Renormalized Meshfree Schemes II: Convergence for Scalar Conservation Laws

Abstract

We establish a convergence rate of the renormalized meshfree numerical scheme applied to scalar conservation laws. Renormalization is a tool introduced in order to eliminate the smoothed particle hydrodynamics (SPH) lack of consistency. A conservative scheme, the weak renormalized scheme, is derived from the general conservation laws weak formulation and is time discretized by using finite volume techniques. Because of the new form of the derivative approximation, the convergence proof of the classical SPH method cannot be applied. Thus, we use a Kruskov technique to obtain an $L^1$ norm comparison between the approximate solution and the solution of a regularized conservation law, the pseudoviscous problem.