Exact Non-Reflecting Boundary Conditions with an FDTD Scheme for the Scalar Wave Equation in Waveguide Problems

Modeling wave propagation often requires a truncation of the computational domain to a smaller subdomain to keep computational cost reasonable. The mere volume of papers on absorbing boundary conditions indicates that a perfect solution is not available. A method is proposed that is numerically exact, at least in the case of a time-domain finite-difference scheme for the scalar wave equation. The word 'exact' is used in the sense that there is no difference between a computation on the truncated domain with this method and one on an enlarged domain with reflecting boundaries that are placed so far away that their reflections cannot reach the original domain within the modeled time span. Numerical tests in 1D produce stable results with central difference schemes from order 2 to 24 for the spatial discretization. The difference with a reference solution computed on an enlarged domain with the boundary moved sufficiently far away only contains accumulated numerical round-off errors. Generalization to more than one space dimension is feasible if there is a single non-reflecting boundary on one side of a rectangular domain or two non-reflecting boundaries at opposing sides, but not for a corner connecting non-reflecting boundaries. The reason is that the method involves recursion based on translation invariance in the direction perpendicular to the boundary, which does not hold in the last case. This limits the applicability of the method to, for instance, modeling waveguides, or exactness has to be partially given up.