elliptic partial differential equations - math for...
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9/14/2011 http://numericalmethods.eng.usf.edu 1
Elliptic Partial Differential Equations
http://numericalmethods.eng.usf.eduTransforming Numerical Methods Education for STEM Undergraduates
Defining Elliptic PDE’s The general form for a second order linear PDE with two independent
variables ( ) and one dependent variable ( ) is
Recall the criteria for an equation of this type to be considered elliptic
For example, examine the Laplace equation given by
then
thus allowing us to classify this equation as elliptic.
02
22
2
2
=+∂∂
+∂∂
∂+
∂∂ D
yuC
yxuB
xuA
042 <− ACB
02
2
2
2
=∂∂
+∂∂
yT
xT
, where , , and 1=A 0=B 1=C 0=D
04)1)(1(4042
<−=−=− ACB
yx, u
Physical Example of an Elliptic PDE
bT
lT
tT
rT
L
W
x
y
Schematic diagram of a plate with specified temperature boundary conditions
The Laplace equation governs the temperature: 02
2
2
2
=∂∂
+∂∂
yT
xT
Discretizing the Elliptic PDEtT
rT
x
y
),( ji ),1( ji +
)1,( −ji
),1( ji −
)1,( +ji
)0,0()0,(m
),0( n
bT
lT ),( ji
x∆
y∆x∆y∆
If we define and ,we can then write the finite difference
approximation of the partial derivatives at a general interior node ( ) as
mLx =∆
nWy =∆
ji,
( )2,1,,1
,2
2 2x
TTTxT jijiji
ji ∆
+−≅
∂∂ −+
and ( )21,,1,
,2
2 2y
TTTyT jijiji
ji ∆
+−≅
∂∂ −+
Substituting these approximations into the Laplace equation yields:
if,
the Laplace equation can be rewritten as
Discretizing the Elliptic PDE
( ) ( )0
222
1,,1,2
,1,,1 =∆
+−+
∆
+− −+−+
yTTT
xTTT jijijijijiji
yx ∆=∆
04 ,1,1,,1,1 =−+++ −+−+ jijijijiji TTTTT
02
2
2
2
=∂∂
+∂∂
yT
xT
Once the governing equation has been discretized there are several numerical methods that can be used to solve the problem.
We will examine the:•Direct Method•Gauss-Seidel Method•Lieberman Method
Discretizing the Elliptic PDE
04 ,1,1,,1,1 =−+++ −+−+ jijijijiji TTTTT
Example 1: Direct MethodConsider a plate that is subjected to the boundary conditions shown below. Find the temperature at the interior nodes using a square grid with a length of by using the direct method.
C°50
C°75
C°300
C°100
m4.2
m0.3
x
y
mm 0.34.2 ×
m6.0
Example 1: Direct MethodWe can discretize the plate by taking,
myx 6.0=∆=∆
x
y
0,0T0,1T 0,2T 0,3T 0,4T
1,0T
2,0T
3,0T
4,0T
5,0T
1,1T 1,2T 1,3T 1,4T
2,1T 2,2T 2,3T 2,4T
3,1T 3,2T 3,3T 3,4T
4,1T 4,2T4,3T 4,4T
5,1T 5,2T 5,3T 5,4T
x
y
0,0T0,1T 0,2T 0,3T 0,4T
1,0T
2,0T
3,0T
4,0T
5,0T
1,1T 1,2T 1,3T 1,4T
2,1T 2,2T 2,3T 2,4T
3,1T 3,2T 3,3T 3,4T
4,1T 4,2T4,3T 4,4T
5,1T 5,2T 5,3T 5,4T
Example 1: Direct MethodThe nodal temperatures at the boundary nodes are given by:
3,2,1,3003,2,1,50
4,3,2,1,100
4,3,2,1,75
5,
0,
,4
,0
==
==
==
==
iTiT
jTjT
i
i
j
j
C°300
C°100
C°50
C°75
Example 1: Direct Method
x
y
0,0T0,1T 0,2T 0,3T 0,4T
1,0T
2,0T
3,0T
4,0T
5,0T
1,1T 1,2T 1,3T 1,4T
2,1T 2,2T 2,3T 2,4T
3,1T 3,3T 3,4T
4,1T 4,2T4,3T 4,4T
5,1T 5,2T 5,3T 5,4T
3,2T
Here we develop the equation for the temperature at the node (2,3)
04 3,22,24,23,13,3 =−+++ TTTTT04 3,34,23,22,23,1 =++−+ TTTTT
i=2 and j=3 04 ,1,1,,1,1 =−+++ −+−+ jijijijiji TTTTT
Example 1: Direct MethodWe can develop similar equations for every interior node leaving us with an equal number of equations and unknowns.
