Numerical Simulation of the Convection–Diffusion PDEs on a Sphere with RBF-FD and RBF-QR Methods

Abstract

The aim of this paper is to investigate the application of radial basis function-generated finite difference (RBF-FD) methods for convection–diffusion partial differential equations (PDEs) on a sphere. In the application of RBF-FD method, choosing a reasonable value of shape parameter is important to the computation of PDEs. The work is devoted to the numerical study of the range of near optimal shape parameters for the convection–diffusion equations. Because the RBF-FD Direct method often leads to ill-conditioned problems for small shape parameters, the RBF-QR method is applied locally to overcome the ill-conditioning in the context of RBF-FD mode. Additionally, for convection-dominated problems, it can be found that the results of using central-type stencil present spurious oscillations. Therefore, we propose an upwind RBF-FD (URBF-FD) scheme to overcome the problems, which is well adapted to the problems on the sphere and easy to be implemented. Further numerical results show that the proposed URBF-FD method is stable and effective for convection-dominated PDEs on the sphere.