Finite Difference Solutionsto the ADE

t

c

x

cv

x

cD

2

2Simplest form of the ADE

t

c

x

cv

Even Simpler form

Plug FlowPlug Source

Flow Equationt

hS

x

hT

2

2

Effect ofNumerical Errors

(overshoot)

(MT3DMS manual)

t

c

x

cv

(See Zheng & Bennett, p. 174-181)

v

j-1 j j+1

x

x

t

cc

x

ccv

nj

nj

nj

nj

11 )(Explicit approximation

with upstream weighting

t

c

x

cv

t

cc

x

ccv

nj

nj

nj

nj

11 )(Explicit;

Upstream weighting

(See Zheng & Bennett, p. 174-181)

v

j-1 j j+1

x

x

Example from Zheng &Bennett

v = 100 cm/h

l = 100 cm

C1= 100 mg/l

C2= 10 mg/l

With no dispersion,breakthrough occursat t = v/l = 1 hour

nj

nj

nj

nj ccc

l

tvc

)( 1

1

t

cc

x

ccv

nj

nj

nj

nj

11 )(

v = 100 cm/hrl = 100 cmC1= 100 mg/lC2= 10 mg/lt = 0.1 hr

Explicit approximation with upstream weighting

t

cc

x

ccv

nj

nj

nj

nj

111

11 )

2(

Implicit;central differences

t

cc

x

ccv

nj

nj

nj

nj

1111 )(

t

cc

x

ccv

nj

nj

nj

nj

111

1

)(

Implicit;upstream weighting

Implicit Approximations

= Finite Element Method

t

c

x

cv

x

cD

2

2

Governing Equationfor Ogata and Banks solution

j-1 j j+1

x

x

j-1/2 j+1/2

t

c

x

cv

x

cD

2

2Governing Equationfor Ogata and Banks solution

t

cc

x

ccv

x

cccD

nj

nj

nj

nj

nj

nj

nj

11

2

11)()

)(

2(

Finite difference formula:explicit with upstream weighting, assuming v >0

)()2()(

11121 n

jnj

nj

nj

nj

nj

nj cc

x

tvccc

x

tDcc

Solve for cj n+1

Stability Constraints for the 1D Explicit Solution(Z&B, equations 7.15, 7.16, 7.36, 7.40)

Courant Numberx

tvCr

Cr < 1

1)(

22

x

tv

x

tDStability Criterion

Peclet Numberx

D

xvPe

Controls

numerical dispersion& oscillation, see Fig.7.5

Co

Boundary Conditions

a “free massoutflow” boundary(Z&B, p. 285)

Specifiedconcentrationboundary

Cb= Co Cb= Cjj j+1j-1 j j+1j-1

Spreadsheet solution(on course homepage)

We want to write a general formof the finite difference equation allowing foreither upstream weighting (v either + or –) or central differences.

j-1 j j+1

x

x

j-1/2 j+1/2

Upstream weighting:

In general:

jjj ccc 12/1 1(

See equations7.11 and 7.17 inZheng & Bennett