Question: How many equations would this generate?
x
y
0,0T0,1T 0,2T 0,3T 0,4T
1,0T
2,0T
3,0T
4,0T
5,0T
1,1T 1,2T 1,3T 1,4T
2,1T 2,2T 2,3T 2,4T
3,1T 3,2T 3,3T 3,4T
4,1T 4,2T4,3T 4,4T
5,1T 5,2T 5,3T 5,4T
Example 1: Direct MethodWe can develop similar equations for every interior node leaving us with an equal number of equations and unknowns.
Question: How many equations would this generate? Answer: 12
Solving yields: C
TTTTTTTTTTTT
°
=
446.182271.131389.104
9833.82512.198248.138302.103
5443.77355.173907.119
0252.938924.73
4,3
3,3
2,3
1,3
4,2
3,2
2,2
1,2
4,1
3,1
2,1
1,1
x
y
75
75
75
75
50 50 50
100
100
100
100
300300300
173
120
93
74
199
138
103
78
182
131
104
83
The Gauss-Seidel Method Recall the discretized equation
This can be rewritten as
For the Gauss-Seidel Method, this equation is solved iteratively for all interior nodes until a pre-specified tolerance is met.
04 ,1,1,,1,1 =−+++ −+−+ jijijijiji TTTTT
41,1,,1,1
,−+−+ +++
= jijijijiji
TTTTT
Example 2: Gauss-Seidel MethodConsider a plate that is subjected to the boundary conditions shown below. Find the temperature at the interior nodes using a square grid with a length of using the Gauss-Siedel method. Assume the initial temperature at all interior nodes to be .
C°50
C°75
C°300
C°100
m4.2
m0.3
x
y
mm 0.34.2 ×
m6.0C°0
Example 2: Gauss-Seidel MethodWe can discretize the plate by taking
myx 6.0=∆=∆
x
y
0,0T0,1T 0,2T 0,3T 0,4T
1,0T
2,0T
3,0T
4,0T
5,0T
1,1T 1,2T 1,3T 1,4T
2,1T 2,2T 2,3T 2,4T
3,1T 3,2T 3,3T 3,4T
4,1T 4,2T4,3T 4,4T
5,1T 5,2T 5,3T 5,4T
x
y
0,0T0,1T 0,2T 0,3T 0,4T
1,0T
2,0T
3,0T
4,0T
5,0T
1,1T 1,2T 1,3T 1,4T
2,1T 2,2T 2,3T 2,4T
3,1T 3,2T 3,3T 3,4T
4,1T 4,2T4,3T 4,4T
5,1T 5,2T 5,3T 5,4T
Example 2: Gauss-Seidel MethodThe nodal temperatures at the boundary nodes are given by:
3,2,1,3003,2,1,50
4,3,2,1,100
4,3,2,1,75
5,
0,
,4
,0
==
==
==
==
iTiT
jTjT
i
i
j
j
C°300
C°100
C°50
C°75
Example 2: Gauss-Seidel Method•Now we can begin to solve for the temperature at each interior node using
•Assume all internal nodes to have an initial temperature of zero.
