Consistency, accuracy and entropy behaviour of remeshed particle methods (Q2836453)

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.