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Computational Fluid Dynamics IPPPPIIIIWWWW

Numerical Methods forParabolic Equations

Instructor: Hong G. ImUniversity of Michigan

Fall 2001


One-Dimensional Problems• Explicit, implicit, Crank-Nicolson• Accuracy, stability• Various schemes• Keller Box method and block tridiagonal system

Multi-Dimensional Problems• Alternating Direction Implicit (ADI)• Approximate Factorization of Crank-Nicolson

Outline

Solution Methods for Parabolic Equations


Numerical Methods forOne-Dimensional Heat Equations


bxatxf

tf <<>

∂∂=

∂∂ ,0;

2

2

α

which is a parabolic equation requiring

)()0,( 0 xfxf =

In this lecture, we consider a model equation

Initial Condition

)(),();(),( ttbfttaf ba φφ == Boundary Condition(Dirichlet)

)(),();(),( ttbxftta

xf

ba ϕϕ =∂∂=

∂∂ Boundary Condition

(Neumann)

or

and


211

1 2h

ffft

ff nj

nj

nj

nj

nj −++ +−

=∆−

α

Explicit: FTCS

( )nj

nj

nj

nj

nj fff

htff 112

1 2 −++ +−∆+= α

j−1 j j+1n

n+1

Explicit Method: FTCS - 1

Computational Fluid Dynamics IPPPPIIIIWWWWModified Equation

xxxxxxxxxx fhthtOfrhxf

tf ),,()61(

12422

2

2

2

∆∆+−=∂∂−

∂∂ αα

2htr ∆= αwhere

- Accuracy ),( 2htO ∆

- If then ),( 22 htO ∆,6/1=r- No odd derivatives; dissipative


Computational Fluid Dynamics IPPPPIIIIWWWWStability: von Neumann Analysis

(Recall Lecture 3, p. 20)

14112<∆−<−

htα

210

2<∆≤

htα

Fourier Condition


[ ])2/(sin412

sin41 222

1

βεε rGhk

htD

n

n

−=∆−=+

Computational Fluid Dynamics IPPPPIIIIWWWWDomain of Dependence for Explicit Scheme

BC BC

x

t

Initial Data

h

P∆t

Boundary effect is not felt at P for many time steps

This may result inunphysical solutionbehavior



2

11

111

1 2h

ffft

ff nj

nj

nj

nj

nj

+−

+++

+ +−=

∆−

α

Implicit Method: Laasonen (1949)

( ) ( )211

111 /12 htrfrffrrf n

jnj

nj

nj ∆==−−+− +

−++

+ α

j−1 j j+1n

n+1

Tri-diagonal matrix system

Implicit Method - 1


- The + sign suggests that implicit method may beless accurate than a carefully implemented explicitmethod.

Modified Equation

xxxxxxxxxx fhthtOfrhxf

tf ),,()61(

12422

2

2

2

∆∆++=∂∂−

∂∂ αα

Implicit Method - 2

Amplification Factor (von Neumann analysis)

[ ] 1)cos1(21 −−+= βrGUnconditionally stable


+−+

+−=

∆− −+

+−

+++

+

211

2

11

111

1 222 h

fffh

ffft

ff nj

nj

nj

nj

nj

nj

nj

nj α

Crank-Nicolson Method (1947)

j−1 j j+1n

n+1

Tri-diagonal matrix system

Crank-Nicolson - 1

( ) ( ) nj

nj

nj

nj

nj

nj rffrrfrffrrf 11

11

111 1212 −+

+−

+++ +−−=−++−


- Second-order accuracy

Modified Equation

xxxxxxxxxx fhtfhxf

tf

+∆+=

∂∂−

∂∂ 423

2

2

2

3601

121

12αααα

Crank-Nicolson - 2

Amplification Factor (von Neumann analysis)

