Chapter 6 Parabolic Equation OUTLINE 6-1 General...

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.

6 1


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

6 2


6-2


(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

6 3


6-3


(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.

6 4


6-4


(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.

6 5


6-5


(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

6 6


2/tΔ6-6


6 7


6-7


(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

6 8


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


If Δx=Δy, i.e., dx=dy=d, then 2/1≤d 6-9


6 10


6-10


(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( −=++++−+⇒ +

++−

++−

++

6 11


6-11


(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

6 12


6-12


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

6 13


6-13


6 14


6-14

Chapter 6 Parabolic Equation OUTLINE 6-1 General...

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