Computational Fluid Dynamics

Computational Fluid Dynamics

http://www.nd.edu/~gtryggva/CFD-Course/

Grétar Tryggvason

Lecture 12March 6, 2017

Computational Fluid Dynamics

Numerical Methods forParabolic Equations

http://www.nd.edu/~gtryggva/CFD-Course/

Computational Fluid Dynamics

One-Dimensional Problems•  Explicit, implicit, Crank-Nicolson•  Accuracy, stability•  Various schemes

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

Splitting

Outline

Solution Methods for Parabolic Equations

Computational Fluid Dynamics

Numerical Methods forOne-Dimensional Heat

Equations

Computational Fluid Dynamics

bxatxf

tf <<>

∂∂=

∂∂ ,0;

2

2

α

which is a parabolic equation requiring

)()0,( 0 xfxf =

Consider the diffusion equation

Initial Condition

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

)(),();(),( ttbxf

ttaxf

ba ϕϕ =∂∂=

∂∂ Boundary Condition

(Neumann)

or

and

Computational Fluid Dynamics

Parabolic equations can be viewed as the limit of a hyperbolic equation with two characteristics as the signal speed goes to infinity

Increasing signal speed

x

t

Computational Fluid Dynamics

211

1 2h

ffftff n

jnj

nj

nj

nj −++ +−

=Δ−

α

Explicit: FTCS

f jn+1 = f j

n +αΔth2

f j+1n − 2 f j

n + f j−1n( )

j-1 j j+1n

n+1

Explicit Method: FTCS - 1

∂ 2 f∂x2

⎞

⎠⎟ j

n

=f j+1

n − 2 f jn + f j−1

n

h2

∂ f∂t

⎞⎠⎟ j

n

=f j

n+1 − f jn

Δt

∂f∂t

⎛⎝⎜

⎞⎠⎟ j

n

= α ∂2 f∂x2

⎛

⎝⎜⎞

⎠⎟ j

n

Computational Fluid Dynamics

Modified Equation

xxxxxxxxxx fhthtOfrhxf

tf ),,()61(

12422

2

2

2

ΔΔ+−=∂∂−

∂∂ αα

2htr Δ= αwhere

- Accuracy ),( 2htO Δ

- No odd derivatives; dissipative

Explicit Method: FTCS - 2

Computational Fluid Dynamics

Stability: von Neumann Analysis

14112<Δ−<−

htα

210

2<Δ≤

htα

Fourier Condition

ε n+1

ε n= 1− 4αΔt

h2sin2 k h

2G = 1− 4r sin2 (β / 2)⎡⎣ ⎤⎦

Explicit Method: FTCS - 3

β = khwhere

Computational Fluid Dynamics

Domain of Dependence for Explicit Scheme

BC

x

t

Initial Data

BCh

P Δt

In the limit of Δt and Δx going to zero, the computational domain contains the physical domain of dependence since Δt ~ Δx2 for stability

Explicit Method: FTCS - 4

Boundary effect is not felt at P for many time steps

This may result in unphysical solution behavior

Computational Fluid Dynamics

Implicit Method for the One-Dimensional Heat

Equation

Computational Fluid Dynamics

2

11

111

1 2h

ffftff n

jnj

nj

nj

nj

+−

+++

+ +−=

Δ−

α

Implicit Method: Backward Euler

j-1 j j+1n

n+1Tri-diagonal matrix system

Implicit Method

fin+1 − fi

n =αΔth2

fi+1n+1 − 2 fi

n+1 + fi−1n+1( )

�

a j−1 f j−1 − d j f j + c j+1 f j+1 = bj

rfi+1n+1 − 1+ 2r( ) fin+1 + rfi−1n+1 = − fin r = αΔt

h2

Computational Fluid Dynamics

aj f j−1 − dj f j + cj f j+1 = bj

�

d1 f1 + c1 f2 = b1a2 f1 + d2 f2 + c2 f3 = b2

aN−1 fN−2 + dN−1 fN−1 + cN−1 fN = bN−1

aN fN−1 + dN fN = bN

�

b1 = −a1 f0bN = −cN fN +1

Write out

If the endpoints are given

Implicit MethodComputational Fluid Dynamics

Gaussian Elimination-  Pivoting: rearranging equations to put the largest coefficient on the main diagonal.-  Eliminate the column below main diagonal.-  Repeat until the last equation is reached.-  Back-substitution

-  Special case: tri-diagonal matrix - Thomas algorithm

nnn cfafa

cfafacfafa

=++

=++=++

2211

2222121

1212111

nnnn cfa

cfacfafa

=′

=+′=++

2222

1212111

è

Implicit Method

Computational Fluid Dynamics

% solving a tridiagonal system nx=50;r=0.1; a=zeros(nx,1)+(1+r);d=zeros(nx,1)-2;c=zeros(nx,1)+(1-r); b=zeros(nx,1); x=zeros(nx,1); b(nx)=-(1-r); % forward elimination for j=2:nx d(j)=d(j)-(a(j)/d(j-1))*c(j-1); b(j)=b(j)-(a(j)/d(j-1))*b(j-1); x(j)=b(j)/d(j); end % backward substitution for j=nx-1:-1:1 x(j)=(b(j)-c(j)*x(j+1))/d(j); end plot(x)

Implicit MethodComputational Fluid Dynamics

Matlab functions

For simple problems MATLAB has a number of functions to deal with matrices.

