A Novel Interpolation-Based Method for Solving the One-Dimensional Wave Equation on a Domain with a Moving Boundary

TL;DR

This paper introduces a new interpolation-based numerical method for efficiently solving the one-dimensional wave equation with a moving boundary, outperforming existing methods in speed while maintaining accuracy.

Contribution

The paper presents a novel interpolation approach that works for fast boundary dynamics and improves computational speed compared to current state-of-the-art methods.

Findings

The method is faster when evaluating solutions at many points.

It maintains high accuracy comparable to existing methods.

It is applicable to boundary movements with complex dynamics.

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.