Asymptotic-preserving hybridizable discontinuous Galerkin method for the Westervelt quasilinear wave equation

Abstract

We discuss the asymptotic-preserving properties of a hybridizable discontinuous Galerkin method for the Westervelt model of ultrasound waves. More precisely, we show that the proposed method is robust with respect to small values of the sound diffusivity damping parameter $\delta$ by deriving low- and high-order energy stability estimates, and \emph{a priori} error bounds that are independent of $\delta$. Such bounds are then used to show that, when $\delta \rightarrow 0^+$, the method remains stable and the discrete acoustic velocity potential $\psi_h^{(\delta)}$ converges to $\psi_h^{(0)}$, where the latter is the singular vanishing dissipation limit. Moreover, we prove optimal convergence rates for the approximation of the acoustic particle velocity variable $\boldsymbol{v} = \nabla \psi$. The established theoretical results are illustrated with some numerical experiments.