Abstract
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.
Highlights
Conservation laws are used to describe flow processes in many different areas of physics, e.g., in fluid dynamics, electromagnetics and geophysics, and are of relevance in many engineering applications
Deriving numerical methods for the treatment of non-linear conservation laws that are accurate and robust is a vibrant field of research that has received considerable attention in the last decades
One technique used to ensure that a method is entropy stable is to construct the scheme such that it satisfies a discrete entropy inequality, mimicking the underlying continuous entropy inequality, see e.g., Fisher and Carpenter [7] for an entropy stable high-order WENO finite difference method (FDM)
Summary
Conservation laws are used to describe flow processes in many different areas of physics, e.g., in fluid dynamics, electromagnetics and geophysics, and are of relevance in many engineering applications. The viscosity loses regularity close to shocks and sharp discontinuities To overcome this issue Szepessy [43] proposed constructing the artificial viscosity term proportional to the residual of the differential equation. The main contribution of the present work is extending the RV method to the finite difference SBP-SAT framework, focusing on scalar conservation laws. As pointed out in Guermond et al [12, Sec. 2.4] and Upperman and Yamaleev [45, Sec. 10.2], the residual has a highly oscillatory behavior, which can result in small scale oscillations in smooth regions of the numerical solution To overcome this issue, we combine the upwind SBP finite difference operators presented in Mattsson [28] with the proposed RV method.
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