( )( )β

βcos11cos11

−+−−=

rrG

Unconditionally stable

),( 22 htO ∆


+−−+

+−=

∆− −+

+−

+++

+

211

2

11

111

1 2)1(

2h

fffh

ffft

ff nj

nj

nj

nj

nj

nj

nj

nj θθα

Generalization

j−1 j j+1n

n+1

Combined Method A - 1

−=

NicolsonCrank1/2Implicit1

(FTCS)Explicit0θ

θ×( )θ−× 1


Other Special Cases (for further accuracy improvement):


),(12

121 42 htO

r∆⇒−=θ(a) If

r121

21 −=θ(b) If and ),(20/1 62 htOr ∆⇒=

Modified Equation

xxxxxxt fhtff

+∆

−=−

1221 2

2 ααθα

xxxxxxfhtht

+∆

−+∆

+−+ 422232

3601

21

61

31 ααθαθθ


⇒ unconditionally stable


121 ≤≤θ

Stability Property

210 <≤θ ⇒ stable only if

θ4210−

≤≤ r


( )2

11

111

11 21

hfff

tff

tff n

jnj

nj

nj

nj

nj

nj

+−

+++

−+ +−=

∆−

−∆−

+ αθθ

Generalized Three-Time-Level Implicit Scheme:Richtmyer and Morton (1967)

j−1 j j+1

n

n+1

Combined Method B - 1

=implicitfully level-Three1/2

Implicit0θ

( )θ+× 1θ×

n−1

Computational Fluid Dynamics IPPPPIIIIWWWWModified Equation:

Combined Method B - 2

!+∆+

+∆

−−=− )(

1221 2

22 tOfhtff xxxxxxt

ααθα

Special Cases:

),(21 22 htO ∆⇒=θ(a) If

r121

21 +=θ(b) If ),( 42 htO ∆⇒


211

11 22 h

ffftff n

jnj

nj

nj

nj −+

−+ +−=

∆−

α

Richardson Method: A Case of Failure

Richardson Method

),( 22 htO ∆

Similar to Leapfrog

but unconditionally unstable!

j−1 j j+1

n

n+1

n−1


21

111

11

2 hffff

tff n

jnj

nj

nj

nj

nj −

−++

−+ +−−=

∆−

α

The Richardson method can be made stable by splitting by time average

j−1 j j+1

n

n+1

DuFort-Frankel - 1

n−1

njf ( ) 2/11 −+ + n

jnj ff

( ) ( )nj

nj

nj

nj

nj fffrfrf 1

11

11 221 −−

+−+ +−+=+


xxxxxxt fhthff

∆−=−

2

232

121 ααα

Modified Equation (Recall Homework #1)

DuFort-Frankel - 2

xxxxxxfhtth

∆+∆−+

4

45234 2

31

3601 ααα

Amplification factor

rrr

G21

sin41cos2 22

+−±

=ββ Unconditionally

stable

Conditionally consistent


UnconditionallyUnstable

Richardson

UnconditionallyStable

Crank-Nicolson


BTCS

Stable forFTCS

xxt ff α=

211

1 2h

ffft

ff nj

nj

nj

nj

nj −++ +−

=∆−

α ( ) xxxxfrh 6112

2

−α

Parabolic Equation - Summary

21≤r

2

11

111

1 2h

ffft

ff nj

nj

nj

nj

nj

+−

+++

+ +−=

∆−

α ( ) xxxxfrh 6112

2

+α

( )[( )]n

jnj

nj

nj

nj

nj

nj

nj

fff

fffht

ff

11

11

1112

1

2

22

−+

+−

+++

+

+−+

+−=∆− α

xxxxxxxxxx ftfh1212

232 ∆+αα

211

11 22 h

ffftff n

jnj

nj

nj

nj −+

−+ +−=

∆−

α ),( 22 htO ∆



3-Level Implicit

UnconditionallyStable,

ConditionallyConsistent

DuFort-Frankel

xxt ff α=

( ) xxxxfrh 22

12112

−α

Parabolic Equation - Summary

2

11

111

11

2243

hfff

tfff

nj

nj

nj

nj

nj

nj

+−

+++

−+

+−

=∆

+−

αxxxxfh

12

2α

21

111

11

2 hffff

tff n

jnj

nj

nj

nj

nj −

−++

−+ +−−=

∆−

α

Computational Fluid Dynamics IPPPPIIIIWWWWThe Keller Box Method (1970)

