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Abstract:

Abstract. We propose a family of algorithms for solving numerically a Cauchy problem for the threedimensional wave equation. The sources that drive the equation #i.e., the right-hand side # are compactly supported in space for any given time; they, however, may actually moveinspace with a subsonic speed. The solution is calculated inside a #nite domain #e.g., sphere # that also moves with a subsonic speed and always contains the support of the right-hand side. The algorithms employ a standard consistent and stable explicit #nite-di#erence scheme for the wave equation. They allow one to calculate the solution for arbitrarily long time intervals without error accumulation and with the #xed non-growing amount of the CPU time and memory required for advancing one time step. The algorithms are inherently three-dimensional; they rely on the presence of lacunae in the solutions of the wave equation in oddly dimensional spaces. The methodology presented in the paper is, in fact, a building block for constructing the nonlocal highly accurate unsteady arti#cial boundary conditions to be used for the numerical simulation of waves propagating with #nite speed over unbounded domains. Key words. wave equation, lacunae, #nite-di#erence approximation, explicit numerical integration, arbitrarily long time intervals, non-accumulation of error, uniform error bounds, #xed expenses per time

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