A residual-based artificial viscosity finite difference method for scalar conservation laws

In this paper, we present an accurate, stable and robust shock-capturing finite difference method for solving scalar non-linear conservation laws. The spatial discretization uses high-order accurate upwind summation-by-parts finite difference operators combined with weakly imposed boundary conditions via simultaneous-approximation-terms. The method is an extension of the residual-based artificial viscosity methods developed in the finite- and spectral element communities to the finite difference setting. The three main ingredients of the proposed method are: (i) shock detection provided by a residual-based error estimator; (ii) first-order viscosity applied in regions with strong discontinuities; (iii) additional dampening of spurious oscillations provided by high-order dissipation from the upwind finite difference operators. The method is shown to be stable for skew-symmetric discretizations of the advective flux. Accuracy and robustness are shown by solving several benchmark problems in 2D for convex and non-convex fluxes. (C) 2020 The Author(s). Published by Elsevier Inc.

Automated summary

This paper presents an accurate, stable, and robust shock-capturing finite difference method for solving scalar non-linear conservation laws. The method utilizes high-order accurate upwind finite difference operators, residual-based error estimators for shock detection, and first-order viscosity in regions with strong discontinuities. The method also includes additional damping of spurious oscillations from high-order dissipation, showing stability for skew-symmetric discretizations. Accuracy and robustness are demonstrated through solving benchmark problems with convex and non-convex fluxes in 2D.