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

¹ Department of Mathematics and Applications, University of Milano-Bicocca, 20125 Milan, Italy
² Department of Mathematics, Radboud University, Heyendaalseweg 135, 6525 AJ Nijmegen, The Netherlands
³ Institute of Mathematics, Czech Academy of Sciences, Prague, Czech Republic

^* Corresponding author: sergio.gomezmacias@unimib.it

Received: 6 May 2024
Accepted: 29 December 2024

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 δ by deriving low- and high-order energy stability estimates, and a priori error bounds that are independent of δ. Such bounds are then used to show that, when δ → 0⁺, the method remains stable and the discrete acoustic velocity potential ψ_h^(δ) converges to ψ_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 υ=∇ψ. The established theoretical results are illustrated with some numerical experiments.