The Double Absorbing Boundary method

A new approach is devised for solving wave problems in unbounded domains. It has common features to each of two types of existing techniques: local high-order Absorbing Boundary Conditions (ABC) and Perfectly Matched Layers (PML). However, it is different from both and enjoys relative advantages with respect to both. The new method, called the Double Absorbing Boundary (DAB) method, is based on truncating the unbounded domain to produce a finite computational domain Ω, and on applying a local high-order ABC on two parallel artificial boundaries, which are a small distance apart, and thus form a thin non-reflecting layer. Auxiliary variables are defined on the two boundaries and inside the layer bounded by them, and participate in the numerical scheme. The DAB method is first introduced in general terms, using the 2D scalar time-dependent wave equation as a model. Then it is applied to the 1D Klein-Gordon equation, using finite difference discretization in space and time, and to the 2D wave equation in a wave guide, using finite element discretization in space and dissipative time stepping. The computational aspects of the method are discussed, and numerical experiments demonstrate its performance.