A high-order discontinuous Galerkin method for nonlinear sound waves

TL;DR

This paper introduces a high-order discontinuous Galerkin method for nonlinear acoustic wave equations, providing stability, convergence analysis, and numerical validation in 2D and 3D settings.

Contribution

It develops a novel high-order DG scheme for nonlinear sound waves on polytopic meshes, with rigorous error estimates and hybrid implementation in 3D.

Findings

The method achieves optimal convergence rates in energy norm.

Numerical experiments confirm theoretical error estimates.

Efficient in complex 3D media with varying parameters.

Abstract

We propose a high-order discontinuous Galerkin scheme for nonlinear acoustic waves on polytopic meshes. To model sound propagation with and without losses, we use Westervelt's nonlinear wave equation with and without strong damping. Challenges in the numerical analysis lie in handling the nonlinearity in the model, which involves the derivatives in time of the acoustic velocity potential, and in preventing the equation from degenerating. We rely in our approach on the Banach fixed-point theorem combined with a stability and convergence analysis of a linear wave equation with a variable coefficient in front of the second time derivative. By doing so, we derive an a priori error estimate for Westervelt's equation in a suitable energy norm for the polynomial degree $p \geq 2$. Numerical experiments carried out in two-dimensional settings illustrate the theoretical convergence results. In addition, we demonstrate efficiency of the method in a three-dimensional domain with varying medium parameters, where we use the discontinuous Galerkin approach in a hybrid way.