Exact and High-Order Boundary Conditions in the Time Domain

Abstract

The problem of imposing accurate radiation boundary conditions at the artificial boundary is generally the primary accuracy bottleneck in the numerical solution of scattering problems in the time domain. An impressive number of schemes have been proposed to overcome this difficulty. Nonetheless, the ideal of solving the problem on a fixed domain and achieving convergence by some systematic improvement of the boundary treatment and/or simple mesh refinement seems rarely to have been used. In this talk we advocate such an approach for the basic constant coefficient equations of wave propagation — the wave equation, Maxwell’s equations, the convective wave equation and the compressible Euler equations linearized about a uniform flow. We first develop representations of the exact boundary conditions for special boundaries using separation of variables. The difficult, that is temporally nonlocal, part of the exact condition is expressed as a temporal convolution operator. Approximate boundary treatments, both new and old, are viewed as approximations to the exact convolution kernel, allowing a fairly easy identification of convergence rates and the development of complexity estimates.