A Flux Conserving Meshfree Method for Conservation Laws

TL;DR

This paper introduces a novel meshfree method that conserves flux locally, improving the accuracy of conservation laws in meshfree finite difference methods without relying on a global mesh.

Contribution

It proposes a new modification to classical GFDMs to include local flux balances, enhancing conservation properties within a meshfree framework.

Findings

Significantly reduces conservation errors in simulations.

Successfully applied to advection-diffusion and Navier-Stokes equations.

Demonstrates improved flux conservation compared to traditional GFDMs.

Abstract

Lack of conservation has been the biggest drawback in meshfree generalized finite difference methods (GFDMs). In this paper, we present a novel modification of classical meshfree GFDMs to include local balances which produce an approximate conservation of numerical fluxes. This numerical flux conservation is done within the usual moving least squares framework. Unlike Finite Volume Methods, it is based on locally defined control cells, rather than a globally defined mesh. We present the application of this method to an advection diffusion equation and the incompressible Navier - Stokes equations. Our simulations show that the introduction of flux conservation significantly reduces the errors in conservation in meshfree GFDMs.