Element centered smooth artificial viscosity in discontinuous Galerkin method for propagation of acoustic shock waves on unstructured meshes

Abstract

This work aims at developing a high-order numerical method for the propagation of acoustic shock waves using the discontinuous Galerkin method. High order methods tend to amplify the formation of spurious oscillations (Gibbs phenomenon) around the discontinuities/shocks, associated to the relative importance of higher-harmonics resulting from nonlinear propagation (in our case). To handle this critical issue, a new shock sensor is introduced for the sub-cell shock capturing. Thereafter, an element-centered smooth artificial viscosity is introduced into the system wherever an acoustic shock wave is sensed. Validation tests in 1D and 2D configurations show that the method is well-suited for the propagation of acoustic shock waves along with other physical effects like geometrical spreading and diffraction.

A high-order discontinuous Galerkin method for nonlinear sound waves

“…Starting from the semi-discrete equation (4.4), this section describes the numerical treatment and solution process involving the assembly of an equation in matrix-vector form and the time-integration scheme that is used. Computational discontinuous Galerkin approaches for nonlinear acoustic waves that are based on developing a first-order conservative system of equations are investigated in, e.g., [34,49,50].…”

The nonlinear ultrasound needle pulse

“…Several algorithms have been developed to solve the KZK equation (Aanonsen et al, 1984;Lee and Hamilton, 1995;Cleveland et al, 1996;Pishchal'Nikov et al, 1996;Varslot and Taraldsen, 2005;Jing and Cleveland, 2007;Dagrau et al, 2011;Tripathi et al, 2018). According to Qiao et al (2016): "In order to capture full diffraction a number of approaches have been reported.…”