A high-order discontinuous Galerkin method for nonlinear sound waves

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.