A Novel Interpolation-Based Method for Solving the One-Dimensional Wave Equation on a Domain with a Moving Boundary
Abstract
We revisit the problem of solving the one-dimensional wave equation on a domain with moving boundary. In J. Math. Phys. 11, 2679 (1970), Moore introduced an interesting method to do so. As only in rare cases, a closed analytical solution is possible, one must turn to perturbative expansions of Moore's method. We investigate the then made minimal assumption for convergence of the perturbation series, namely that the boundary position should be an analytic function of time. Though, we prove here that the latter requirement is not a sufficient condition for Moore's method to converge. We then introduce a novel numerical approach based on interpolation which also works for fast boundary dynamics. In comparison with other state-of-the-art numerical methods, our method offers greater speed if the wave solution needs to be evaluated at many points in time or space, whilst preserving accuracy. We discuss two variants of our method, either based on a conformal coordinate transformation or on the method of characteristics, together with interpolation.
Cite
@article{arxiv.2408.16483,
title = {A Novel Interpolation-Based Method for Solving the One-Dimensional Wave Equation on a Domain with a Moving Boundary},
author = {Michiel Lassuyt and Emma Vancayseele and Wouter Deleersnyder and David Dudal and Sebbe Stouten and Koen Van Den Abeele},
journal= {arXiv preprint arXiv:2408.16483},
year = {2025}
}
Comments
Preprint, 18 pages. This preprint has not undergone peer review (when applicable) or any post-submission improvements or corrections. The Version of Record of this article is published in Zeitschrift f\"ur angewandte Mathematik und Physik, and is available online at https://doi.org/10.1007/s00033-025-02499-6