x
y
0,0T0,1T 0,2T 0,3T 0,4T
1,0T
2,0T
3,0T
4,0T
5,0T
1,1T 1,2T 1,3T 1,4T
2,1T 2,2T 2,3T 2,4T
3,1T 3,2T 3,3T 3,4T
4,1T 4,2T4,3T 4,4T
5,1T 5,2T 5,3T 5,4T i=1 and j=14
0,12,11,01,21,1
TTTTT
+++=
4500750 +++
=
C°= 2500.31
i=1 and j=2 41,13,12,02,2
2,1
TTTTT
+++=
42500.310750 +++
=
C°= 5625.26
Iteration #1
41,1,,1,1
,−+−+ +++
= jijijijiji
TTTTT
Example 2: Gauss-Seidel MethodAfter the first iteration, the temperatures are as follows. These will now be used as the nodal temperatures for the second iteration.
x
y
31 20 43
27 12 39
25 3.9 37
100 102 135
C°300
C°100
C°50
C°75
Example 2: Gauss-Seidel Method
i=1 and j=1
Iteration #2
40,12,11,01,2
1,1
TTTTT
+++=
4505625.26753125.20 +++
=
C°= 9688.42 %27.27
1009688.42
2500.319688.42
1001,1
1,11,11,1
=
×−
=
×−
= present
previouspresent
a TTT
ε
Example 2: Gauss-Seidel Method
x
y
300 300 300
75
75
75
75
133 156 161 100
56 56 87 100
39 29 56 100
43 37 56 100
50 50 50x
y
%25 %34 %16
%54 %83 %58
%31 %62 %32
%27 %45 %24
The figures below show the temperature distribution and absolute relative error distribution in the plate after two iterations:
Temperature Distribution
Absolute Relative Approximate
Error Distribution
NodeTemperature Distribution in the Plate (°C)
Number of Iterations
1 2 10 Exact
31.2500 42.9688 73.0239
26.5625 38.7695 91.9585
25.3906 55.7861 119.0976
100.0977 133.2825 172.9755
20.3125 36.8164 76.6127
11.7188 30.8594 102.1577
9.2773 56.4880 137.3802
102.3438 156.1493 198.1055
42.5781 56.3477 82.4837
38.5742 56.0425 103.7757
36.9629 86.8393 130.8056
134.8267 160.7471 182.2278
Example 2: Gauss-Seidel Method
1,1T
2,1T
3,1T4,1T
1,2T
2,2T3,2T
4,2T
1,3T
2,3T
3,3T
4,3T
The Lieberman Method Recall the equation used in the Gauss-
Siedel Method,
Because the Guass-Siedel Method is guaranteed to converge, we can accelerate the process by using over- relaxation. In this case,
41,1,,1,1
,−+−+ +++
= jijijijiji
TTTTT
oldji
newji
relaxedji TTT ,,, )1( λλ −+=
Example 3: Lieberman MethodConsider a plate that is subjected to the boundary conditions shown below. Find the temperature at the interior nodes using a square grid with a length of . Use a weighting factor of 1.4 in the Lieberman method. Assume the initial temperature at all interior nodes to be .
C°50
C°75
C°300
C°100
m4.2
m0.3
x
y
mm 0.34.2 ×
m6.0C°0
Example 3: Lieberman MethodWe can discretize the plate by taking
myx 6.0=∆=∆
x
y
0,0T0,1T 0,2T 0,3T 0,4T
1,0T
2,0T
3,0T
4,0T
5,0T
1,1T 1,2T 1,3T 1,4T
2,1T 2,2T 2,3T 2,4T
3,1T 3,2T 3,3T 3,4T
4,1T 4,2T4,3T 4,4T
5,1T 5,2T 5,3T 5,4T
x
y
0,0T0,1T 0,2T 0,3T 0,4T
1,0T
2,0T
3,0T
4,0T
5,0T
1,1T 1,2T 1,3T 1,4T
2,1T 2,2T 2,3T 2,4T
3,1T 3,2T 3,3T 3,4T
4,1T 4,2T4,3T 4,4T
5,1T 5,2T 5,3T 5,4T
Example 3: Lieberman MethodWe can also develop equations for the boundary conditions to define the temperature of the exterior nodes.