Keller Box - 1

2

2

xf

tf

∂∂=

∂∂ α

- Implicit with ),( 22 htO ∆

Basic Concept:

Define xfvfu∂∂== ;

yielding v

xu =∂∂

xv

tu

∂∂=

∂∂ α

Computational Fluid Dynamics IPPPPIIIIWWWWIn discretized form

Keller Box - 2

12/1

11

1+−

+−

+

=− n

jj

nj

nj v

huu

j

nj

nj

n

nj

nj

hvv

tuu 2/1

12/1

1

2/11

2/1+−

+

+

−+− −

=∆−

αj−1 j

n+1

n

∆tn+1

hj

12/1

+−

njv 1+n

junju

j−1 j

n+1

n

2/11+−

njv 2/1+n

jvnju 2/1−

12/1

+−

nju

where

2;

2

12/1

11

11

2/1

nj

njn

j

nj

njn

j

vvv

uuu

+=

+=

++

+−

++−

Computational Fluid Dynamics IPPPPIIIIWWWWSubstituting

Keller Box - 3

2

11

111

1 +−

++−

+ +=

− nj

nj

j

nj

nj vv

huu

j

nj

nj

n

nj

nj

j

nj

nj

n

nj

nj

hvv

tuu

hvv

tuu 1

1

111

1

1

11

1−

+

−+−

+

+

+−

+ −+

∆+

+−

=∆+

αα

or 0

2211

11

11 =−−− +−

+−

++ nj

jnj

nj

jnj v

huv

hu

( ) ( )nj

nj

nj

nj

n

jnj

nj

n

jnj

nj

n

j vvuut

hvu

th

vut

h11

1

11

11

1

11

1−−

+

+−

+−

+

++

+

−++∆

=+∆

+−∆ ααα

Computational Fluid Dynamics IPPPPIIIIWWWWAdding boundary conditions, e.g.

Keller Box - 4

),,1(; 11

11 JjCuCu J

nJ

n !=== ++

022

:2 12

212

11

211 =−+−−= ++++ nnnn vhuvhuj

( ) ( )nnnn

n

nn

n

nn

n

vvuut

hvut

hvut

h1212

1

212

12

1

211

11

1

2 −++∆

=−∆

++∆ +

++

+

++

+ ααα

11

1:1 Cuj n == +

022

: 1111

11 =−+−−= +++

−+−

nj

jnj

nj

jnj v

huv

hujj

( ) ( )nj

nj

nj

nj

n

jnj

nj

n

jnj

nj

n

j vvuut

hvu

th

vut

h11

1

11

1

11

11

1−−

+

++

+

+−

+−

+

−++∆

=−∆

++∆ ααα

JnJ CuJj == +1:

2R≡

jR≡jη≡

Computational Fluid Dynamics IPPPPIIIIWWWWIn Matrix Form

Keller Box - 5

=

−−−−

−−−−

−−−−

0

0

0

00

010000000011000000

2/12/1000000

000000011000002/12/1000000011000002/12/1000000001

3

2

3

3

2

2

1

1

33

33

22

22

J

J

JJJ

JJ

R

R

R

vu

vuvuvu

hh

hh

hh

"

"

"

"""""""""""

""""""

!

!

!

!

!

ηη

ηη

ηη

[ ] [ ] [ ] CFFF =++ ++

++−

11

111

nj

nj

nj ADB

D1 A1

A2D2B2

B3 D3 A3

DJBJ

Block Tridiagonal Matrix ⇒ Appendix, Linpack

Computational Fluid Dynamics IPPPPIIIIWWWWModified Keller Box Method – Express in terms of

Keller Box - 6

v

Starting with original Keller box method:

u

2

11

111

1 +−

++−

+ +=

− nj

nj

j

nj

nj vv

huu

j

nj

nj

n

nj

nj

j

nj

nj

n

nj

nj

hvv

tuu

hvv

tuu 1

1

111

1

1

11

1−

+

−+−

+

+

+−

+ −+

∆+

+−

=∆+

αα

Eliminating from (b) using (a) nj

nj vv 1

11 , −+−

(a)

