An adaptive meshfree diffusion wavelet method for partial differential equations on the sphere

Abstract

An adaptive meshfree diffusion wavelet method for solving partial differential equations (PDEs) on the sphere is developed. Approximation formulae for Laplacian–Beltrami (∇2) and gradient (∇→) operators are derived using radial basis functions (RBFs), and the convergence of these approximations to ∇2 and ∇→ is verified for two test functions. The matrix approximating the Laplace–Beltrami operator is used for the construction of the diffusion wavelet. The diffusion wavelet is used for the adaptation of node arrangement as well as for the fast computation of dyadic powers of the matrices involved in the numerical solution of the PDE. The efficiency of the method is that the same operator is used for the construction of the diffusion wavelet and for the approximation of the differential operators. As a part of the wavelet method the behaviour of the compression error with respect to different parameters involved in the construction of the diffusion wavelet is tested. The CPU time taken by the proposed method is compared with the CPU time taken by the RBF based collocation method and it is observed that the proposed method performs better. The method is tested on three test problems namely spherical diffusion equation (linear), problem of computing a moving steep front (nonlinear) and problem of Turing patterns (system of nonlinear reaction–diffusion equations).