Approximate solutions of scalar conservation laws

TL;DR

This paper investigates the properties of approximate solutions to scalar conservation laws using BGK-type schemes with a focus on how microscopic interactions and a deviation threshold influence the solutions, especially near shocks.

Contribution

It introduces a new analysis of BGK-type schemes with a deviation threshold, providing insights into their convergence and properties in the vanishing relaxation time limit.

Findings

Solutions are close to the original flux within order epsilon

The schemes produce approximate shock solutions for Burger's equation

Several properties of the approximate solutions are established

Abstract

We study compactness properties of time-discrete and continuous time BGK-type schemes for scalar conservation laws, in which microscopic interactions occur only when the state of a system deviates significantly from an equilibrium distribution. The threshold deviation, $\epsilon,$ is a parameter of the problem. In the vanishing relaxation time limit we obtain solutions of a conservation law in which flux is pointwisely close (of order $\epsilon$) to the flux of the original equation and derive several other properties of such solutions, including an example of approximate solution to a shock for Burger's equation.