Help matfun: generalHelp sparfun:sparse matrices

Implicit Method

Computational Fluid Dynamics

- The + sign suggests that implicit method may be less accurate than a carefully implemented explicit method.

Modified Equation∂f∂t

−α ∂2 f∂x2

=αh2

12(1+ 6r) fxxxx +O(Δt

2 ,h2Δt,h4 ) fxxxxxx

Amplification Factor (von Neumann analysis)

G = 1+ 2r(1− cosβ)[ ]−1

Unconditionally stable

Implicit MethodComputational Fluid Dynamics

⎟⎟⎠

⎞⎜⎜⎝

⎛ +−+

+−=

Δ− −+

+−

+++

+

211

2

11

111

1 222 h

fffh

ffftff n

jnj

nj

nj

nj

nj

nj

nj α

Crank-Nicolson Method (1947)

j-1 j j+1n

n+1

Tri-diagonal matrix system

−rf j+1n+1 + 2 r +1( ) f jn+1 − rf j−1n+1 = rf j+1n − 2 r −1( ) f jn + rf j−1n

Crank-Nicolson - 1

Computational Fluid Dynamics

- Second-order accuracy

Modified Equation

xxxxxxxxxx fhtfh

xf

tf

⎥⎦⎤

⎢⎣⎡ +Δ+=

∂∂−

∂∂ 423

2

2

2

3601

121

12αααα

Crank-Nicolson - 2

Amplification Factor (von Neumann analysis)

( )( )β

βcos11cos11

−+−−=

rrG

Unconditionally stable

),( 22 htO Δ

Computational Fluid Dynamics

⎟⎟⎠

⎞⎜⎜⎝

⎛ +−−+

+−=

Δ− −+

+−

+++

+

211

2

11

111

1 2)1(

2h

fffh

ffftff n

jnj

nj

nj

nj

nj

nj

nj θθα

Generalization

j-1 j j+1n

n+1

⎪⎩

⎪⎨⎧

−=

NicolsonCrank1/2Implicit1

(FTCS)Explicit0θ

θ×( )θ−× 1

Combined Method A - 1

Computational Fluid Dynamics

( )2

11

111

11 21

hfff

tff

tff n

jnj

nj

nj

nj

nj

nj

+−

+++

−+ +−=

Δ−

−Δ−

+ αθθ

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

j-1 j j+1

n

n+1

⎩⎨⎧

=implicitfully level-Three1/2

Implicit0θ

( )θ+× 1θ×

n-1

Combined Method B - 1Computational Fluid Dynamics

Modified Equation:

Combined Method B - 2

+Δ+⎥⎦⎤

⎢⎣⎡ +Δ⎟

⎠⎞⎜

⎝⎛ −−=− )(

1221 2

22 tOfhtff xxxxxxt

ααθα

Special Cases:

),(21 22 htO Δ⇒=θ

r121

21 +=θ ),( 42 htO Δ⇒

Computational Fluid Dynamics

211

11 22 h

ffftff n

jnj

nj

nj

nj −+

−+ +−=

Δ−

α

Richardson’s Method

Richardson Method

),( 22 htO Δ

Similar to Leapfrog

but unconditionally unstable!

