Utilizing Time-Reversibility for Shock Capturing in Nonlinear Hyperbolic Conservation Laws

TL;DR

This paper presents a new shock capturing method for nonlinear hyperbolic conservation laws that leverages time-reversibility properties near shocks, providing a system-specific artificial viscosity without extra equations or prior knowledge.

Contribution

It introduces a novel, parameter-efficient artificial viscosity approach based on time-reversibility, adaptable to different system components and independent of mesh and order.

Findings

Effective in multi-dimensional hyperbolic systems

Compatible with high-order spectral element methods

Handles systems with mixed discontinuities

Abstract

In this work, we introduce a novel approach to formulating an artificial viscosity for shock capturing in nonlinear hyperbolic systems by utilizing the property that the solutions of hyperbolic conservation laws are not reversible in time in the vicinity of shocks. The proposed approach does not require any additional governing equations or a priori knowledge of the hyperbolic system in question, is independent of the mesh and approximation order, and requires the use of only one tunable parameter. The primary novelty is that the resulting artificial viscosity is unique for each component of the conservation law which is advantageous for systems in which some components exhibit discontinuities while others do not. The efficacy of the method is shown in numerical experiments of multi-dimensional hyperbolic conservation laws such as nonlinear transport, Euler equations, and ideal magnetohydrodynamics using a high-order discontinuous spectral element method on unstructured grids.