6 finite difference methods
TRANSCRIPT
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5. Finite difference methods
Finite difference formulae Dirichlet and Neumannproblems Elliptic, parabolic and hyperbolic equations
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1. Finite difference operators
Recall finite difference operators from numerical
differentiations Central difference
Forward / Backward difference
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2. Rectangular even grid
Grid size
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3. Taylors series expansion
3
,
33
2
,
22,
,,1
3
,
33
2
,
22,
,,1
!3!2!1
!3!2!1
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jijiji
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),(
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3.1 Mixed second-order derivative
hk
khO
hk
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khOkyhxukyhxu
kyhxukyhxuhku
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jijijijiji
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2
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],[],[4
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!1],[],[
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4. Uneven grid system
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5. Difference equations
in uneven grid system
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6. Solution technique
Replace derivatives by difference equations
Lay the derivatives on the grid Substitute the required BC
Solve the algebraic difference equation system
02
2
2
2
y
u
x
u
.04,if
022
,1,1,,1,1
2
1,,1,
2
,1,,1
jijijijiji
jijijijijiji
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7. Computational molecule
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2
1,,1,
2
,1,,1
jijijijiji
jijijijijiji
uuuuuhk
k
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uuu
Grid generation (even/uneven)
Apply computational molecule at each node Setup algebraic equations for entire grid
Insert/Deal with boundary conditions
Solve algebraic equations for each node
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7.1 Liemanns method
(Dirichlets problem)
.40,3
,20,0
,8,3
,10,0
;10
3
2
2
2
2
Ty
Ty
yTx
Tx
xyy
T
x
T
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1
104
2
,1,1,,1,1
kh
yxh
TTTTTii
jijijijiji
),2)(2()1(104:P
),2)(1()1(104:P
),1)(2()1(104:P
),1)(1()1(104:P
2
222332211222
2
121322110212
2
212231201121
2111221100111
TTTTT
TTTTT
TTTTT
TTTTT
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.40,20,64,8,10
conditionsboundary
23132010
32310201
TTTTTTTT
667.22
583.10
083.16
667.11
64
8
30
10
4110
1401
1041
0114
22
21
12
11
22
21
12
11
T
T
T
T
T
T
T
T
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7.2 Liemanns problem (Neumann BC)
.4]1,[
,0]0,[
,0],5.1[,0],0[
;02
2
2
2
xxy
u
xu
yuyu
y
u
x
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, , , 1
12 11 22 2112 22
11
12
21
22
11 12 21 22
Method 1: backward difference
P : 2 P : 40.5 0.5
4 1 1 0 0
1 1 0 0 1
1 0 4 1 0
0 0 1 1 2
0.625 1.625 0.875 2.875
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443:P
243:P
2
43
differencebackward:2Method
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42
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nodesVirtual:3Method
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7.3 Accuracy of various method
1 22 3
2sinh
3
2sin
3
2cosh
cos18],[
n
ynxn
nn
nyxu exact
Analytical Method 1 Method 2 Method 3
u11 0.4320 44.7% 17.9% 10.9%
u21 0.5248 66.7% 31.7% 10.1%
u12 1.1870 36.9% 13.4% 10.0%
u22 1.8037 59.4% 25.0% 16.7%
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Most numerical solutions involve systems of much larger
size, impossible to solve using Gaussian elimination
To accelerate, over-relaxation or successive-over-relaxation (SOR) is used
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7.4 Successive-over-relaxation
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7.5 Alternating-direction-implicit (ADI)
04 ,1,1,,1,1 jijijijiji uuuuu
Row junknown Row iknown from
