2 A Strategy for Bioremediation of Marine Shorelines …
37
that relates a variation δ J i (φ) in the mean concentration of nutrients in Ω i with
variations δ Q j and δφ 0 in the emission rates Q j and initial distribution of nutrient
φ 0 . It makes the estimates (2.25) and (2.35) rather efficient and computationally
economical, because the solutions g i (r j , t), once found, can be re-used in these
formulas for different values of Q j , r j or φ 0 (r ).
The effect of changing the position of sources from r j to r ´
j , j = 1, . . . , N , is
estimated by the formula
δ J i (φ) =
N
j=1
T
0
g i (r
´
j , t) − g i (r j , t)
Q j (t) dt.
(2.36)
Finally, we give without proof a general sensitivity formula
δ J i (φ) =
N
j=1
T
0
g i (r j , t)δ Q j (t) dt +
D
g i (r, 0)δφ
0
(r ) dr
−
T
0
S T
g i (r, t)φ(r, t)δζ (r, t) d Sdt −
T
0
D
g i (r, t)B(r, t) drdt,
(2.37)
where
B(r, t) = δU · ∇φ − ∇ · δμ∇φ + δσ φ + δv s
∂φ
∂z
,
cf. [43], taking into account arbitrary variations δ Q j (t) and δφ 0 (r ), and small variations δU, δσ , δμ, δv s and δζ in the domain D. Unlike the previous formulas, estimate
(2.37) is more complicated, because it uses solutions of both problems (2.4)–(2.11)
and (2.18)–(2.24) and linearised equations for perturbations.
2.5 Main and Adjoint Numerical Schemes of the Dispersion
Problem
In this section, balanced and absolutely stable second-order finite diference schemes
based on the application of the splitting method by Marchuk [22] and Crank-Nicolson
schemes [8] are developed to solve numerically the dispersion model (2.4)–(2.11)
and its adjoint formulation (2.18)–(2.24). Since they were described in detail in Skiba
[41], we give here only basic results.
Using the continuity Eq. (2.11), the operator A of Eq. (2.4) can be written as
A = A 1 + A 2 + A 3 , where
37
that relates a variation δ J i (φ) in the mean concentration of nutrients in Ω i with
variations δ Q j and δφ 0 in the emission rates Q j and initial distribution of nutrient
φ 0 . It makes the estimates (2.25) and (2.35) rather efficient and computationally
economical, because the solutions g i (r j , t), once found, can be re-used in these
formulas for different values of Q j , r j or φ 0 (r ).
The effect of changing the position of sources from r j to r ´
j , j = 1, . . . , N , is
estimated by the formula
δ J i (φ) =
N
j=1
T
0
g i (r
´
j , t) − g i (r j , t)
Q j (t) dt.
(2.36)
Finally, we give without proof a general sensitivity formula
δ J i (φ) =
N
j=1
T
0
g i (r j , t)δ Q j (t) dt +
D
g i (r, 0)δφ
0
(r ) dr
−
T
0
S T
g i (r, t)φ(r, t)δζ (r, t) d Sdt −
T
0
D
g i (r, t)B(r, t) drdt,
(2.37)
where
B(r, t) = δU · ∇φ − ∇ · δμ∇φ + δσ φ + δv s
∂φ
∂z
,
cf. [43], taking into account arbitrary variations δ Q j (t) and δφ 0 (r ), and small variations δU, δσ , δμ, δv s and δζ in the domain D. Unlike the previous formulas, estimate
(2.37) is more complicated, because it uses solutions of both problems (2.4)–(2.11)
and (2.18)–(2.24) and linearised equations for perturbations.
2.5 Main and Adjoint Numerical Schemes of the Dispersion
Problem
In this section, balanced and absolutely stable second-order finite diference schemes
based on the application of the splitting method by Marchuk [22] and Crank-Nicolson
schemes [8] are developed to solve numerically the dispersion model (2.4)–(2.11)
and its adjoint formulation (2.18)–(2.24). Since they were described in detail in Skiba
[41], we give here only basic results.
Using the continuity Eq. (2.11), the operator A of Eq. (2.4) can be written as
A = A 1 + A 2 + A 3 , where