Explicit Local Time-Stepping for the Inhomogeneous Wave Equation with Optimal Convergence
<|reference_start|>Explicit Local Time-Stepping for the Inhomogeneous Wave Equation with Optimal Convergence: Adaptivity and local mesh refinement are crucial for the efficient numerical simulation of wave phenomena in complex geometry. Local mesh refinement, however, can impose a tiny time-step across the entire computational domain when using explicit time integration. By taking smaller time-steps yet only inside locally refined regions, local time-stepping methods overcome the stringent CFL stability restriction imposed on the global time-step by a small fraction of the elements without sacrificing explicitness. In [21], a leapfrog based local time-stepping method was proposed for the inhomogeneous wave equation, which applies standard leapfrog time-marching with a smaller time-step inside the refined region. Here, to remove potential instability at certain time-steps, a stabilized version is proposed which leads to optimal L2-error estimates under a CFL condition independent of the coarse-to-fine mesh ratio. Moreover, a weighted transition is introduced to restore optimal H1-convergence when the source is nonzero across the coarse-to-fine mesh interface. Numerical experiments corroborate the theoretical error estimates and illustrate the usefulness of these improvements.<|reference_end|>
@article{grote2024explicit,
    title={Explicit Local Time-Stepping for the Inhomogeneous Wave Equation with
  Optimal Convergence},
    author={Marcus J. Grote, Simon R. J. Michel, Stefan A. Sauter},
    journal={arXiv preprint arXiv:2409.18085},
    year={2024},
    archivePrefix={arXiv},
    eprint={2409.18085},
    primaryClass={math.NA cs.NA}
}