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

TL;DR

This work addresses solving the one-dimensional wave equation on a domain with a moving boundary, where Moore's perturbation method has limited convergence. It introduces two interpolation-based numerical approaches, IMR (conformal mapping) and IMC (method of characteristics), which extend $R(\xi)$ or the solution via interpolation and perform well even for fast boundary dynamics. Compared to the state-of-the-art backtracing method and Moore's perturbative approach, IMR and IMC offer higher accuracy and substantial speedups, especially when multiple evaluations of the solution are required. The methods have potential applications in problems involving dynamic domains and energy calculations in settings like the dynamical Casimir effect.

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.

Figures (10)

• Figure 1: Representation of a wave $u$ on a domain with a variable length.
• Figure 2: Convergence of the function $R$ using Moore's method in the case $L(t) = L_0 + vt$.
• Figure 3: Convergence of the function $R$ using Moore's method in the case $L(t) = e^{-kt}$.
• Figure 4: Illustration of the interpolation method. Blue indication for $\xi$ values represent instances where $R(\xi)$ is known. New values can be found using the relation \ref{['eq:R_condition']}. For example, in order to find the red value at $\xi = t_i + L(t_i)$, it is interpolated at $\xi = t_i - L(t_i)$ based on the blue values, followed by an addition of 2.
• Figure 5: Illustration of the corresponding length evolution $L(t)$ in the case of a hyperbolic sine transformation $R$ as listed in Tab. \ref{['tab:boundary_conditions']}, with $k = \xi = 1$. The blue curve is for $A = 2$ and the red curve for $A = 0.1$.