Local polynomial regression and filtering for a versatile meshfree PDE solver

A high-order, meshfree finite difference method for solving partial differential equations is presented. The spatial domain is sampled by an irregular point cloud, and at each point a local polynomial approximation is formulated. This polynomial is used to approximate both function and its derivatives. The function approximation is used as a filter to regularise the solution and ensure stability of the method.<br/><br/>The local polynomial is obtained by straightforward least squares minimisation, which is solved by using the singular value decomposition method, with truncation. A non-parametric regression with robust M-estimator functions is also investigated and exhibits additional robustness and accuracy. A straightforward semi-discrete form is obtained by adopting an explicit time discretisation, and combined with the local polynomial regression, a modular assembly of complete solver algorithms is possible.<br/><br/>The resulting numerical method is accurate, simple and versatile. Challenging numerical benchmark tests are investigated, namely the solution to inviscid Burgers' equation, linearised Euler equations and incompressible Navier-Stokes equations (with the artificial compressibility and vorticity transport formulations) in Eulerian and Lagrangian reference frames.