A Strategy for Bioremediation of Marine Shorelines by Using Several Nutrient Release Points - Mathematical Modelling and Numerical Simulation of Oil Pollution Problems - page 40

n n

A 3

n n

1
3

1
3

1
3

1
3

ʦ + − ʦ − =− ˄ ʦ + + ʦ − + 2 ˄ q [ n ]

(2.45)

n n

=− 2 A i

n

4 − i 3 − i 4 − i

ʦ + − ʦ + ʦ +

3 3 3

n

3 − i

+ ʦ + ( i = 2 , 1 )

3

and
G n G n A i ) ∗ G n

3 − i 4 − i

=− 2 (

3 − i

+ − + +

3 3 3

G n

4 − i

+ + ( i = 1 , 2 )

3

G n G n A 3 ) ∗ G n G n

1
3

1
3

1
3

1
3

− − + =− ˄( − + + + 2 ˄ p [ n ]

(2.46)

G n G n A i ) ∗ G n

4 − i 3 − i

=− 2 (

4 − i

− − − −

3 3 3

G n

3 − i

+ − ( i = 2 , 1 ),

3

where and g

at fractional time steps, and q and p are the vectors representing the grid values of
functions Q and P at moment t n , respectively [ 41 ]. The discretization in time of
each one-dimensional split problem is performed with the Crank-Nicolson scheme,
and the resulting discrete problem is efficiently solved by the Thomas' factorization
method for the tridiagonal matrices [ 24 ]. The unconditional stability of the numerical
schemes ( 2.45 ) and ( 2.46 ) directly follows from the inequalities

ʦ and G are the vectors representing the grid values of solutions ˆ

ʦ [ n + 1 ] ≤ ʦ [ n − 1 ] + 2 ˄ q [ n ] (2.47)

and

G [ n − 1 ] ≤ G [ n + 1 ] + 2 ˄ p [ n ] , (2.48)

where is the Euclidean vector norm [ 41 ]. The use of Lagrange identity leads to

the equation

·

ʦ

n n

1
3

1
3

G ∗ [ p ∗ [

+ ] ʦ [ + ]+ ˄ ] + + ʦ −

n 1 n 1 n

G ∗ n

G ∗ n

q [ n ]

1
3

1
3

G ∗ [ = ˄ + − − + n − 1 ] ʦ [ n − 1 ]

(2.49)