A local hermitian RBF meshless numerical method for the solution of multi-zone problems

Abstract

The local Hermitian interpolation (LHI) method is a strong-form meshless numerical technique in which the solution domain is covered by a series of small and heavily overlapping radial basis function (RBF) interpolation systems. Aside from its meshless nature and the ability to work on very large scattered datasets, the main strength of the LHI method lies in the formation of local interpolations, which themselves satisfy both boundary and governing PDE operators, leading to an accurate and stable reconstruction of partial derivatives without the need for artificial upwinding or adaptive stencil selection. In this work, an extension is proposed to the LHI formulation which allows the accurate capture of solution profiles across discontinuities in governing equation parameters. Continuity of solution value and mass flux is enforced between otherwise disconnected interpolation systems, at the location of the discontinuity. In contrast to other local meshless methods, due to the robustness of the Hermite RBF formulation, it is possible to impose both matching conditions simultaneously at the interface nodes. The procedure is demonstrated for 1D and 3D convection–diffusion problems, both steady and unsteady, with discontinuities in various PDE properties. The analytical solution profiles for these problems, which experience discontinuities in their first derivatives, are replicated to a high degree of accuracy. The technique has been developed as a tool for solving flow and transport problems around geological layers, as experienced in groundwater flow problems. The accuracy of the captured solution profiles, in scenarios where the local convective velocities exceed those typically encountered in such Darcy flow problems, suggests that the technique is indeed suitable for modeling discontinuities in porous media properties. © 2010 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 27: 1201–1230, 2011