A k-space method for nonlinear wave propagation

TL;DR

This paper introduces a k-space method for simulating nonlinear wave propagation in absorptive media, demonstrating improved computational efficiency over traditional methods for certain conditions, with high accuracy in homogeneous media.

Contribution

The paper develops a novel k-space approach for nonlinear wave equations that is not limited to forward or parabolic approximations, enhancing efficiency and accuracy in specific media.

Findings

Accurate results with as little as two points per wavelength in homogeneous media.

Computational efficiency surpasses finite element and FDTD methods under studied conditions.

Less accurate in strongly inhomogeneous media, with potential remedies discussed.

Abstract

A k-space method for nonlinear wave propagation in absorptive media is presented. The Westervelt equation is first transferred into k-space via Fourier transformation, and is solved by a modified wave-vector time-domain scheme [Mast et al., IEEE Tran. Ultrason. Ferroelectr. Freq. Control 48, 341-354 (2001)]. The present approach is not limited to forward propagation or parabolic approximation. One- and two-dimensional problems are investigated to verify the method by comparing results to the finite element method. It is found that, in order to obtain accurate results in homogeneous media, the grid size can be as little as two points per wavelength, and for a moderately nonlinear problem, the Courant-Friedrichs-Lewy number can be as small as 0.4. As a result, the k-space method for nonlinear wave propagation is shown here to be computationally more efficient than the conventional finite element method or finite-difference time-domain method for the conditions studied here. However, although the present method is highly accurate for weakly inhomogeneous media, it is found to be less accurate for strongly inhomogeneous media. A possible remedy to this limitation is discussed.