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

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.

Asymptotic-preserving finite element analysis of Westervelt-type wave equations

“…Uniform-in-ε analysis of a mixed finite element approximation of the local Westervelt equation in the so-called potential form (with K = δ 0 and a = 1 + ku ε t ) is available in [24]. Analysis of a time-discretization of (1.1) with fractional-type damping has been performed in [3].…”