Numerical Modelling of Nonlinear Diffusion Phenomena on a Sphere - Simulation and Modeling Methodologies, Technologies and Application - page 59

Fig. 3. Experiment 1: numerical solution at several time moments (a nonmonotonic colour map
is used for better visualisation)

Experiment 2. Consider a more intricate problem, e.g., a nonlinear diffusion process
in a heterogeneous medium. For this we set α =1 and complicate the problem, taking
μ ( λ, ϕ ) ∼ sin 3 λ

2 sin 2 ϕ.

(31)

Let the initial condition be the same spot, but now placed on the North pole. The
anisotropy of the medium occurs because the diffusion process is taking place in a lon-
gitudinal sector — since the asymmetry is concentrated in the diffusion factor μ ,weare
expecting intensive diffusion at those λ 's (at a fixed latitude) where μ has a maximum,
while poor diffusion is expected where μ is almost zero (Fig. 5).
The numerical solution shown in Fig. 6 confirms the expectations. Indeed, a strong
diffusion process is being observed at λ ≈ π , while weak diffusion is taking place at
λ ≈ 0 , which is consistent with the profile of the diffusion factor (31).
Experiment 3. The aim of this experiment is to investigate the accuracy of the nu-
merical solution to the split linearised problem that approximates the original unsplit
nonlinear differential equation. Besides, of practical interest is the question of large
gradients of the solutions that may appear in real physical problems. Therefore, we in-
crease the nonlinearity of the problem up to α =2 and compare the numerical solution
versus the analytical one

T ( λ, ϕ, t )= c 1 sin ξ cos ϕ cos 2 t + c 2 , (32)