A stable radial basis function partition of unity method for solving convection–diffusion equations on surfaces

Abstract

The radial basis function partition of unity (RBF-PU) method is used to solve various types of partial differential equations, since it is based on scattered node sets that can be easily implemented. In this paper, a framework for the RBF-PU method is proposed to solve the convection–diffusion equations on closed surfaces. The surface gradient and Laplace–Beltrami operators are discretized based on the RBF-PU method using the polyharmonic spline radial basis functions augmented with polynomials. Since the RBF-PU method is a local meshless method, an advantage is that the differential matrix obtained by discretizing the surface differential operator is sparse. Additionally, solving convection-dominated problems or transport equations directly using RBF-PU method can produce numerical oscillations. For this reason, we add hyperviscosity term to improve the stability of the RBF-PU method for solving the convection–diffusion equations on surfaces, and numerical experiments show that such a combination has higher-order convergence rates. Finally, we solve the discontinuous source problem on the torus and the Turing system with convection term on different surfaces to demonstrate the effectiveness of the proposed method.