An improved numerical ordinary‐differential‐equation solution to the parabolic wave equation

Abstract

The solution of the parabolic wave equation can be approximated by the solution of a stiff Initial Value Problem (IVP) associated with a system of Ordinary Differential Equations. The Generalized Adams Methods (GAM) based upon a stable rational approximation to the matrix exponential has been shown to be an effective method for solving this problem. However, previous implementations of the GAM suffered from severe storage limitations because of the use of an explicit rational approximation to the matrix exponential. In this presentation, we describe a new implementation of the GAM that is based upon an implicit use of a “Restricted‐Pade” approximation to the matrix exponential. This implementation is particularly effective for problems for which the associated matrix is banded, as is the case for the IVP approximating the parabolic wave equation. A problem of propagation in a wedge shaped region is presented to illustrate the strong limitation of the previous implementation and how this limitation is removed in the new implementation. [Work is supported by ONR.]