Analysis and Applications of Leapfrog-Based Local Time-Stepping Methods for the Wave Equation

In a wide range of real-world applications in acoustics, electromagnetism and elasticity, relevance of numerical simulation of wave propagation is immanent and in that context efficiently approximating solutions to the time-dependent wave equation is clearly key. To discretize this second-order partial differential equation, the combination of finite element methods (FEM) in the spatial domain and the leapfrog method for time integration has proved a flexible and highly efficient approach. While FEM are generally well-suited for local mesh refinement originating from local geometric features or heterogeneous media, this situation creates a serious bottleneck for any explicit time-stepping method, since the CFL stability condition enforces a discrete time-step $\Delta t$ proportional to the smallest element in the FE mesh.
Local time-stepping (LTS) methods allow to overcome these constraints by applying different time integration strategies in different parts of the domain. More precisely, Diaz and Grote introduced a LF-based LTS approach [SIAM J. Sci. Comput. 31 (2009), pp. 1985--2014] that is fully explicit, second-order accurate, progresses with an there-term recurrence relation and conserves a discrete energy. Recently, Grote, Mehlin, and Sauter proved optimal convergence rates for this method coupled with a standard Galerkin FE discretization, however under a CFL condition which, in fact, depends on the size of the smallest elements [SIAM J. Numer. Anal. 56 (2018), pp. 994--1021].
Here, we slightly adjust the original LF-LTS (as done recently for LF-Chebyshev methods by Carle, Hochbruck, and Sturm [SIAM J. Numer. Anal. 58 (2020), pp. 2404--2433]) to remove certain discrete values of $\Delta t$, where otherwise instabilities can occur, while nonetheless maintaining all important properties. For this new stabilized LF-LTS method, we prove optimal convergence rates under a CFL condition independent of the mesh size inside the locally refined region. Numerical experiments illustrate these results and verify that the method also preserves the optimal rates on meshes suitably graded towards a reentrant corner.
The effectiveness of these LF-LTS methods is displayed further in an application to uncertainty quantification. A very popular approach to estimate statistics of quantities of interest are the robust and non-intrusive multilevel Monte Carlo methods (MLMC), which efficiently distribute computation of samples across a hierarchy of discretizations. However, inside a given domain, some elements may be forced to remain small across all levels, thereby, for time-dependent problems, inducing a tiny time-step on every level, if explicit time-stepping is used. By adapting the time-step to the locally refined elements on each level, LTS methods permit to restore the efficiency of MLMC methods even in the presence of complex geometry without sacrificing the explicitness and inherent parallelism.