An exact particle method for scalar conservation laws and its application to stiff reaction kinetics

TL;DR

This paper introduces an exact particle-based method for solving scalar hyperbolic conservation laws, extending it to stiff reaction kinetics and detonation wave modeling, achieving high accuracy without resolving internal wave structures.

Contribution

The paper presents a novel exact particle method for scalar conservation laws, including an extension to stiff reaction kinetics and detonation wave evolution, improving accuracy and efficiency over traditional methods.

Findings

The particle method achieves high accuracy comparable to ODE solvers.

It effectively models detonation waves at correct velocities.

Compared to finite volume methods, it offers improved accuracy for conservation laws and reaction kinetics.

Abstract

An "exact" method for scalar one-dimensional hyperbolic conservation laws is presented. The approach is based on the evolution of shock particles, separated by local similarity solutions. The numerical solution is defined everywhere, and is as accurate as the applied ODE solver. Furthermore, the method is extended to stiff balance laws. A special correction approach yields a method that evolves detonation waves at correct velocities, without resolving their internal dynamics. The particle approach is compared to a classical finite volume method in terms of numerical accuracy, both for conservation laws and for an application in reaction kinetics.