Multi-level explicit local time-stepping methods for second-order wave equations

Julien Diaz (1, 2) , Marcus J. Grote (3)

1 LMAP - Laboratoire de Mathématiques et de leurs Applications [Pau]
2 Magique 3D - Advanced 3D Numerical Modeling in Geophysics
3 Department of Mathematics [Basel]

Local mesh refinement severely 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 Diaz and Grote (2009), 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.

Keywords: Wave propagation; finite element methods; adaptivity; explicit time integration; leap-frog method; local time-stepping; multirate methods

Domains: Analysis of PDEs [math.AP]; Numerical Analysis [math.NA]

Julien Diaz, Marcus J. Grote. Multi-level explicit local time-stepping methods for second-order wave equations. Computer Methods in Applied Mechanics and Engineering, 2015, 291, pp.240-265. ⟨10.1016/j.cma.2015.03.027⟩. ⟨hal-01184090⟩