A rarefaction-tracking method for hyperbolic conservation laws

TL;DR

This paper introduces a novel numerical method for scalar conservation laws that uses particles representing similarity solutions, effectively capturing rarefaction and compression waves without numerical dissipation.

Contribution

The method employs particles based on similarity solutions to accurately track rarefaction and compression waves, avoiding dissipation and explicitly resolving shocks through particle merging.

Findings

Accurately captures rarefaction and compression waves.

No numerical dissipation away from shocks.

Shocks are precisely located via particle merging.

Abstract

We present a numerical method for scalar conservation laws in one space dimension. The solution is approximated by local similarity solutions. While many commonly used approaches are based on shocks, the presented method uses rarefaction and compression waves. The solution is represented by particles that carry function values and move according to the method of characteristics. Between two neighboring particles, an interpolation is defined by an analytical similarity solution of the conservation law. An interaction of particles represents a collision of characteristics. The resulting shock is resolved by merging particles so that the total area under the function is conserved. The method is variation diminishing, nevertheless, it has no numerical dissipation away from shocks. Although shocks are not explicitly tracked, they can be located accurately. We present numerical examples, and outline specific applications and extensions of the approach.