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

Michel, Simon René Jonas. Analysis and Applications of Leapfrog-Based Local Time-Stepping Methods for the Wave Equation. 2021, Doctoral Thesis, University of Basel, Faculty of Science.

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Official URL: https://edoc.unibas.ch/85754/

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.
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.