Solving One Dimensional Scalar Conservation Laws by Particle Management

TL;DR

This paper introduces a meshfree particle management method for solving one-dimensional scalar conservation laws that maintains sharp shocks, is conservative, and improves accuracy over existing methods like CLAWPACK.

Contribution

The paper presents a novel particle management approach that avoids global remeshing, preserves sharp shocks, and enhances accuracy in solving scalar conservation laws.

Findings

Method is TVD and entropy decreasing

Achieves higher accuracy than CLAWPACK at similar resolutions

Handles shocks effectively without numerical dissipation

Abstract

We present a meshfree numerical solver for scalar conservation laws in one space dimension. Points representing the solution are moved according to their characteristic velocities. Particle interaction is resolved by purely local particle management. Since no global remeshing is required, shocks stay sharp and propagate at the correct speed, while rarefaction waves are created where appropriate. The method is TVD, entropy decreasing, exactly conservative, and has no numerical dissipation. Difficulties involving transonic points do not occur, however inflection points of the flux function pose a slight challenge, which can be overcome by a special treatment. Away from shocks the method is second order accurate, while shocks are resolved with first order accuracy. A postprocessing step can recover the second order accuracy. The method is compared to CLAWPACK in test cases and is found to yield an increase in accuracy for comparable resolutions.