesh refinement itself contains a sub-region of even smaller elements, any local time-step again will be overly restricted. To remedy the repeated bottleneck caused by hierarchical mesh refinement, multi-level local time-stepping methods are proposed, which permit the use of the appropriate time-step at every level of mesh refinement. Based on the LTS methods from [1], these multi-level LTS methods are explicit, yield arbitrarily high accuracy and conserve the energy. Numerical experiments illustrate the theoretical properties and the usefulness of these methods.