Chapter 6 Parabolic Equation
OUTLINEOUTLINE6-1 General Remarks6-2 Finite Difference Formulation6-3 Parabolic Equations in Two-Space Dimensions6 3 Parabolic Equations in Two Space Dimensions
Chapter 6: Parabolic EquationApplied Computational Fluid DynamicsY.C. Shih Spring 2009
6-1 General Remarks
Equations of motion in fluid mechanics are frequently reduced to parabolicEquations of motion in fluid mechanics are frequently reduced to parabolic formulations. Boundary layer equations are examples of such formulations.In addition, the unsteady heat conduction equation is also parabolic.
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Chapter 6: Parabolic EquationApplied Computational Fluid DynamicsY.C. Shih Spring 2009
6-1
6-2 Finite Difference Formulations (1)
A typical parabolic second-order PDE is the unsteady heat conduction equation, which is considered first in one-space dimension. It has the following form
2 uu ∂∂
(1) FTCS (forward time/central space) method:
2xu
tu
∂∂
=∂∂ α
∂(i) is expressed by a forward difference approximation whichis of order :
tu
∂∂
tΔ
( ) )16(...1
−Δ+Δ−
=∂∂ +
tot
uutu n
ini
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Chapter 6: Parabolic EquationApplied Computational Fluid DynamicsY.C. Shih Spring 2009
6-2
6-2 Finite Difference Formulations (2)
(ii) Using the second-order central differencing of orderf h diff i ( ) b i d
( )2xΔ
for the diffusion term , eq. (6-1) can be approximatedby
( )( )
22
111
Δ+−
=Δ− −+
+ ni
ni
ni
ni
ni
xuuu
tuu α
(iii) Eq (6-2) is also called explicit formulation
( )( )
( ) )26(...2 1121 −+−
ΔΔ
+=⇒ −++ n
ini
ni
ni
ni uuu
xtuu α
(iii) Eq. (6 2) is also called explicit formulation, which is of order . It will beshown that the solution is stable for
( ) ( )[ ]2, xt ΔΔ
( ) 21
2 ≤ΔΔ tαshown that the solution is stable for ( ) 22Δ x
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Chapter 6: Parabolic EquationApplied Computational Fluid DynamicsY.C. Shih Spring 2009
6-3
6-2 Finite Difference Formulations (3)
(2) BTCS (backward/central space) method:(i)
( )ni
ni
ni
ni
ni
xuuu
tuu
Δ+−
=Δ− +
−++
++
2
11
111
1 2α( )
( ) ( ) ( )ni
ni
ni
ni uu
xtu
xtu
xt
xt
−−=ΔΔ
+⎥⎦
⎤⎢⎣
⎡
ΔΔ
+−ΔΔ
⇒
ΔΔ
++
++−
112
12
112 )36(...21 ααα
( ) ( ) ( )ni
ni
ni
ni
ni
ni
ni Ducubua
xxx
=++⇒
Δ⎦⎣ ΔΔ++
++−
11
111
The above equation can be solved by TDMA.
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Chapter 6: Parabolic EquationApplied Computational Fluid DynamicsY.C. Shih Spring 2009
6-4
6-2 Finite Difference Formulations (4)
(ii) Eq. (6-3) is defined as being implicit, since more than one unknown appears in the finite difference equation. As a result,a set of simultaneous equations needs to be solved, whichrequire more computation time per time step Implicit methodsrequire more computation time per time step. Implicit methodsgreater advantage on the stability of the finite differenceequations, since most are unditionally stable. Therefore, aq , y ,larger step size in time is permitted.
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Chapter 6: Parabolic EquationApplied Computational Fluid DynamicsY.C. Shih Spring 2009
6-5
6-2 Finite Difference Formulations (5)
(3) CTCS (central time/central space) method: (The Crank-Nicolson method)(i) If the diffusion term of eq. (6-1) is replaced by the average
of the central differences at time levels n and n+1, thediscretized equation would be of the form:discretized equation would be of the form:
ni
ni
ni
ni
ni
ni
ni
ni uuuuuuuu 11
11
111
1
)46(221 −++−
+++
+
⎥⎤
⎢⎡ +−
++−
⎟⎞
⎜⎛−
( ) ( )( ) ( ) n
ini
ni
ni
ni
ni
iiiiiiii
ruurruruurru
xxt
111
111
1
211
211
22
)46(...2
+++
++
++
+−+=−++−⇒
−⎥⎦
⎢⎣ Δ
+Δ
⎟⎠⎞
⎜⎝⎛=
Δα
( ) ( ) iiiiii ruurruruurru 1111 22 −++− ++++⇒
Note: The left side of eq. (6-4) is a central difference of
