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 for the approximation of the acoustic particle velocity variable~$\bv = \nabla \psi$. The established theoretical results are illustrated with some numerical experiments.

翻译：本文讨论用于超声波Westervelt模型的混合间断Galerkin方法的渐近保持特性。具体而言，通过推导与声扩散阻尼参数$\delta$无关的低阶和高阶能量稳定性估计以及先验误差界，证明所提方法对$\delta$的较小取值具有鲁棒性。这些误差界进而表明，当$\delta \rightarrow 0^+$时，该方法保持稳定，且离散声速势$\psi_h^{(\delta)}$收敛于$\psi_h^{(0)}$，其中后者为奇异消失耗散极限。此外，我们证明了声粒子速度变量$\bv = \nabla \psi$近似的最优收敛性。数值实验验证了建立的理论结果。