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

TL;DR

This work develops an asymptotic-preserving HDG method for the Westervelt quasilinear wave equation, ensuring stability and accuracy as the damping parameter $\delta$ tends to $0$. By formulating a semidiscrete HDG scheme with variables $(\psi_h, \boldsymbol{v}_h, \lambda_h)$ and conducting a linearized, then nonlinear, analysis via a fixed-point argument, the authors establish $\delta$-robust energy estimates and optimal $h$-convergence in energy norms, along with convergence to the vanishing-viscosity limit and postprocessing-induced superconvergence for the primal variable. Theoretical results are corroborated by fully discrete numerical experiments that demonstrate $\mathcal{O}(h^{p+1})$ convergence for $\psi_h$ and $\mathcal{O}(h^{p+2})$ postprocessed rates when $p>0$, as well as $\mathcal{O}(\delta)$-convergence in the $\delta\to0^+$ regime. The approach provides a robust, efficient framework for nonlinear acoustic wave simulations with small damping, with potential extensions to Kuznetsov-type models and higher-order time integrators.

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 \emph{a priori} error bounds that are independent of $δ$. Such bounds are then used to show that, when $δ\rightarrow 0^+$, 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 $\boldsymbol{v} = \nabla ψ$. The established theoretical results are illustrated with some numerical experiments.

Figures (5)

• Figure 1: Asymptotic-preserving commutative diagram for the Westervelt equation. This diagram represents the connections between $\psi_h^{(\delta)}$ and $\psi^{(\delta)}$ as $h \rightarrow 0^+$ (even in the limit case $\delta = 0$) as well as between $\psi_h^{(\delta)}$ and $\psi_h^{(0)}$ as $\delta \to 0^+$. The superscript $(\delta)$ is used to emphasize the dependence on the parameter $\delta$ of the continuous solution and its numerical approximation.
• Figure 2: First panel: Example of the simplicial meshes used in the numerical examples. Remaining panels:$h$-convergence of the errors in \ref{['EQN::h-ERRORS']} at the final time $T = 1$s for the test case with exact solution \ref{['EQN::EXACT-SOLUTION']}. The numbers in the yellow rectangles denote the experimental rates of convergence.
• Figure 3: $\delta$-convergence of the errors in \ref{['EQN::DELTA-ERRORS']} at the final time $T = 1$s for the test case in Section \ref{['SECT::DELTA-ERROR']}.
• Figure 4: Results obtained at $t = 5\times 10^{-5}$s (first row) and $t = 2\times 10^{-4}$s (second row) for the test case in Section \ref{['SECT::WAVEFRONT']}. Left panels: Approximation of $\partial_t \psi$ obtained for $p = 5$ and $k = -10\text{s}^2\text{m}^{-2}$. Right panels: Comparison of the approximations obtained for the Westervelt equation (black lines) and the linear damped wave equation (red lines) along the line $y = 0.5$.
• Figure 5: $\delta$-convergence of the errors in \ref{['EQN::DELTA-ERRORS']} at $t = 10^{-4}$s for the test case in Section \ref{['SECT::WAVEFRONT']} with degree of approximation $p = 5$.

Theorems & Definitions (26)

• Remark 2.1: Structure of $\mathcal{N}_h(\cdot, \cdot)$
• Remark 2.2: Linear case
• Remark 3.1: Linearization argument
• Theorem 3.2: Energy estimates for the discrete linearized problem
• proof
• Remark 3.3: Regularity of $\underaccent{\bar{}}{\boldsymbol{\Upsilon}}$
• Remark 3.4: Stabilization parameter
• Lemma 3.5: Approximation properties of $\Pi_{\sf{HDG}}$
• Lemma 3.6
• proof