The use of PDE centres in the local RBF Hermitian method for 3D convective-diffusion problems

Abstract

In this work an extension is proposed to the Local Hermitian Interpolation (LHI) method; a meshless numerical method based on interpolation with small and heavily overlapping radial basis function (RBF) systems. This extension to the LHI method uses interpolation functions which themselves satisfy the partial differential equation (PDE) to be solved. In this way, a much improved reconstruction of partial derivatives can be obtained, resulting in significantly improved accuracy in many cases. The implementation algorithm is described, and is validated via three convection-diffusion-reaction problems, for steady and transient situations. A Crank–Nicolson implicit time stepping technique is used for the time-dependent problems. In the proposed approach, a form of ‘analytical upwinding’ is implicitly implemented by the use of the partial differential operator of the governing equation in the interpolation function, which includes the desired information about the convective velocity field. The implicit upwinding scheme intrinsic to the proposed numerical approach is tested by solving a one-dimensional travelling front problem at Péclet numbers of 500, 1000, 2000, 5000 and infinity, which corresponds to a shock front in the case of infinity. In addition, the accuracy of the numerical scheme is validated against a one-dimensional steady state solution exhibiting strong boundary layer effects, and also against a steady and a transient three-dimensional convection-diffusion problem on irregular datasets. All the test cases are validated against the corresponding analytical solutions. Finally, the effect of various interpolation stencil configurations is investigated, and some important limitations on local data-centre distribution are identified.