Taylor Expansion and Discretization Errors in Gaussian Beam Superposition

TL;DR

This paper analyzes the accuracy of Gaussian beam superposition methods for high frequency wave fields, focusing on errors from discretization and Taylor expansion, and reveals error cancellation effects specific to odd order beams.

Contribution

It provides new error estimates for Gaussian beam superposition, highlighting the role of beam order and challenging assumptions about beam width and accuracy.

Findings

Odd order beams exhibit error cancellation effects.

Error estimates show no direct relation between beam width and accuracy.

Numerical examples confirm theoretical error bounds.

Abstract

The Gaussian beam superposition method is an asymptotic method for computing high frequency wave fields in smoothly varying inhomogeneous media. In this paper we study the accuracy of the Gaussian beam superposition method and derive error estimates related to the discretization of the superposition integral and the Taylor expansion of the phase and amplitude off the center of the beam. We show that in the case of odd order beams, the error is smaller than a simple analysis would indicate because of error cancellation effects between the beams. Since the cancellation happens only when odd order beams are used, there is no remarkable gain in using even order beams. Moreover, applying the error estimate to the problem with constant speed of propagation, we show that in this case the local beam width is not a good indicator of accuracy, and there is no direct relation between the error and the beam width. We present numerical examples to verify the error estimates.