A Flux Conserving Meshfree Method for Conservation Laws
It addresses the lack of conservation in meshfree methods, a key drawback for computational scientists using these techniques.
This paper introduces a modification to meshfree generalized finite difference methods to enforce approximate conservation of numerical fluxes, reducing conservation errors in simulations of advection-diffusion and incompressible Navier-Stokes equations.
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.