# On Meshfree GFDM Solvers for the Incompressible Navier-Stokes Equations

## TL;DR

This paper critically examines meshfree GFDM schemes for incompressible Navier-Stokes equations, identifies their limitations in local mass conservation, and proposes a modified scheme that improves accuracy and divergence approximation.

## Contribution

It introduces a novel modification to a monolithic GFDM scheme that enhances local mass conservation and velocity divergence approximation.

## Key findings

- The new scheme outperforms existing methods in accuracy.
- Identifies key drawbacks of conventional meshfree schemes.
- Provides numerical evidence of improved divergence approximation.

## Abstract

Meshfree solution schemes for the incompressible Navier--Stokes equations are usually based on algorithms commonly used in finite volume methods, such as projection methods, SIMPLE and PISO algorithms. However, drawbacks of these algorithms that are specific to meshfree methods have often been overlooked. In this paper, we study the drawbacks of conventionally used meshfree Generalized Finite Difference Method~(GFDM) schemes for Lagrangian incompressible Navier-Stokes equations, both operator splitting schemes and monolithic schemes. The major drawback of most of these schemes is inaccurate local approximations to the mass conservation condition. Further, we propose a new modification of a commonly used monolithic scheme that overcomes these problems and shows a better approximation for the velocity divergence condition. We then perform a numerical comparison which shows the new monolithic scheme to be more accurate than existing schemes.