Numerical Modelling of Nonlinear Diffusion Phenomena on a Sphere - Simulation and Modeling Methodologies, Technologies and Application - page 56

where D = D in (12) and D = D| cos ϕ| in (13), as well as

D i± 1 + D i
2

D i± 1 / 2 := . (14)

(We omitted the nonvarying index l in (12) and the nonvarying index k in (13) for
clarity.) As for the function T kl , we involve the Crank-Nicolson approximation

T n + 2

kl + T n + p− 1

2

kl

T kl := , (15)

2

where p =1 for (12) and p =2 for (13). Substituting (15) into (12)-(13), we come to
the systems of linear algebraic equations in λ

1

τ + m k

− T n + 2
k +1

m k +1 + T n + 2 − T n + 2

k− 1

m k− 1

k

= T k +1 m k +1 + T k 1

τ − m k

f n + 2
k

+ T k− 1 m k− 1 + , (16)

2

where

D k +1 / 2 + D k− 1 / 2
2 R 2 cos 2 ϕ l Δλ 2

D k + j/ 2
2 R 2 cos 2 ϕ l Δλ 2 ,

m k = , k + j = j = ± 1 , (17)

and in ϕ

1

τ + m l

− T n +1
l +1

m l +1 + T n +1 − T n +1

l− 1

m l− 1

l

1

τ − m l

f n + 2
l

= T n + 2
l +1

m l +1 + T n + 2 + T n + 2

l− 1

m l− 1 + , (18)

l

2

where

D l +1 / 2 + D l− 1 / 2
2 R 2 | cos ϕ l |Δϕ 2

D l + j/ 2
2 R 2 | cos ϕ l |Δϕ 2 ,

m l = , l + j = j = ± 1 .

(19)

Using the procedure of bicyclic splitting [4]

T n + 4
kl

f n + 4
k

− T kl
τ/ 2

= A Δλ T kl + , (20)

2

T n + 4
kl

− T n + 4
kl

f n + 4
l

= A Δϕ T kl + ,

(21)

τ/ 2 2

T n + 4
kl

− T n + 4
kl

f n + 4
l

= A Δϕ T kl + , (22)

τ/ 2 2

− T n + 4
kl

f n + 4
k

T n +1
kl

= A Δλ T kl + , (23)

τ/ 2 2