Working around the Corner Problem in Numerically Exact Non-Reflecting Boundary Conditions for the Wave Equation

Recently introduced non-reflecting boundary conditions are numerically exact: the solution on a given domain is the same as a subset of one on an enlarged domain where boundary reflections do not have time to reach the original domain. In 1D with second- or higher-order finite differences, a recurrence relation based on translation invariance provides the boundary conditions. In 2D or 3D, a recurrence relation was only found for a non-reflecting boundary on one or two opposing sides of the domain and zero Dirichlet or Neumann boundaries elsewhere. Otherwise, corners cause translation invariance to be lost.

The proposed workaround restores translation invariance with classic, approximately non-reflecting boundary conditions on the other sides. As a proof of principle, the method is applied to the 2-D constant-density acoustic wave equation, discretized on a rectangular domain with a second-order finite-difference scheme, first-order Enquist-Majda boundary conditions as approximate ones, and numerically exact boundary conditions in the horizontal direction. The method is computationally costly but has the advantage that it can be reused on a sequence of problems as long as the time step and the sound speed values next to the boundary are kept fixed.