Numerical Modelling of Nonlinear Diffusion Phenomena on a Sphere - Simulation and Modeling Methodologies, Technologies and Application - page 52

Numerical Modelling of Nonlinear Diffusion Phenomena
on a Sphere

Yuri N. Skiba and Denis M. Filatov

1 Centro de Ciencias de la Atmosfera (CCA), Universidad Nacional Aut onoma de Mexico
(UNAM), Av. Universidad #3000, Cd. Universitaria, C.P. 04510, Mexico D.F., Mexico
2 Centro de Investigaci on en Computaci on (CIC), Instituto Politecnico Nacional (IPN),
Av. Juan de Dios B atiz s/n, esq. Miguel Othon de Mendizabal, C.P. 07738, Mexico D.F., Mexico
[e-mail], [e-mail]

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 proce-
dure 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 do-
main. 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 dif-
ference scheme keeps all the substantial properties of the corresponding differen-
tial problem: the spatial finite difference operator is negative definite, whereas the
scheme itself is balanced and dissipative. The results of several numerical sim-
ulations are presented and thoroughly analysed. Increase of the accuracy of the
finite difference schemes to the fourth approximation order in space is discussed.

1 Introduction

The diffusion equation is relevant to a wide range of important physical phenomena and
has many applications [1,2,3,8]. The classical problem is the heat or pollution transfer
in the atmosphere. A more sophisticated example is the single-particle Schr odinger
equation which, formally speaking, is diffusion-like.
In many practical applications the diffusion equation has to be studied on a sphere,
and besides, in the most general case it is nonlinear. Hence, consider the nonlinear
diffusion equation on a sphere S = { ( λ, ϕ ): λ ∈ [0 , 2 π ) ,ϕ∈ ( −

π
2

, π

2 ) }

∂
∂λ

μT α
R cos ϕ

μT α

∂T
∂t =

1
R cos ϕ

∂T
∂λ

∂
∂ϕ

cos ϕ
R

∂T
∂ϕ

+ + f (1)

equipped with a suitable initial condition. Here T = T ( λ, ϕ, t ) ≥ 0 is the unknown
function, μT α is the diffusion coefficient, μ = μ ( λ, ϕ, t ) ≥ 0 is the amplification