Renormalized Meshfree Schemes I: Consistency, Stability, and Hybrid Methods for Conservation Laws

Abstract

This paper is devoted to the study of a new kind of meshfree scheme based on a new class of meshfree derivatives: the renormalized meshfree derivatives, which improve the consistency of the original weighted particle methods. The weak renormalized meshfree scheme, built from the weak formulation of general conservation laws, turns out to be $L^2$ stable under some geometrical conditions on the distribution of particles and some regularity conditions of the transport field. A time discretization is then performed by analogy with finite volume methods, and the $L^1$, $L^\infty$, and $BV$ stabilities of the obtained time discretized scheme are studied. From the same analogy with finite volume methods, a hybrid particle scheme is built using the Godunov method and is numerically compared to the weak renormalized scheme.