A high order meshless method with compact support

Abstract

In-spite of their advantages in handling complex geometries, meshless methods have been restricted very much to applications at low spatial order, typically of the second-order. This is because the conventional meshless methods would require a very large supporting set of nodes when high-order interpolants are used, and this could pose problems for neighborhood and nodal selection in complex domains. In this work, a new meshless method is proposed for obtaining high-order approximation on a highly compact nodal set. Whereas conventional meshless methods utilize the lowest-order solution data to reconstruct the solution and its derivatives in the neighborhood of the reference node, the present method increases the usage of available information at the nodes by interpolating over nodal set of vector data. The utilization of the increased degrees of freedom and corresponding high-order information at nodes results in a significant reduction in the number of support nodes needed for solution reconstruction. High-order accuracy of up to the 6th order is demonstrated for problems in linear advection, diffusion and Poisson equation. The scheme is also applied to incompressible viscous fluid flows governed by the nonlinear system of Navier–Stokes equations. In all cases, nine support nodes (including the reference node) are used for data interpolation and spatial reconstruction.