Stencil and kernel optimisation for mesh-free very high-order generalised finite difference method

Abstract

Generalised Finite Difference Methods and similar mesh-free methods (Pointset method, Multipoint method) are based on three main ingredients: a stencil around the reference node, a polynomial reconstruction and a weighted functional to provide the relations between the derivatives at the reference node and the nodes of the stencil. Very few studies were dedicated to the optimal choice of the stencil together with the other parameters that could reduce the global conditioning of the system and bring more stability and better accuracy. We propose a detailed construction of the very high-order polynomial representation and define a functional that assesses the quality of the reconstruction. We propose and implement several techniques of optimisation and demonstrate the advantages in terms of accuracy and stability.