A semi-Lagrangian radial basis function partition of unity closest point method for advection-diffusion equations on surfaces

Abstract

A semi-Lagrangian radial basis function partition of unity (RBF-PU) closest point method is designed for solving advection-diffusion equations on surfaces. This new meshfree method combines the semi-Lagrangian method with the RBF-PU closest point method. The semi-Lagrangian RBF-PU closest point method traces the departure point backwards along the velocity field based on patches at each time step. Therefore, it saves the computational cost compared with the semi-Lagrangian radial basis function finite difference (RBF-FD) closest point method. Our proposed RBF-PU closest point method for approximating the Laplace-Beltrami operator has two main advantages over the RBF-FD closest point method. Firstly, the RBF-PU closest point method to construct the local influence domain is not required to identify the size of the computational tube. Therefore, our method is more concise and easier to implement. Secondly, the RBF-PU closest point method not only improves the accuracy but also saves computational costs, and we confirm the advantage by testing differential accuracy. Numerical experiments verify the convergence and effectiveness of the semi-Lagrangian RBF-PU closest point method.