Jack R. C. King

Lucas Gerken Starepravo, Georgios Fourtakas, Steven Lind et al.

Mesh-free numerical methods provide flexible discretisations for complex geometries; however, classical meshless discrete differential operators typically trade low computational cost for limited accuracy or high accuracy for substantial per-stencil computation. We introduce a parametrised framework for learning mesh-free discrete differential operators using a graph neural network trained via polynomial moment constraints derived from truncated Taylor expansions. The model maps local stencils relative positions directly to discrete operator weights. The current work demonstrates that neural networks can learn classical polynomial consistency while retaining robustness to irregular neighbourhood geometry. The learned operators depend only on local geometry, are resolution-agnostic, and can be reused across particle configurations and governing equations. We evaluate the framework using standard numerical analysis diagnostics, showing improved accuracy over Smoothed Particle Hydrodynamics, and a favourable accuracy-cost trade-off relative to a representative high-order consistent mesh-free method in the moderate-accuracy regime. Applicability is demonstrated by solving the weakly compressible Navier-Stokes equations using the learned operators.

Henry M. Broadley, Steven J. Lind, Jack R. C. King

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.