Numerical Modelling of Nonlinear Diffusion Phenomena on a Sphere - Simulation and Modeling Methodologies, Technologies and Application - page 58

Theorem 3. Finite difference schemes (12)-(13) are dissipative.

Proof. Consider (13). Multiplying both sides by RΔϕT l | cos ϕ l |

and summing all over
the nodes l 's, given the Crank-Nicolson approximation (15) we obtain

f n + 2
l

||T n +1 || 2 −||T n + 2 || 2
2 τ

= A Δϕ T l ,T l + ,T l . (29)

2

denotes the scalar product on S (2) Here

Δλ,Δϕ at a fixed λ k . Due to the negative
definiteness of the operator A Δϕ (Theorem 2) the first summand on the right-hand side
of (29) is less than or equal to zero. Consequently, if f n + 2
l

·, ·

=0 then

||T n +1 || 2 ≤||T n + 2 || 2 ,

(30)

that is the solution's L 2 -norm decays in time.
Analogous calculations can be performed for (12).

Corollary 1. Finite difference schemes (12)-(13) on grids (10)-(11) are absolutely sta-
ble in the corresponding L 2 -norms.

All the statements of the aforegiven theorems and corollary 1 are true for problem
(20)-(23).

3 Numerical Experiments

Our purpose is now to test the developed method. For this we shall numerically sim-
ulate several diffusion phenomena. We shall start from the simplest linear model, then
consider a little more complicated nonlinear case, and finally prove our schemes on a
highly nonlinear diffusion model.
Experiment 1. First we have to verify how the idea of the map swap works — whether
nonphysical (purely computational) effects appear near the poles due to the convergence
of meridians. To do this, we set α =0 , μ constant, f =0 and take the initial condition
in the form of a disc-like spot located near a pole. From the theory it is known that the
spot should isotropically propagate from the centre in all possible directions without
any disturbance in the disc's shape.
The numerical solution on the grid 6 ◦ × 6 ◦ at a few time moments is shown in
Fig. 3. In Fig. 4 we also plot the solution's L 2 -norm in time. The solution is seen to be
consistent with what we have been expecting — the spot is isotropically spreading over
the sphere and no visual disturbance of the spot's shape is observed while passing over
the North pole. Yet, the graph of the L 2 -norm proves the dissipativity of the schemes
(cf. Theorem 3).
Therefore, we may conclude that the procedure of the map swap is mathematically
correct and the diffusion process through the poles is being simulated physically ade-
quate on the shifted grids (10)-(11).