Solution of the Fourth-moment Equation by an Adaptive Grid Method

Abstract

The parabolic fourth moment equation for a plane wave is integrated numerically using an adaptive grid algorithm . Results are compared with those from an accurate, but far less efficient, operator splitting method, and agree very closely . The adaptive approach has several advantages over current fixed grid methods . The primary one is that significant reductions in computer memory usage can be obtained without a concomitant increase in run time . It is also ideally suited for accurate integration of the fourth-moment equation for an extremely large range of values of both the scattering parameter, F, and the wave propagation distance .

Numerical Solution of the Fourth-moment Equation for a Point Source

“…For an incident plane wave, accurate numerical solutions [1,2] have confirmed and extended the analytical results of [3,4] . The more important problem of propagation from a point source in a two-dimensional plane, however, has proved far more intractable .…”

“…The main method that has been developed uses an adaptive grid algorithm . This has been demonstrated to give accurate and efficient solutions to the fourth-moment equation [2] . Brief descriptions of the adaptive grid and operator splitting methods, as applied to the point source equation, (9), are presented below .…”

“…Here, the grid points are constrained to move on lines parallel to thẽ -and q -axes . Such a method has already been developed to solve the fourthmoment equation for a plane wave [2] . The adaptive grid algorithm (AGA) which is used here is an extension of that method .…”

“…Some modification is required to cope with the increased concentration of the fourth moment along the axes for the case of a point source. The parameter a, in equation (12) of [2], which is a lower bound on the ratio of the smallest to largest grid spacings, is now specified as a function of range . Near the source it was found that for small values of a, extremely small range steps had to be taken in order to maintain the chosen error tolerance .…”

“…Numerical results for a beamed source [5] have been presented only for very small values of scattering strength F. The purpose of this paper is to present an accurate, efficient numerical solution of the fourth-moment equation for a point source, valid for a wide range of scattering strengths . The difficulties are overcome by means of an adaptive grid technique, developed for this purpose and described in [2] . The accuracy of the results is confirmed by comparison with the operator splitting method [1], which is cumbersome and inefficient when applied to a point source .…”