step, i.e., , which is ( )2/2
1
tuu
tu n
ini
Δ−
=∂∂ +
( )2to Δ
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Chapter 6: Parabolic EquationApplied Computational Fluid DynamicsY.C. Shih Spring 2009
2/tΔ6-6
6-2 Finite Difference Formulations (6)
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Chapter 6: Parabolic EquationApplied Computational Fluid DynamicsY.C. Shih Spring 2009
6-7
6-2 Finite Difference Formulations (7)
(ii) The method may be thought of as the addition of two step computations as follows:Using the explicit method,
)6(2 112/1 uuuuu n
ini
ni
ni
ni +−− ++
while using the implicit method,( )
)56(...22/ 2
11 ax
uuut
uu iiiii −Δ
+=
Δ−+α
1112/11
( ))56(...2
2/ 2
11
111
2/11
bx
uuutuu n
ini
ni
ni
ni −
Δ+−
=Δ− +
−++
+++
α
Adding eqs. (6-5a) and (6-5b), we can get eq. (6-4).(iii) This implicit method is unconditionally stable and is of
order that is a second order accurate scheme( ) ( )[ ]22t ΔΔorder , that is a second-order accurate scheme.Example
( ) ( )[ ], xt ΔΔ
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Chapter 6: Parabolic EquationApplied Computational Fluid DynamicsY.C. Shih Spring 2009
6-8
6-3 Parabolic Equations in Two-Space Dimensions (1)
Consider the model equationq
⎟⎟⎠
⎞⎜⎜⎝
⎛∂∂
+∂∂
=∂∂
2
2
2
2
yu
xu
tu α
(1) FTCS (or Explicit) method:
⎤⎡1
( ) ( ) ⎥⎥⎦
⎤
⎢⎢⎣
⎡
Δ
++
Δ
+=
Δ
− −−+−−++
21,,1,
2,1,,1,
1, 22
yuuu
xuuu
tuu n
jin
jin
jin
jin
jin
jin
jin
ji α
which is of order Stability analysis indicates that the method is stable for
( ) ( ) ( )[ ]22 ,, yxt ΔΔΔ
2/1≤+ yx ddwhere
If Δx=Δy i e dx=dy=d then( ) ( )22 ,
ytd
xtd yx Δ
Δ=
ΔΔ
=αα
2/1≤d 6 9
Chapter 6: Parabolic EquationApplied Computational Fluid DynamicsY.C. Shih Spring 2009
If Δx=Δy, i.e., dx=dy=d, then 2/1≤d 6-9
6-3 Parabolic Equations in Two-Space Dimensions (2)
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Chapter 6: Parabolic EquationApplied Computational Fluid DynamicsY.C. Shih Spring 2009
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6-3 Parabolic Equations in Two-Space Dimensions (3)
(2) Implicit (BTCS) method:(i) Consider an implicit formulation for which the FDE is
⎤⎡ +++++++ 1111111 22 nnnnnnnn
( ) ( ) ⎥⎥⎦
⎤
⎢⎢⎣
⎡
Δ
+−+
Δ
+=
Δ
− +−
+++
+−
++−+
+
2
11,
1,
11,
2
1,1
1,
1,1,
1, 22
yuuu
xuuu
tuu n
jin
jin
jin
jin
jin
jin
jin
ji α
nji
njiy
njiy
njiyx
njix
njix uudududdudud ,
11,
11,
1,
1,1
1,1 )122( −=++++−+⇒ +
++−
++−
++
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Chapter 6: Parabolic EquationApplied Computational Fluid DynamicsY.C. Shih Spring 2009
6-11
6-3 Parabolic Equations in Two-Space Dimensions (4)
(ii) The 2-D FDEs in the ADI formulation are
( ) ( ) ( ) ⎥⎤
⎢⎡ +−
++
=− −+
+−
++−+
+
21,,1,
2
2/1,1
2/1,
2/1,1,
2/1, 22 uuuuuuuu n
jin
jin
jin
jin
jin
jin
jin
ji α( ) ( ) ( ) ⎥⎥⎦⎢
⎢⎣ Δ
+ΔΔ 222/ yxt
α
dand
( ) ( ) ( ) ⎥⎥⎦
⎤
⎢⎢⎣
⎡
Δ
+−+
Δ
+=
Δ
− +−
+++
+−
++−+
+
2
11,
1,
11,
2
2/1,1
2/1,
2/1,1
2/,
1, 22
2/ yuuu
xuuu
tuu n
jin
jin
jin
jin
jin
jin
jin
ji α
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Chapter 6: Parabolic EquationApplied Computational Fluid DynamicsY.C. Shih Spring 2009
6-12
6-3 Parabolic Equations in Two-Space Dimensions (5)
This method is of order and is unconditionally( ) ( ) ( )[ ]222 ,, yxt ΔΔΔ
stable. The above equations are written in the tridiagonal form as
2/12/12/1 nji
nji
nji
nji
nji
nji udududududud 1,2,21,2
2/1,11
2/1,1
2/1,11 )21()21( −+
++
++− +−+=−++−
2/12/12/1111 )21()21( ++++++ +−+=−++− nnnnnn udududududud ,11,1,111,2,21,2 )21()21( −++− ++=++ jijijijijiji udududududud
where
( ) ( )2221 21
21,
21
21
ytdd
xtdd yx Δ
Δ==
ΔΔ
==αα
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Chapter 6: Parabolic EquationApplied Computational Fluid DynamicsY.C. Shih Spring 2009
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