Research Note: A workaround for the corner problem in numerically exact non-reflecting boundary conditions

Simulations of wave propagation in the Earth usually require truncation of a larger domain to the region of interest to keep computational cost acceptable. This introduces artificial boundaries that should not generate reflected waves. Most existing boundary conditions are not able to completely suppress all the reflected energy, but suffice in practice except when modelling subtle events such as interbed multiples. Exact boundary conditions promise better performance but are usually formulated in terms of the governing wave equation and, after discretization, still may produce unwanted artefacts. Numerically exact non-reflecting boundary conditions are instead formulated in terms of the discretized wave equation. They have the property that the numerical solution computed on a given domain is the same as one on a domain enlarged to the extent that waves reflected from the boundary do not have the time to reach the original truncated domain. With a second- or higher-order finite-difference scheme for the one-dimensional wave equation, these boundary conditions follow from a recurrence relation. In its generalization to two or three dimensions, a recurrence relation was only found for a single non-reflecting boundary on one side of the domain or two of them at opposing ends. The other boundaries should then be zero Dirichlet or Neumann. If two non-reflecting boundaries meet at a corner, translation invariance is lost and a simple recurrence relation could not be found. Here, a workaround is presented that restores translation invariance by imposing classic, approximately non-reflecting boundary conditions on the other sides and numerically exact ones on the two opposing sides that otherwise would create the strongest reflected waves with the classic condition. The exact ones can also be applied independently. As a proof of principle, the method is applied to the two-dimensional acoustic wave equation, discretized on a rectangular domain with a second-order finite-difference scheme and first-order Enquist–Majda boundary conditions as approximate ones. 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.