A compact 9 point stencil based on integrated RBFs for the convection-diffusion equation

摘要

In this paper, a high-order compact stencil for solving the convection-diffusion equation in two dimensions is proposed. The convection and diffusion terms are both approximated by means of radial basis functions (RBFs) that are constructed over 3×3 rectangular stencils. Salient features here are that (ⅰ) integration is employed to construct local RBF approximations; and (ⅱ) through the constants of integration, values of the convection-diffusion equation at several selected nodes on the stencil are also enforced. Numerical results indicate that (ⅰ) the inclusion of the governing equation into the stencil leads to a significant improvement in accuracy; (ⅱ) when the convection dominates, accurate solutions are obtained at a regime of the RBF width which makes the RBFs peaked; and (ⅲ) high levels of accuracy are achieved using relatively coarse grids.