Fast Sweeping Methods for Hyperbolic Systems of Conservation Laws at Steady State

TL;DR

This paper introduces a fast sweeping method for efficiently computing steady state solutions of hyperbolic conservation laws, capturing shocks accurately by enforcing Rankine-Hugoniot conditions, with demonstrated convergence and numerical results.

Contribution

It adapts fast sweeping techniques from Hamilton-Jacobi equations to hyperbolic conservation laws, enabling efficient and accurate steady state computations.

Findings

Method computes solutions efficiently by exploiting characteristic flow.

Sharp shock capturing through direct enforcement of Rankine-Hugoniot conditions.

Numerical experiments confirm convergence and effectiveness in 1D and 2D cases.

Abstract

Fast sweeping methods have become a useful tool for computing the solutions of static Hamilton-Jacobi equations. By adapting the main idea behind these methods, we describe a new approach for computing steady state solutions to systems of conservation laws. By exploiting the flow of information along characteristics, these fast sweeping methods can compute solutions very efficiently. Furthermore, the methods capture shocks sharply by directly imposing the Rankine-Hugoniot shock conditions. We present convergence analysis and numerics for several one- and two-dimensional examples to illustrate the use and advantages of this approach.