Explicit Approximation
description
Transcript of Explicit Approximation
t
hh
T
S
x
hhh ni
ni
ni
ni
ni
1
211
)(
2
Explicit Approximation
t
h
T
S
x
h
2
2
t
hh
T
S
x
hhh ni
niiii
1
211
)(
2
t
hh
T
S
x
hhh ni
ni
ni
ni
ni
1
211 2
Explicit Solution
))(
2(
2111
x
hhh
S
tThh
ni
ni
nin
ini
Eqn. 4.11(W&A)
Everything on the RHS of the equation is known.Solve explicitly for ; no iteration is needed.1n
ih
Explicit approximations are unstable unless small time steps are used.
Problems with explicit solution:1. Requires small time step2. Unnatural propagation of boundary effect3. Large mass balance error for some time steps suggests t = 5 minutes is too large.
211
2
11
111
)(
21(
)(
2
x
hhh
x
hhh ni
ni
ni
ni
ni
ni
2
2
x
h
where = 1 for fully implicit = 0.5 for Crank-Nicolson = 0 for explicit
In general:
t
hh
T
S
x
hhh ni
ni
ni
ni
ni
1
2
11
111
)(
2
Implicit Approximation
t
h
T
S
x
h
2
2
t
hh
T
S
x
hhh ni
niiii
1
211 2
t
hh
T
S
x
hhh ni
ni
ni
ni
ni
1
2
11
111 2
Solve for 1nih and use Gauss-Seidel iteration.
11)( mnih },)(,){( 1
111
1ni
mni
mni hhhfunction
Implicit Solution
tIterationplanes
n
n+1
m+2
m+1
m+3
Implicit Solution
t = 5
Note: at t=5 min, the boundary effect is propagatedpast the first node near the boundary.
Implicit solution
Explicit solution
Computational molecules
t = 10
Compare with matrix solution givenin directions for Problem Set 3.
t = 5
Note: at t=5 min, the boundary effect is propagatedpast the first node near the boundary.
t = 1
t = 0.5
t Implicit
Solution
10. 14.09
5.0 13.89
1.0 13.68
0.5 13.65
0.1 13.62
t = 10 minutes
t Implicit
Solution
5.0 14.66
1.0 14.44
0.5 14.41
0.1 14.38
0.01 14.37
t = 5 minutes
Sensitivity to time step at x = 90 m
Explicit
solution
13.50
14.28
14.33
14.36
14.36
Explicit
solution
--
13.50
13.56
13.59
13.61
Note small water balance errort = 5
Use of a time step multiplier
Most transient problems will “shock” the system atthe beginning of the simulation. The shock could be adrop in water level or the start of pumping, for example.
The system will respond rapidly to the “shock” and tocapture this rapid response it is necessary to use smalltime steps.
Such small time steps are not necessary later in thesimulation. Hence, a time step multiplier increasesthe size of the time step as the solution progresses.
tnew = told x MULT
where MULT is the time step multiplier,e.g., 1.2
Use of a time step multiplier
Could also solve the implicitfinite difference equation using SOR iteration.
])()[( )()( 111111 mni
mni
mni
mni hhhh
Gauss-Seidel value
Another option for solving the implicit finite difference eqn.is a “direct solution” using matrix methods. All known terms are on the RHS; all unknown terms are on the LHS.
ni
ni
ni
ni
ni hhhhh
xS
tT
11
111
12]2[
)(
t
hh
T
S
x
hhh ni
ni
ni
ni
ni
1
2
11
111 2
ni
ni
ni
ni h
tT
xShh
tT
xSh
21
11
21
1)(
))(
2(
ni
ni
ni
ni
ni h
tT
xSh
tT
xShhh
21
21
111
1)()(
]2[
Solve this equation using matrix methods. See W&A, p. 95