3,2,1,3003,2,1,50
4,3,2,1,100
4,3,2,1,75
5,
0,
,4
,0
==
==
==
==
iTiT
jTjT
i
i
j
j
C°300
C°100
C°50
C°75
Example 3: Lieberman Method•Now we can begin to solve for the temperature at each interior node using the rewritten Laplace equation from the Gauss-Siedel method.
•Once we have the temperature value for each node we will apply the over relaxation equation of the Lieberman method
•Assume all internal nodes to have an initial temperature of zero.
i=1 and j=14
0,12,11,01,21,1
TTTTT
+++=
4500750 +++
=
C°= 2500.31
Iteration #1
1,1 1,1 1,1(1 )relaxed new oldT T Tλ λ= + −
C°=−+=
7500.430)4.11()2500.31(4.1
Iteration #2
i=1 and j=24
1,13,12,02,22,1
TTTTT
+++=
475.430750 +++
=
C°= 6875.29
1,1 1,1 1,1(1 )relaxed new oldT T Tλ λ= + −
C°=−+=
5625.410)4.11()6875.29(4.1
Example 3: Lieberman MethodAfter the first iteration the temperatures are as follows. These will be used as the initial nodal temperatures during the second iteration.
x
y
44 33 64
42 26 67
41 23 66
146 164 221
C°300
C°100
C°50
C°75
Example 3: Lieberman Method
i=1 and j=1
Iteration #2
40,12,11,01,2
1,1
TTTTT
+++=
C°=
+++=
8438.494
505625.41758125.32
%32.16
1002813.52
7500.432813.52
1001,1
1,11,11,1
=
×−
=
×−
= present
previouspresent
a TTT
ε
1,1 1,1 1,1(1 )relaxed new oldT T Tλ λ= + −
C°=−+=
2813.5275.43)4.11()8438.49(4.1
Example 3: Lieberman Method
x
y
%6.9 %24 %22
%53 %81 %57
%19 %55 %13
%16 %39 %5.7
The figures below show the temperature distribution and absolute relative error distribution in the plate after two iterations:
Temperature Distribution
x
y
300 300 300
75
75
75
75
161 216 181 100
87 122 155 100
51 58 76 100
52 54 69 100
50 50 50
Absolute Relative Approximate
Error Distribution
Example 3: Lieberman Method
NodeTemperature Distribution in the Plate (°C)
Number of Iterations
1 2 9 Exact
43.7500 52.2813 73.7832
41.5625 51.3133 92.9758
40.7969 87.0125 119.9378
145.5289 160.9353 173.3937
32.8125 54.1789 77.5449
26.0313 57.9731 103.3285
23.3898 122.0937 138.3236
164.1216 215.6582 198.5498
63.9844 69.1458 82.9805
66.5055 76.1516 104.3815
66.4634 155.0472 131.2525
220.7047 181.4650 182.4230
1,1T
2,1T
3,1T4,1T
1,2T
2,2T3,2T
4,2T
1,3T
2,3T
3,3T
4,3T
Alternative Boundary Conditions In Examples 1-3, the boundary conditions on the plate had a
specified temperature on each edge. What if the conditions are different ? For example, what if one of the edges of the plate is insulated.
In this case, the boundary condition would be the derivative of the temperature. Because if the right edge of the plate is insulated, then the temperatures on the right edge nodes also become unknowns.
