Numerical Modelling of Nonlinear Diffusion Phenomena on a Sphere

Abstract

A new method for the numerical modelling of physical phenomena described by nonlinear diffusion equations on a sphere is developed. The key point of the method is the splitting of the differential equation by coordinates that reduces the original 2D problem to a pair of 1D problems. Due to the splitting, while solving the 1D problems separately one from another we involve the procedure of map swap — the same sphere is covered by either one or another of two different coordinate grids, which allows employing periodic boundary conditions for both 1D problems, despite the sphere is, actually, not a doubly periodic domain. Hence, we avoid the necessity of dealing with cumbersome mathematical procedures, such as the construction of artificial boundary conditions at the poles, etc. As a result, second-order finite difference schemes for the one-dimentional problems implemented as systems of linear algebraic equations with tridiagonal matrices are constructed. It is essential that each split one-dimentional finite difference scheme keeps all the substantial properties of the corresponding differential problem: the spatial finite difference operator is negative definite, whereas the scheme itself is balanced and dissipative. The results of several numerical simulations are presented and thoroughly analysed. Increase of the accuracy of the finite difference schemes to the fourth approximation order in space is discussed.