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<p>We revisit the problem of solving the one-dimensional wave equation on a domain with moving boundary. Moore (J Math Phys 11:2679, 1970) 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 moving boundaries. 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, while 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.</p>

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A novel interpolation-based method for solving the one-dimensional wave equation on a domain with a moving boundary

  • 期刊论文

  • 发表日期：

    2025年5月24日

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  • Michiel Lassuyt,
  • Emma Vancayseele,
  • Wouter Deleersnyder,
  • David Dudal,
  • Sebbe Stouten,
  • Koen Van Den Abeele

摘要

We revisit the problem of solving the one-dimensional wave equation on a domain with moving boundary. Moore (J Math Phys 11:2679, 1970) 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 moving boundaries. 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, while 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.