Comparisons of Two RBF-FD Approaches Based on Polyharmonic Splines for Solving the Shallow Water Equations on the Sphere

In the last decade, the radial basis function-generated finite difference (RBF-FD) method has been widely used for numerical solutions of partial differential equations on smooth surfaces. In this paper, we present a comparison between two approaches for approximating spherical differential operators using the RBF-FD method based on polyharmonic splines. First, we compare common differential operators, including the surface gradient and the Laplacian operator on the sphere, employing polyharmonic splines with different degrees of augmented multivariate polynomials. Second, we examine the approximation errors for various node layouts and spatial resolutions. In most cases, the RBF-FD method with a constant along the normal direction yields more accurate results. To maintain consistency with the high-order RBF-FD spatial discretization, we employ high-order time integration methods, including two Runge–Kutta schemes and exponential Rosenbrock schemes. Finally, we apply the method to solve the spherical shallow water equations using a series of standard test cases, such as steady geostrophic flow, zonal flow over an isolated mountain, Rossby–Haurwitz wave propagation and perturbed zonal jet flow. The simulation results demonstrate that the proposed method is capable of reproducing the evolution of nonlinear dynamics successfully.