An exactly conservative particle method for one dimensional scalar conservation laws

TL;DR

This paper introduces an exactly conservative particle method for 1D scalar conservation laws that accurately captures shocks and rarefactions, maintaining sharpness and entropy conditions with second order accuracy.

Contribution

The paper presents a novel particle scheme that conserves area exactly, handles shocks and rarefactions effectively, and achieves second order accuracy, improving upon existing methods.

Findings

Method is exactly conservative and entropy decreasing.

Achieves second order accuracy in solutions.

Comparable or better accuracy than CLAWPACK.

Abstract

A particle scheme for scalar conservation laws in one space dimension is presented. Particles representing the solution are moved according to their characteristic velocities. Particle interaction is resolved locally, satisfying exact conservation of area. Shocks stay sharp and propagate at correct speeds, while rarefaction waves are created where appropriate. The method is variation diminishing, entropy decreasing, exactly conservative, and has no numerical dissipation away from shocks. Solutions, including the location of shocks, are approximated with second order accuracy. Source terms can be included. The method is compared to CLAWPACK in various examples, and found to yield a comparable or better accuracy for similar resolutions.