Towards Smooth Particle Methods Without Smoothing

Abstract

In this article we present a new class of particle methods which aim at being accurate in the uniform norm with a minimal amount of smoothing. The crux of our approach is to compute local polynomial expansions of the characteristic flow to transport the particle shapes with improved accuracy. In the first order case the method consists of representing the transported density with linearly-transformed particles, the second order version transports quadratically-transformed particles, and so on. For practical purposes we provide discrete versions of the resulting LTP and QTP schemes that only involve pointwise evaluations of the forward characteristic flow, and we propose local indicators for the associated transport error. On a theoretical level we extend these particle schemes up to arbitrary polynomial orders and show by a rigorous analysis that for smooth flows the resulting methods converge in \(L^\infty \) without requiring remappings, extended overlapping or vanishing moments for the particles. Numerical tests using different passive transport problems demonstrate the accuracy of the proposed methods compared to basic particle schemes, and they establish their robustness with respect to the remapping period. In particular, it is shown that QTP particles can be transported without remappings on very long periods of time, without hampering the accuracy of the numerical solutions. Finally, a dynamic criterion is proposed to automatically select the time steps where the particles should be remapped. The strategy is a by-product of our error analysis, and it is validated by numerical experiments.