finite difference solutions to the ade. simplest form of the ade even simpler form plug flow plug...
DESCRIPTION
Effect of Numerical Errors (overshoot) (MT3DMS manual)TRANSCRIPT
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