Numerical analysis of a time-stepping method for the Westervelt equation with time-fractional damping

TL;DR

This paper introduces a numerical method for the Westervelt equation with non-local time damping, analyzing its stability and error, and confirming results through extensive numerical experiments.

Contribution

A novel semi-discrete numerical scheme using trapezoidal rule and convolution quadrature for the Westervelt equation with time-fractional damping, including stability and error analysis.

Findings

Numerical scheme is stable and convergent.

Error analysis accounts for initial singularity.

Numerical experiments validate theoretical results.

Abstract

We develop a numerical method for the Westervelt equation, an important equation in nonlinear acoustics, in the form where the attenuation is represented by a class of non-local in time operators. A semi-discretisation in time based on the trapezoidal rule and A-stable convolution quadrature is stated and analysed. Existence and regularity analysis of the continuous equations informs the stability and error analysis of the semi-discrete system. The error analysis includes the consideration of the singularity at $t = 0$ which is addressed by the use of a correction in the numerical scheme. Extensive numerical experiments confirm the theory.