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