(b)

21

1

12

11

11

1

11

1

2222j

nj

nj

j

nj

n

nj

nj

j

nj

nj

j

nj

n

nj

nj

huu

hv

tuu

huu

hv

tuu −

+

−+−

++

+

+−

+ −++

∆+

+−

+=∆+

αααα

Computational Fluid Dynamics IPPPPIIIIWWWWFurther elimination of yields (Tannehill, p. 136)

Keller Box - 7

jnjj

njj

njj CuAuDuB =++ +

+++

−11

111

jn

jj ht

hB α2

1

−∆

=+

Tridiagonal Matrix ⇒ Thomas Algorithm

nj

nj vv ,1+

11

1 2

++

+ −∆

=jn

jj ht

hA α

11

1

1

22

++

+

+

++∆

+∆

=jjn

j

n

jj hht

hth

D αα

( ) ( )1

11

11

1

11 22+

++

+−

+

+−

∆++

∆++

−+

−=

n

jnj

nj

n

jnj

nj

j

nj

nj

j

nj

nj

j th

uuth

uuh

uuh

uuC αα


Notes on Keller Box Method

Keller Box - 8

1. Second-order accurate in time and space

2. Accuracy is preserved for nonuniform grids

3. More operations per timestep compared to Crank-Nicolson


Numerical Methods forMulti-Dimensional Heat Equations


∂∂+

∂∂=

∂∂

2

2

2

2

yf

xf

tf α

Applying forward Euler scheme:

∆

+−+

∆+−

=∆− −+−+

+

21,,1,

2,1,,1,

1, 22

yfff

xfff

tff n

jinji

nji

nji

nji

nji

nji

nji α

Consider a 2-D heat equation

hyx =∆=∆If

( )nji

nji

nji

nji

nji

nji

nji fffff

htff

,1,1,,1,12,

1, 4−+++=∆−

−+−+

+ α

Explicit Method - 1

Computational Fluid Dynamics IPPPPIIIIWWWWIn matrix form

−

−−

−

∆+

=

−−+

+−

+

+

+

+

+

nJI

nJI

n

n

nJ

n

n

nJI

nJI

n

n

nJ

n

n

nJI

nJI

n

n

nJ

n

n

ff

fff

ff

ht

ff

fff

ff

ff

fff

ff

,

1,

2,2

1,2

,1

2,1

1,1

2

,

1,

2,2

1,2

,1

2,1

1,1

1,

11,

12,2

11,2

1,1

12,1

11,1

410

1001001

101410100141010014

"

"

"

"

"

!!!!

!!!!

!!!!

"

"

"

"

"

"

α

Explicit Method - 2

Computational Fluid Dynamics IPPPPIIIIWWWWExplicit Method - 3

Von Neumann Analysis imyikxnnj eeεε =

( ) nimhimhikhikhnn eeeeh

t εαεε 421 −+++∆+= −−+

( )4cos2cos21 2

1

−+∆+=+

mhkhh

tn

n αεε

+∆−=

2sin

2sin41 22

2

mhkhh

tα

Worst case

1811 2 ≤∆−≤−h

tα41

2 ≤∆h

tα


Explicit method for multi-dimensional heat equation is not desirable.

Explicit Method - 4

Stability Condition for Heat Equation

41

2 ≤∆h

tα (2-D)61

2 ≤∆h

tα (3-D)21

2 ≤∆h

tα (1-D)


Too expensive!!

