1. Explicit Approximation

2. Explicit Solution Eqn. 4.11 (W&A)
Everything on the RHS of the equation is known.
Solve explicitly for ; no iteration is needed.
Explicit approximations are unstable unless small time steps are used.

3. Problems with explicit solution:
Requires small time step
Unnatural propagation of boundary effect
Large mass balance error for some time steps suggests
t = 5 minutes is too large.

4. In general: where  = 1 for fully implicit  = 0.5 for Crank-Nicolson
 = for explicit

5. Implicit Approximation

6. Implicit Solution
Solve for
and use Gauss-Seidel iteration.

7. n+1 t
m+3
Iteration
planes
m+2
m+1 n

8. Implicit Solution
Spreadsheet

9. t = 5
Note: at t=5 min, the boundary effect is propagated
past the first node near the boundary.

10. Computational molecules
Explicit solution
Implicit solution

11. t = 10
Compare with matrix solution given
in directions for Problem Set 3.

12. t = 5
Note: at t=5 min, the boundary effect is propagated
past the first node near the boundary.

13. t = 1

14. t = 0.5

15. Sensitivity to time step at x = 90 m
t = 5 minutes
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
Explicit
solution
13.50
14.28
14.33
14.36 t
Implicit
Solution
10.
14.09
5.0
13.89
1.0
13.68
0.5
13.65
0.1
13.62
Explicit
solution --
13.50
13.56
13.59
13.61

16. Note small water balance error

17. Use of a time step multiplier
Most transient problems will “shock” the system at
the beginning of the simulation. The shock could be a
drop in water level or the start of pumping, for example.
The system will respond rapidly to the “shock” and to
capture this rapid response it is necessary to use small
time steps.
Such small time steps are not necessary later in the
simulation. Hence, a time step multiplier increases
the size of the time step as the solution progresses.

18. Use of a time step multiplier
tnew = told x MULT
where MULT is the time step multiplier,
e.g., 1.2

19. Could also solve the implicit
finite difference equation using SOR iteration.
Gauss-Seidel value

20. 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.
Solve this equation using matrix methods. See W&A, p. 95