previous iteration
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1,
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1,
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previous iteration
Rearranging
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,1
k
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k
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ji uuuuu
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Row junknown Row iknown from
previous iteration
)1(
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,
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1,
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1, )2()2(
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previous iteration
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,1 )2()2( k
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k
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k
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k
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k
ji uuuuuu
incorporating with the SOR method
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7 7 ADI th d ( l )
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7.7 ADI method (example)
02 u
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89.011.3
89.011.3
k
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80 oC
80 oC
0 oC
0 oC
11.1x
y
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)(
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21
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Row junknown
8089.0
808089.089.0
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111.3000011.31
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Row unknown
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Column unknown
8089.0
89.0
8089.080
89.080
11.3100
111.300
0011.31
00111.3
)1(
22
)1(
12
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21
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11
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22
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12
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21
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11
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21
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12
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11
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k
k
k
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u
u
u
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8089.0
808089.0
89.0
8089.0
11.3100
111.300
0011.31
00111.3
)0(
22
)0(
21
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)0(
22
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21
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12
)0(
11
)1(
22
)1(12
)1(
21
)1(
11
uu
uu
uu
uu
u
u
u
u
First loop
140.47
604.66
225.9
690.28
)1(
22
)1(
12
)1(21
)1(
11
u
u
u
u
Second loop
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8089.0
89.0
8089.080
89.080
11.3100
111.300
0011.31
00111.3
)1(
22
)1(
12
)1(21
)1(11
)1(
22
)1(
12
)1(
21
)1(
11
)2(
22
)2(12
)2(
21
)2(
11
uu
uu
uu
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u
u
u
u
891.39
412.19371.60
891.38
)2(
22
)2(
21
)2(
12
)2(
11
u
uu
u
Second loop
k u11 u21 u12 u22
0 0.000 0.000 0.000 0.000
1 39.891 19.412 60.371 39.891
2 40.000 20.011 59.988 40.000
3 40.000 20.000 60.000 40.000
8 P b li ti
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8. Parabolic equations
2
22
x
uc
t
u
8 1 N i l i t bilit
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8.1 Numerical instability
Occurs due to marching:Accumulation of error
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8.2 Forward time central space
(FTCS) method
2
1
:
21
2
2
2
,,1,11,
2
,1,,12,1,
2
22
h
kc
F
uFuuFu
huuuc
kuu
x
uc
t
u
nininini
ninininini
Fourier number
8 3 FTCS (e ample)
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nininini uuuu
kFh
xxu
yuyu
x
u
t
u
,,1,11,
2
2
5.025.0
01.025.0,2.0
sin]0,[
,0],1[],0[
;
8.3 FTCS (example)
