  Mesh-free methods have significant potential for simulations in complex
geometries, as the time consuming process of mesh-generation is avoided.
Smoothed Particle Hydrodynamics (SPH) is the most widely used mesh-free method,
but suffers from a lack of consistency. High order, consistent, and local
(using compact computational stencils) mesh-free methods are particularly
desirable. Here we present a novel framework for generating local high order
difference operators for arbitrary node distributions, referred to as the Local
Anisotropic Basis Function Method (LABFM). Weights are constructed from linear
sums of anisotropic basis functions (ABFs), chosen to ensure exact reproduction
of polynomial fields up to a given order. The ABFs are based on a fundamental
Radial Basis Function (RBF), and the choice of fundamental RBF has small effect
on accuracy, but influences stability. LABFM is able to generate high order
difference operators with compact computational stencils (4th order with 25
nodes, 8th order with 60 nodes in two dimensions). At domain boundaries (with
incomplete support) LABFM automatically provides one-sided differences of the
same order as the internal scheme, up to 4th order. We use the method to solve
elliptic, parabolic and mixed hyperbolic-parabolic PDEs, showing up to 8th
order convergence. The inclusion of hyperviscosity is straightforward, and can
effectively provide stability when solving hyperbolic problems. LABFM is a
promising new mesh-free method for the numerical solution of PDEs in complex
geometries. The method is highly scalable, and for Eulerian schemes, the
computational efficiency is competitive with RBF-FD for a given accuracy. A
particularly attractive feature is that in the low order limit, LABFM collapses
to SPH, and there is potential for Arbitrary Lagrangian-Eulerian schemes with
natural adaptivity of resolution and accuracy.
