A correction function method for the wave equation with interface jump conditions

Abstract

In this paper a novel method to solve the constant coefficient wave equation, subject to interface jump conditions, is presented. In general, such problems pose issues for standard finite difference solvers, as the inherent discontinuity in the solution results in erroneous derivative information wherever the stencils straddle the given interface. Here, however, the recently proposed Correction Function Method (CFM) is used, in which correction terms are computed from the interface conditions, and added to affected nodes to compensate for the discontinuity. In contrast to existing methods, these corrections are not simply defined at affected nodes, but rather generalized to a continuous function within a small region surrounding the interface. As a result, the correction function may be defined in terms of its own governing partial differential equation (PDE) which may be solved, in principle, to arbitrary order of accuracy. The resulting scheme is not only arbitrarily high order, but also robust, having already seen application to Poisson problems and the heat equation. By extending the CFM to this new class of PDEs, the treatment of wave interface discontinuities in homogeneous media becomes possible. This allows, for example, for the straightforward treatment of infinitesimal source terms and sharp boundaries, free of staircasing errors. Additionally, new modifications to the CFM are derived, allowing compatibility with explicit multi-step methods, such as Runge-Kutta (RK4), without a reduction in accuracy. These results are then verified through numerous numerical experiments in one and two spatial dimensions.

FDTD schemes for Maxwell's equations with embedded perfect electric conductors based on the correction function method

“…The approximation of D at each node (xi, t) comes from the evaluation of a space-time polynomial of degree k, which has been computed using either discrete problem (3) or (10). It has been shown in [16] that a direct interpolation in time of the correction function at each stage 200 of an one-step method might lead to a suboptimal order of convergence. However, one could find the correction that needs to be apply for each stage of an one-step method using a Taylor's expansion of the correction function D. This approach can be cumbersome to compute for high-order one-step methods.…”

“…This numerical strategy has been applied to Poisson's equation with discontinuous piecewise coefficients [14,15]. It also been applied to the wave equation [16] and Maxwell's equations [10], but with constant coefficients. Briefly, the underline assumption of the CFM based strategy is that jumps on the interface can be smoothly extended in the vicinity of the interface.…”

“…We then focus on the discretization of time derivatives for time-dependent problems. As it has been shown in [16], an one-step method when applied to CFM-FD scheme might need a modification of the correction function at each stage to retain its order of convergence. In this work, we propose a general approach to avoid such an explicit modification of an one-step method.…”

Imposing jump conditions on nonconforming interfaces for the Correction Function Method: A least squares approach

“…[2], and is the subject of current research by the authors. The CFM has also been extended to other classes of differential equations, such as the heat equation [2], the Navier-Stokes equations [2], and the wave equation [39].…”

Treatment of Complex Interfaces for Maxwell’s Equations with Continuous Coefficients Using the Correction Function Method

“…This allows us to compute approximations of the correction function to correct the finite difference (FD) scheme in the vicinity of an interface. The CFM was applied on Poisson's equation with piecewise constant coefficients [16] and on the wave equation with constant coefficients [2].…”