C°50
C°75
C°300
m4.2
m0.3
x
y
Insulated
Alternative Boundary Conditions The finite difference equation in this case for the right edge for
the nodes for
However the node is not inside the plate. The derivative boundary condition needs to be used to account for these additional unknown nodal temperatures on the right edge. This is done by approximating the derivative at the edge node as
1,..3,2 −= nj
04 ,1,1,,1,1 =−+++ +−−+ jmjmjmjmjm TTTTT
),1( jm +
),( jm
)(2,1,1
, xTT
xT jmjm
jm ∆
−≅
∂∂ −+
C°50
C°75
C°300
m4.2
m0.3
x
y
Insulated
),( jm
Alternative Boundary Conditions Rearranging this approximation gives us,
We can then substitute this into the original equation gives us,
Recall that is the edge is insulated then,
Substituting this again yields,
jmjmjm x
TxTT,
,1,1 )(2∂∂
∆+= −+
04)(22 ,1,1,,
,1 =−++∂∂
∆+ +−− jmjmjmjm
jm TTTxTxT
0,
=∂∂
jmxT
042 ,1,1,,1 =−++ +−− jmjmjmjm TTTT
Example 3: Alternative Boundary Conditions
A plate is subjected to the temperatures and insulated boundary conditions as shown in Fig. 12. Use a square grid length of . Assume the initial temperatures at all of the interior nodes to be . Find the temperatures at the interior nodes using the direct method.
mm 0.34.2 ×m6.0
C°0
C°50
C°75
C°300
m4.2
m0.3
x
y
Insulated
Example 3: Alternative Boundary Conditions
We can discretize the plate taking,
myx 6.0=∆=∆
x
y
0,0T0,1T 0,2T 0,3T 0,4T
1,0T
2,0T
3,0T
4,0T
5,0T
1,1T 1,2T 1,3T 1,4T
2,1T 2,2T 2,3T 2,4T
3,1T 3,2T 3,3T 3,4T
4,1T 4,2T4,3T 4,4T
5,1T 5,2T 5,3T 5,4T
x
y
0,0T0,1T 0,2T 0,3T 0,4T
1,0T
2,0T
3,0T
4,0T
5,0T
1,1T 1,2T 1,3T 1,4T
2,1T 2,2T 2,3T 2,4T
3,1T 3,2T 3,3T 3,4T
4,1T 4,2T4,3T 4,4T
5,1T 5,2T 5,3T 5,4T
Example 3: Alternative Boundary Conditions
We can also develop equations for the boundary conditions to define the temperature of the exterior nodes.
C°300
C°50
C°75
4,3,2,1;0
4,3,2,1;3004,3,2,1;50
4,3,2,1;75
,4
5,
0,
,0
==∂∂
==
==
==
jxT
iTiTjT
j
i
i
j
Insulated
Example 3: Alternative Boundary Conditions
x
y
0,0T0,1T 0,2T 0,3T 0,4T
1,0T
2,0T
3,0T
4,0T
5,0T
1,1T 1,2T 1,3T 1,4T
2,1T 2,2T 2,3T 2,4T
3,1T 3,3T
4,1T 4,2T4,3T 4,4T
5,1T 5,2T 5,3T 5,4T
3,2T3,4T
Here we develop the equation for the temperature at the node (4,3), to show the effects of the alternative boundary condition.i=4 and j=3 042 3,44,42,43,3 =−++ TTTT
042 4,43,42,43,3 =+−+ TTTT
Example 3: Alternative Boundary Conditions
The addition of the equations for the boundary conditions gives us a system of 16 equations with 16 unknowns.
Solving yields: C
TTTTTTTTTTTTTTTT
°
=
738.232830.178617.130
7882.88060.232483.174426.127
2678.87021.218614.159335.117
8571.82410.180617.128
4444.998254.76
4,4
3,4
2,4
1,4
4,3
3,3
2,3
1,3
4,2
3,2
2,2
1,2
4,1
3,1
2,1
1,1
x
y
300 300 300
75
75
75
75
180 218 232 233
129 160 174 179
99 117 127 131
77 83 87 89
50 50 50
THE END