A Conservative Mesh-Free Scheme and Generalized Framework for Conservation Laws

Abstract

We present a novel mesh-free scheme for solving partial differential equations. We first derive a conservative and stable formulation of mesh-free first derivatives. We then show that this formulation is a special case of a general conservative mesh-free framework that allows flexible choices of flux schemes. Necessary conditions and algorithms for calculating the coefficients for our mesh-free schemes that satisfy these conditions are also discussed. We include numerical examples of solving the one- and two-dimensional inviscid advection equations, demonstrating the stability and convergence of our scheme and the potential of using the general mesh-free framework to extend finite volume discretization to a mesh-free context.