Finite Difference Solutionsto the ADE

tc

xcv

xcD

2

2Simplest form of the ADE

tc

xcv

Even Simpler form

Plug FlowPlug Source

Flow EquationthS

xhT

2

2

Effect ofNumerical Errors

(overshoot)

(MT3DMS manual)

tc

xcv

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

vj-1 j j+1

x

x

tcc

xcc

vnj

nj

nj

nj

11 )(Explicit approximation

with upstream weighting

tc

xcv

tcc

xcc

vnj

nj

nj

nj

11 )(Explicit;

Upstream weighting

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

vj-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 = l/v = 1 hour

nj

nj

nj

nj ccc

ltvc

)( 1

1

tcc

xcc

vnj

nj

nj

nj

11 )(

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

Explicit approximation with upstream weighting

tcc

xcc

vnj

nj

nj

nj

111

11 )

2(

Implicit;central differences

tcc

xcc

vnj

nj

nj

nj

1111 )(

tcc

xcc

vnj

nj

nj

nj

111

1

)(

Implicit;upstream weighting

Implicit Approximations

= Finite Element Method

tc

xcv

xcD

2

2

Governing Equationfor Ogata and Banks solution

j-1 j j+1

x

x

j-1/2 j+1/2

Central difference approximation

tc

xcv

xcD

2

2Governing Equationfor Ogata and Banks solution

tcc

xcc

vx

cccD

nj

nj

nj

nj

nj

nj

nj

11

211 )()

)(

2(

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

)()2()( 1112

1 nj

nj

nj

nj

nj

nj

nj cc

xtvccc

xtDcc

Solve for cj n+1

21

)( 2 xtD

1xtv

1)(

22

xtv

xtD

Stability Criterion for Explicit Approximation

For dispersion alone

For advection alone(Courant number)

For both

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

Courant NumberxtvCr

Cr < 1

1)(

22

xtv

xtD

Stability Criterion

Also need to minimize numerical dispersion.

Numerical Dispersion controlled by theCourant Number and the Peclet Number

for all numerical solutions (both explicit and implicit approximations)

Courant NumberxtvCr

Cr < 1

Peclet Numberx

DxvPe

Controlsnumerical dispersion& oscillation, see Fig.7.5

2Pe

Co

Boundary Conditions

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

SpecifiedconcentrationboundaryCb= Co Cb= Cj

j j+1j-1 j j+1j-1

Spreadsheet solution(on course homepage)

Co a “free massoutflow” boundary

SpecifiedconcentrationboundaryCb= Co

Cb= Cj

tc

xcv

xcD

2

2

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