Compact LABFM: a framework for meshless methods with spectral-like resolving power

TL;DR

This paper introduces a compact, meshless numerical scheme based on LABFM that enhances the resolving power and accuracy of PDE solutions in complex geometries, achieving spectral-like precision.

Contribution

The paper presents a novel compact LABFM-based meshless method that mimics finite-difference compact schemes, improving accuracy and resolving power for PDEs in complex geometries.

Findings

Significant accuracy improvements in high wavenumber solutions

Enhanced resolving power comparable to spectral methods

Validated through convergence tests and PDE solutions

Abstract

Meshless methods are often used in numerical simulations of systems of partial differential equations (PDEs), particularly those which involve complex geometries or free surfaces. Here we present a novel compact scheme based on the local anisotropic basis function method (LABFM), a meshless method which provides approximations to spatial operators to arbitrary polynomial consistency. Our approach mimics compact finite-differences by using implicit stencils to optimise the resolving power of each operator, whilst retaining diagonal dominance of the resulting global sparse linear system. The new method is demonstrated to provide improved approximations by a series of convergence tests and resolving power analysis, before solutions to canonical PDEs are computed. Significant gains in accuracy are observed, in particular for solutions containing high wavenumber components. Our compact meshfree method provides a pathway to high-order simulations of PDEs in complex geometries with spectral-like resolving power, and has the potential to lead to a step-change in the accuracy of numerical solutions to such problems.