SIC^2: Simulation and Integration of Control for Canals

Semi-implicit discretization

David Dorchies — 2009-10-22T15:33:51Z

Saint Venant's equations have no known analytical solution in real geometry. They are solved numerically by discretizing the equations: the partial derivatives are replaced by finite differences. Various solution schemes may be used to provide a solution to these equations. The discretization scheme chosen in the SIC model is a four-point semi-implicit scheme known as Preissmann's scheme (Figure 26).
This scheme is implicit because the values of the variables at the unknown time step also (...)

- Semi-implicit discretization

General equations of the scheme

David Dorchies — 2009-10-22T15:33:51Z

Figure 26

Let:
Dfi = fA' - fA fA = fi

Dfj = fB' - fB fB = fj

The expression of a function in M may be written as:

fM = (1-Q) + Q

fM = + (Dfi + Dfj) [31]

The derivative () at point M is written as:

()M = (1-Q) + Q

=> ()M = + Q [32]

The derivative () at point M is written as:

()M = ( + )

=> ()M = [33]

Equations [31], [32] and [33] are used to discretize St. Venant's equations.

Continuity equations

David Dorchies — 2009-10-22T15:33:51Z

Discretize each term of equation [28]:
= DAi » Bi.DZi
=> =
= + Q
q = = qij
Equation [28], then becomes:
+ + Q =
+ + DQj - DQi = (qi + qj)
DQi - Bi.DZi = DQj + Bj.DZj + - (qi + qj)
which is of the form:
A21.DQi + A22.DZi = B21.DQj + B22.DZj + B23 [34]
A21 = 1 B21 = 1
A22 = - Bi B22 = Bj
B23 = - (qi + (...)

- Semi-implicit discretization

Dynamic equation

David Dorchies — 2009-10-22T15:33:51Z

Discretize each term of the equation [30]:

* =

calculation of :

= [ - ] + [D( ) - D( )]

= [ - ] + [( - DAj) - ( - DAi)]

= [ - ] + DQj - DQi + BiDZi - BjDZj)

* = a1.DQi + a2.DQj + a3.DZi + a4.DZj + a5

with:
a5 = [ - ]

a4 = - Bj

a3 = Bi

a2 =

a1 = -

Calculation of gA :

gA = g [ + Q].[]

= [(Ai + Aj)(Zj - Zi) + Q(DAi + DAj)(Zj - Zi) + Q(DZj - DZi)(Ai + Aj)]

= [(Ai + Aj)(Zj-Zi) + Q(BiDZi + BjDZj)(Zj-Zi) + Q(DZj - DZi)(Ai + Aj)]

* gA = [a6DZi + a7DZj + a8]

with: a6 = Q[Bi (Zj - Zi) - (Ai + Aj)]

a7 = Q[Bj (Zj - Zi) + (Ai + Aj)]

a8 = (Ai + Aj).(Zj - Zi)

calculation of gASf:

gASf = g [ + [D(AiSfi) + D(AjSfj)]]

= g [ + [SfiDAi + AiDSfi + SfjDAj + AjDSfj]]

Calculate DSf:

Sf = = sgn(Q)

with: sgn(Q) = 1 if Q > 0
sgn(Q) = -1 if Q < 0

=> DSf = 2 sgn(Q).Q - 2 sgn (Q) .DDe

or DSf = 2 |Q| - 2 .DDe

De =

=> DDe = + R-1/3 DR

with:

=> DDe = B.DZ + R-1/3 a DZ

= ae DZ

DSf = 2 |Q| - 2 DDe

= 2 |Q| - 2 ae DZ

=> gASf = g [ + [(SfiBiDZi + SfjBjDZj)

+ Ai (2 |Qi| - 2 aei DZi)

+ Aj (2 |Qj| - 2 aej DZj)]]

* gASf = a9.DZi + a10.DZj + a11.DQi + a12.DQj + a13

with:

a9 = [SfiBi - Ai 2 aei]

a10 = [SfjBj - Aj 2 aej]

a11 = gQ

a12 = gQ

a13 = g []

Calculation of kq :

kq = [qi + qj + Q(D(qi ) + D(qj ))]

= [qi + qj + Q(qi - qi + qj - qj )]

* kq = a14.DZi + a15.DZj + a16.DQi + a17.DQj + a18

with:
a14 = - Q qi Bi

a15 = - Q qj Bj

a16 = Q

a17 = Q

a18 = [qi + qj ]

The dynamic equation [30] can then be written in the form:

A11.DQi + A12.DZi = B11.DQj + B12.DZj + B13 [35]

with:
A11 = + a1 + a11 - a16

A12 = a3 + a6 + a9 - a14

B11 = - ( + a2 + a12 - a17)

B12 = - (a4 + a7 + a10 - a15)

B13 = - (a5 + a8 + a13 - a18)