An implicit direct unconditionally stable numerical algorithm for the solution of advection-diffusion equation on a sphere

A new algorithm, proposed for solving linear and nonlinear advection-diffusion problems on a sphere, is tested with various numerical experiments. The velocity field on the sphere is non-divergent, and assumed to be known. Discretization of the advection-diffusion equation in space is performed by the finite volume method using the Gauss theorem for each grid cell. For the discretization in time, the symmetrized double-cycle componentwise splitting method and the Crank-Nicolson scheme are used. The one-dimensional periodic problems arising at splitting in the longitudinal direction are solved with Sherman-Morrison's formula and Thomas's algorithm. The one-dimensional problems arising at splitting in the latitudinal direction are solved by the bordering method that requires a prior determination of the solution at the poles. The resulting linear systems have tridiagonal matrices and are solved by Thomas's algorithm. The algorithm is of second order approximation in space and time. It is implicit, unconditionally stable, direct (without iterations) and rapid in realization. The theoretical results are confirmed numerically by simulating various linear and nonlinear advection-diffusion processes. The tests show high accuracy and efficiency of the method that correctly describes the advection-diffusion processes and the mass balance of a substance in a forced and dissipative discrete system. In addition, in the absence of external forcing and dissipation, it conserves both the total mass and the norm of solution.