Long-time numerical computation of wave-type solutions driven by moving sources Original Research Article

We propose a methodology for calculating the solution of an initial-value problem for the three-dimensional wave equation over arbitrarily long time intervals. The solution is driven by moving sources that are compactly supported in space for any particular moment of time; the extent of the support is assumed bounded for all times. By a simple change of variables the aforementioned formulation obviously translates into the problem of propagation of waves across a medium in motion, which has multiple applications in unsteady aerodynamics, advective acoustics, and other areas. The algorithm constructed in the paper can employ any appropriate (i.e., consistent and stable) explicit finite-difference scheme for the wave equation. This scheme is used as a core computational technique and modified so that to allow for a non-deteriorating calculation of the solution for as long as necessary. Provided that the original underlying scheme converges in some sense, i.e., in suitable norms with a particular rate, we prove the grid convergence of the new algorithm in the same sense uniformly in time on arbitrarily long intervals. Thus, the new algorithm obviously does not accumulate error in the course of time; besides, it requires only a fixed non-growing amount of computer resources (memory and processor time) per one time step; these amounts are linear with respect to the grid dimension and thus optimal. The algorithm is inherently three-dimensional; it relies on the presence of lacunae in the solutions of the wave equation in odd-dimension spaces. The methodology presented in the paper is, in fact, a building block for constructing the nonlocal highly accurate unsteady artificial boundary conditions to be used for the numerical simulation of waves propagating with finite speed over unbounded domains.