Consistency, accuracy and entropy behaviour of remeshed particle methods
type: Item
instance of: scholarly article
title: Consistency, accuracy and entropy behaviour of remeshed particle methods
zbMATH Open document ID: 1278.65136
author: Adrien Magni
author: Lisl Weynans
publication date: 2013-07-03T00:00:00Z
Mathematics Subject Classification ID: 65M12
Mathematics Subject Classification ID: 65M75
Mathematics Subject Classification ID: 35L65
Mathematics Subject Classification ID: 35Q53
Mathematics Subject Classification ID: 65M15
zbMATH DE Number: 6183182
zbMATH Keywords: particle methods with remeshing
zbMATH Keywords: interpolation kernels
zbMATH Keywords: consistency
zbMATH Keywords: entropy inequalities
zbMATH Keywords: truncation error
zbMATH Keywords: limiters
zbMATH Keywords: convergence
zbMATH Keywords: Burgers equation
zbMATH Keywords: nonlinear scalar transport equation
zbMATH Keywords: infinite domain
zbMATH Keywords: finite difference scheme
zbMATH Keywords: total-variation diminishing-remeshing scheme
zbMATH Keywords: nonlinear conservation law
zbMATH Keywords: Euler equation
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.1051/m2an/2012019
OpenAlex ID: W2140152949
DOI: 10.1051/M2AN/2012019
published in: European Series in Applied and Industrial Mathematics (ESAIM): Mathematical Modelling and Numerical Analysis
review text: Particle methods are Lagrangian techniques designed for physical problems which are dominated by advection. The idea is to discretize the fluid on small masses concentrated on points (the ``particles'') which are then moved according with Lagrangian mechanics. One drawback of these schemes is that the distribution of particles becomes less and less uniform along the time evolution: they tend to accumulate in certain zones and rarefy in others, with the result of a loss of accuracy. The technique used to remedy this problem consists in periodically creating a new uniform distribution of particles by an interpolation of the values of the existing particles, thus leading to the so-called remeshed particle methods. Typically, this is done at each time step.NEWLINENEWLINEIn this paper, an analysis of the consistency and accuracy properties of re\-me\-shed particle methods is carried out when this class of methods are applied to a scalar one-dimensional conservation law, in particular, the one-dimensional nonlinear scalar transport equation in an infinite domain \(\partial_t u + \partial_x (g(u) u) = 0\), \(t\geq 0\), \(-\infty < x <+\infty\). A previous step for this analysis consists in expressing particle methods with remeshing as finite difference schemes. It is shown that the accuracy of the particle scheme depends on the accuracy of the interpolation kernel used. In the nonlinear case, a correction of the evaluation of the particle velocities allows one to get second-order accuracy. The authors also study how total-variation diminishing-remeshing schemes can be built for nonlinear conservation laws with arbitrary sign of the particle velocity and apply the new schemes to Burgers and Euler equations. It is also shown that with this remeshing technique, the particle methods converge to the entropy solution of the scalar conservation law.
reviewed by: Fernando Casas
description: scientific article; zbMATH DE number 6183182