Solution of the linear and nonlinear advection–diffusion problems on a sphere

Abstract

AbstractVarious linear advection–diffusion problems and nonlinear diffusion problems on a sphere are considered and solved using the direct, implicit and unconditionally stable finite‐volume method of second‐order approximation in space and time. In the absence of external forcing and dissipation, the method preserves the total mass of the substance and the norm of the solution. The component wise operator splitting allows us to develop the direct (noniterative) and fast numerical algorithm. The split problems in the longitudinal direction are solved using the Sherman‐Morrison formula and Thomas algorithm. The direct solution of the split problems in the latitudinal direction requires the use of the bordering method for a block matrix, and the preliminary determination of the solution at the poles. The resulting systems with tridiagonal matrices are solved by the Thomas algorithm. The numerical experiments demonstrate that the method correctly describes the local advection–diffusion processes on the sphere, in particular, through the poles, and accurately simulate blow‐up regimes (unlimited growing solutions) of nonlinear combustion, the propagation of nonlinear temperature and spiral waves, and solutions to Gray‐Scott reaction–diffusion model.