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In this paper a novel method to solve the constant coefficient wave equation,subject to interface jump conditions, is presented. In general, such problemspose issues for standard finite difference solvers, as the inherentdiscontinuity in the solution results in erroneous derivative informationwherever the stencils straddle the given interface. Here, however, the recentlyproposed Correction Function Method (CFM) is used, in which correction termsare computed from the interface conditions, and added to affected nodes tocompensate for the discontinuity. In contrast to existing methods, thesecorrections are not simply defined at affected nodes, but rather generalized toa continuous function within a small region surrounding the interface. As aresult, the correction function may be defined in terms of its own governingpartial differential equation (PDE) which may be solved, in principle, toarbitrary order of accuracy. The resulting scheme is not only arbitrarily highorder, but also robust, having already seen application to Poisson problems andthe heat equation. By extending the CFM to this new class of PDEs, thetreatment of wave interface discontinuities in homogeneous media becomespossible. This allows, for example, for the straightforward treatment ofinfinitesimal source terms and sharp boundaries, free of staircasing errors.Additionally, new modifications to the CFM are derived, allowing compatibilitywith explicit multi-step methods, such as Runge-Kutta (RK4), without areduction in accuracy. These results are then verified through numerousnumerical experiments in one and two spatial dimensions.

A Correction Function Method for the Wave Equation with Interface Jump Conditions