Absorbing boundary conditions, difference operators, and stability

In this paper we present a review of some of the methods currently used for solving the absorbing boundary problem for the two-dimensional scalar wave equation. We show the relationship between the methods of Lindman and Clayton and Engquist. Through this relationship we can derive discretizations of any rational approximation to the one-way wave equation. We prove that, for all the cases considered here, which can be solved in a manner similar to Lindman's approach, the bounds imposed on the Courant number for stability at the boundary are no more severe than the bound 1 √2 required for stability of the interior scheme. These bounds are, however, necessary but not sufficient. We also compare the methods reviewed numerically. It is demonstrated that Lindman's scheme is no better than a sixth-order approximation of Halpern and Trefethen. For low-order approximations, Higdon's one-dimensional equations are satisfactory, but as the order increases the two-dimensional form of the equations, as derived by Halpern and Trefethen, is preferable.