High-order Higdon-like boundary conditions for exterior transient wave problems

Recently developed non-reflecting boundary conditions are applied for exterior time-dependent wave problems in unbounded domains. The linear time-dependent wave equation, with or without a dispersive term, is considered in an infinite domain. The infinite domain is truncated via an artificial boundary B, and a high-order non-reflecting boundary condition (NRBC) is imposed on B. Then the problem is solved numerically in the finite domain bounded by B. The new boundary scheme is based on a reformulation of the sequence of NRBCs proposed by Higdon. We consider here two reformulations: one that involves high-order derivatives with a special discretization scheme, and another that does not involve any high derivatives beyond second order. The latter formulation is made possible by introducing special auxiliary variables on B. In both formulations the new NRBCs can easily be used up to any desired order. They can be incorporated in a finite element or a finite difference scheme; in the present paper the latter is used. In contrast to previous papers using similar formulations, here the method is applied to a fully exterior two-dimensional problem, with a rectangular boundary. Numerical examples in infinite domains are used to demonstrate the performance and advantages of the new method. In the auxiliary-variable formulation long-time corner instability is observed, that requires special treatment of the corners (not addressed in this paper). No such difficulties arise in the high-derivative formulation