Efficient Method for Solving the Three-Dimensional Wide Angle Wave Equation

摘要

The authors propose a new method for the solution of the wide angle wave equation in three dimensions. In contrast with standard techniques, our approach requires only solutions of successive tridiagonal systems in the resulting finite difference parabolic equations. The method is based on a simple approximation to the squareroot operator written formally as the square root of I+x+y where x is a partial differential operator with respect to the depth z and Y is a partial differential operator with respect to the azimuthal angle theta. This document the fact that the partial derivative term y with respect to the azimuthal angle is small, but not negligible, as compared with other terms. It is then natural to replace the squareroot operator by an expansion which is of order 2 with respect to the X operator, and of order 1 with respect to the Y operator. An important feature of this approach is that it is then possible to derive a rational function approximation to the exponential of the squareroot operator which has the property of being stable, and accurate. Moreover, the approximation decouples naturally as a product of a (1,1) rational function of y. As a consequence, this will result in a solution technique that requires only two tridiagonal system solutions per step, namely one for the X operator and one for the Y operator. Numerical examples are reported that show the wide angle capability of this method.