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0 1 2 3 4 5
0 0.000 0.588 0.951 0.951 0.588 0.000
1 0.000 0.532 0.860 0.860 0.532 0.000
2 0.000 0.481 0.778 0.778 0.491 0.000
3 0.000 0.435 0.704 0.704 0.435 0.000
4 0.000 0.393 0.637 0.637 0.393 0.000
5 0.000 0.356 0.576 0.576 0.356 0.000
6 0.000 0.322 0.521 0.521 0.322 0.000
7 0.000 0.291 0.471 0.471 0.291 0.000
8 0.000 0.263 0.426 0.426 0.263 0.000
9 0.000 0.238 0.385 0.395 0.238 0.000
10 0.000 0.215 0.349 0.349 0.215 0.000
8 4 Crank Nicolson (CN) scheme
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8.4 Crank-Nicolson (CN) scheme
2,1,,1
2
1,11,1,1
2
,
2
2
1,
2
2
2/1,
2
,1,2/1,
2
2
2
1
2
1
huuu
h
uuu
x
u
x
u
x
u
k
uu
t
u
ninini
ninini
ninini
ninini
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nininininini FuuFFuFuuFFu
,1,,11,11,1,1 )1(2)1(2
ninininini uuuuu
F
,1,11,11,1,1 4
1set
Implicit scheme, cannot obtain solution explicitly fromprevious data
need to solve the algebraic system for each time step
8 5 Crank Nicolson scheme (example)
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8.5 Crank-Nicolson scheme (example)
0 1 2 3 4 5
0 0.000 0.588 0.095 0.095 0.588 0.000
1 0.000 0.399 0.646 0.646 0.399 0.000
2 0.000 0.271 0.439 0.439 0.271 0.000
3 0.000 0.184 0.298 0.298 0.184 0.000
4 0.000 0.125 0.203 0.203 0.125 0.000
5 0.000 0.085 0.138 0.138 0.085 0.000
6 0.000 0.058 0.093 0.093 0.058 0.000
7 0.000 0.039 0.064 0.064 0.039 0.000
8 0.000 0.027 0.043 0.043 0.027 0.000
9 0.000 0.018 0.029 0.029 0.018 0.000
10 0.000 0.012 0.020 0.020 0.012 0.000
04.0
2.0
k
h
8 6 Other schemes for parabolic equations
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8.6 Other schemes for parabolic equations
Richardsons scheme
Central-time
No practical use
Du FortFraenkels scheme
Replace RHS of Richardson by average
0:2
2
1,,,1,11,
2
,1,,121,1,
FuFuuuFu
h
uuuc
k
uu
ninininini
ninininini
0:21
21
21
21,,1,11,
2
,11,1,,121,1,
FuF
Fuu
F
Fu
h
uuuuc
k
uu
nininini
nininininini
8 7 Parabolic equations
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8.7 Parabolic equations
in two-spatial dimensions
ADI + Crank-Nicolsons scheme
2
1,,1,
2
,1,,12
2
,,21
21
21
21
22
2 y
l
ji
l
ji
l
ji
x
l
ji
l
ji
l
ji
k
l
ji
l
ji
h
uuu
h
uuucuu
2
1,,1,
2
1
,1
1
,
1
,12
2
,
1
,21
21
21
21
22
2y
l
ji
l
ji
l
ji
x
l
ji
l
ji
l
ji
k
l
ji
l
ji
h
uuu
h
uuucuu
1ststep
2ndstep
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hx= hy,F= 1
21
21
21
21
21
1,11,
1
,1
1
,
1
,1
,1,11,,1,
4
4
l
ji
l
ji
l
ji
l
ji
l
ji
l
ji
l
ji
l
ji
l
ji
l
ji
uuuuu
uuuuu
9 Hyperbolic equations
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9. Hyperbolic equations
02
22
2
2
x
uc
t
u
9 1 Central time central space (CTCS)
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9.1 Central-time central space (CTCS)
1
)1(2
)(
2
2
0
,
2
,1,1
2
1,1,
2
,1,,12
2
1,,1,
2
22
2
2
hckR
uR
uuRuu
h
uuu
c
k
uuu
x
u
ct
u
ni
nininini
ninini
ninini
Courant number
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][)1()(2
][2
],[
][]0,[],[]0,[
conditionsinitial
)1(2)(
0
0,
2
0,10,1
2
1,
1,1,0,
0,2
0,10,12
1,1,
iiiii
iii
ii
t
iiiii
xkguRuuR
u
xgkuuxfu
xgxuxfxu
uRuuRuu
n
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0 1 2 3 4 5 6
0 0.00000 0.01750 0.03500 0.05250 0.07000 0.03500 0.00000
1 0.00000 0.01750 0.03500 0.05250 0.04375 0.03500 0.00000
2 0.00000 0.01750 0.03500 0.02625 0.01750 0.00875 0.00000
3 0.00000 0.01750 0.08750 0.00000 -0.00875 -0.01750 0.00000
4 0.00000 -0.00875 -0.01750 -0.05250 -0.03500 -0.01750 0.00000
5 0.00000 -0.03500 -0.04375 -0.05250 -0.03500 -0.01750 0.00000
6 0.00000 -0.03500 -0.07000 -0.05250 -0.03500 -0.01750 0.00000
7 0.00000 -0.03500 -0.04375 -0.02620 -0.03500 -0.01750 0.00000
8 0.00000 -0.00875 -0.01750 0.00000 -0.03500 -0.01750 0.00000
9 0.00000 0.01750 0.00875 0.02625 -0.00875 -0.01750 0.00000
10 0.00000 0.01750 0.03500 0.05250 0.01750 0.00875 0.00000
9 3 Transport equation
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9.3 Transport equation
Transport equation
0
02
22
2
2
ux
ctx
ct
x
u
ct
u
0
u
xc
t
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