Crank-Nicolson

Crank-Nicolson Method for 2-D Heat Equation

∂∂+

∂∂+

∂∂+

∂∂=

∆− +++

2

2

2

2

2

12

2

121

2 yf

xf

yf

xf

tff nnnnnn α

( )1,

11,

11,

1,1

1,12,

1, 4

2++

−++

+−

++

+ −+++∆+= nji

nji

nji

nji

nji

nji

nji fffff

htff α

hyx =∆=∆If

( )nji

nji

nji

nji

nji fffff

ht

,1,1,,1,12 42

−+++∆+ −+−+α


Fractional Step:

ADI - 1

A Clever Remedy – Alternating Direction Implicit (ADI)

( ) ( )[ ]nji

nji

nji

nji

nji

nji

nn ffffffh

tff 1,,1,2/1

,12/1

,2/1

,122/1 22

2 −++

−++

++ +−++−∆=− α

( )hyx =∆=∆

( ) ( )[ ]11,

1,

11,

2/1,1

2/1,

2/1,12

2/11 222

+−

+++

+−

+++

++ +−++−∆=− nji

nji

nji

nji

nji

nji

nn ffffffh

tff α

Step 1:

Step 2:

∂∂+

∂∂+

∂∂∆=−

+++

2

2

2

12

2

2/121

21

2 yf

yf

xftff

nnnnn α

Combining the two becomes equivalent to:

Computational Fluid Dynamics IPPPPIIIIWWWWADI - 2

Computational Molecules for ADI Method

n

n+1/2

n+1

ii+1

i−1

j+1

j−1

j


Stability Analysis:

ADI - 3

ADI Method is accurate

Similarly,

( )22 ,htO ∆imyikxnn

j eeεε =

( ) ( )[ ]imhimhnikhikhnnn eeeeh

t −−++ +−++−∆+= 222/12

2/1 εεαεε

2sin21

2sin21

22

222/1

mhh

t

khh

t

n

n

∆+

∆−=

+

α

α

εε

2sin21

2sin21

22

22

2/1

1

khh

t

mhh

t

n

n

∆+

∆−=+

+

α

α

εε

Computational Fluid Dynamics IPPPPIIIIWWWWADI - 4

Combining

Unconditionally stable!

1

2sin21

2sin21

2sin21

2sin21

22

22

22

221

<

∆+

∆−

∆+

∆−=

+

khh

t

mhh

t

mhh

t

khh

t

n

n

α

α

α

α

εε

Unfortunately, a 3-D version does not have the same desirable stability properties.

Computational Fluid Dynamics IPPPPIIIIWWWWApproximate Factorization - 1

The Crank-Nicolson for heat equation becomes

Define ( ) ( ) ( ) ( )[ ]jijijixx x ,1,,12 21+− +−

∆=δ

( ) ( ) ( ) ( )[ ]1,,1,2 21+− +−

∆= jijijiyy y

δ

( ) ( ) ),,(22

222111

yxtOfffft

ff nnyy

nnxx

nn

∆∆∆++++=∆− ++

+

δαδα

),,(22

122

1

222

1

yxttO

fttftt nyyxx

nyyxx

∆∆∆∆+

∆+∆+=

∆−∆− + δαδαδαδα

which can be rewritten as


Factoring each side

),,( 222 yxttO ∆∆∆∆+

=∆−

∆−

∆− ++ 1

221

421

21 n

yyxxn

yyxx ftftt δδαδαδα

or

nyyxx

nyyxx ftftt δδαδαδα

421

21

22∆−

∆+

∆+

nyyxx

nyyxx fttftt

∆+

∆+=

∆−

∆− + δαδαδαδα

21

21

21

21 1

)( tO ∆( ) ),,(

42221

22

yxttOfft nnyyxx ∆∆∆∆+−∆+ +δδα


Final factored form of the discrete equation

at an accuracy of

nyyxx

nyyxx fttftt

∆+

∆+=

∆−

∆− + δαδαδαδα

21

21

21

21 1

),,( 222 yxtO ∆∆∆

Note that the ADI method can be written as

nyy

nxx ftft

∆+=

∆− + δαδα

21

21 2/1

2/11

21

21 ++

∆+=

∆− n

xxyn

yy ftft δαδα

which is equivalent to factorized Crank-Nicolson

Computational Fluid Dynamics IPPPPIIIIWWWWSummary of ADI

Why splitting?

1. Stability limits of 1-D case apply.

2. Different ∆t can be used in x and y directions.

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