j-1 j j+1

n

n+1

n-1

Computational Fluid Dynamics

21

111

11

2 hffff

tff n

jnj

nj

nj

nj

nj −

−++

−+ +−−=

Δ−

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 f j

n+1 + f jn−1( ) / 2

( ) ( )nj

nj

nj

nj

nj fffrfrf 1

11

11 221 −−

+−+ +−+=+

Computational Fluid Dynamics

xxxxxxt fht

hff ⎥⎦⎤

⎢⎣⎡ Δ−=−

2

232

121 ααα

Modified Equation

DuFort-Frankel - 2

xxxxxxfht

th ⎥⎦⎤

⎢⎣⎡ Δ+Δ−+

4

45234 2

31

3601 ααα

Amplification factor

rrr

G21

sin41cos2 22

+−±

=ββ Unconditionally

stable

Conditionally consistent

Computational Fluid Dynamics

FTCSStable for

BTCSUnconditionally

Stable

Crank-NicolsonUnconditionally

Stable

RichardsonUnconditionally

Unstable

xxt ff α=

211

1 2h

ffftff n

jnj

nj

nj

nj −++ +−

=Δ−

α ( ) xxxxfrh 6112

2

−α

Parabolic Equation – Elementary Schemes

21≤r

2

11

111

1 2h

ffftff n

jnj

nj

nj

nj

+−

+++

+ +−=

Δ−

α ( ) xxxxfrh 6112

2

+α

f jn+1 − f j

n

Δt=

α2h2

f j+1n+1 − 2 f j

n+1 + f j−1n+1( )⎡⎣

+ f j+1n − 2 f j

n + f j−1n( )⎤⎦

xxxxxxxxxx ft

fh

1212

232 Δ+αα

211

11 22 h

ffftff n

jnj

nj

nj

nj −+

−+ +−=

Δ−

α ),( 22 htO Δ

Computational Fluid Dynamics

DuFort-Frankel UnconditionallyStable,

ConditionallyConsistent

3-Level ImplicitUnconditionally

Stable

xxt ff α=

( ) xxxxfrh 22

12112

−α

2

11

111

11

2243

hfff

tfff

nj

nj

nj

nj

nj

nj

+−

+++

−+

+−

=Δ

+−

αxxxxf

h12

2α

21

111

11

2 hffff

tff n

jnj

nj

nj

nj

nj −

−++

−+ +−−=

Δ−

α

And others!

Parabolic Equation – Elementary SchemesComputational Fluid Dynamics

Numerical Methods forMulti-Dimensional Heat

Equations

Computational Fluid Dynamics

Two-dimensional grid

j

j-1

j+1

i i+1i-1

⎟⎟⎠

⎞⎜⎜⎝

⎛∂∂+

∂∂=

∂∂

2

2

2

2

yf

xf

tf α

Consider a 2-D heat equation

Computational Fluid Dynamics

⎟⎟⎠

⎞⎜⎜⎝

⎛∂∂+

∂∂=

∂∂

2

2

2

2

yf

xf

tf α

Applying forward Euler scheme:

⎟⎟⎠

⎞⎜⎜⎝

⎛Δ

+−+

Δ+−

=Δ− −+−+

+

21,,1,

2,1,,1,

1, 22

yfff

xfff

tff n

jinji

nji

nji

nji

nji

nji

nji α

2-D heat equation

hyx =Δ=ΔIf

( )njinji

nji

nji

nji

nji

nji fffff

htff

,1,1,,1,12,

1, 4−+++=Δ−

−+−+

+ α

Explicit Method - 1

Computational Fluid Dynamics

In matrix form

f1,1n+1

f1,2n+1

f1,J

n+1

f2,1n+1

f2,2n+1

fI ,J −1n+1

fI ,Jn+1

⎡

⎣

⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢

⎤

⎦

⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥

=

f1,1n

f1,2n

f1,J

n

f2,1n

f2,2n

fI ,J −1n

fI ,Jn

⎡

⎣

⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢

⎤

⎦

⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥

+αΔth2

−4 1 0 0 1 0 1 −4 1 0 0 1 0 1 −4 1 0 1 1 00 10 0 1

0 1 −4

⎡

⎣

⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢

⎤

⎦

⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥

f1,1n

f1,2n

f1,J

n

f2,1n

f2,2n

fI ,J −1n

fI ,Jn

⎡

⎣

⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢

⎤

⎦

⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥

Explicit Method - 2Computational Fluid Dynamics

Explicit Method - 3

Von Neumann Analysis imyikxnnj eeεε =

�

εn+1 = εn + αΔth2

eikh + e− ikh + eimh + e−imh − 4( )εn

�

εn+1

εn=1+ αΔt

h22coskh + 2cosmh − 4( )

⎟⎠⎞⎜

⎝⎛ +Δ−=

2sin

2sin41 22

2

mhkhhtα

Worst case

1811 2 ≤Δ−≤−htα

41

2 ≤Δhtα

Computational Fluid Dynamics

Δtαh2

=12

Δtαh2

=14

�

Δtαh2

= 16

One-dimensional flow

Two-dimensional flow

Three-dimensional flow

Stability limits depend on the dimension of the problems

Different numerical algorithms usually have different stability limits

StabilityComputational Fluid Dynamics

Implicit time integration

Computational Fluid Dynamics

f

i , j

n+1 = fi , j

n +Δtαh2

⎛⎝⎜

⎞⎠⎟

fi+1, jn + fi−1, j

n + fi, j−1n + fi, j+1

n − 4 fi, jn( )

Implicit Methods

f

i , j

n+1 = fi , j

n +Δtαh2

⎛⎝⎜

⎞⎠⎟

fi+1, jn+1 + fi−1, j

n+1 + fi, j−1n+1 + fi, j+1

n+1 − 4 fi, jn+1( )

Evaluate the spatial derivatives at the new time (n+1), instead of at n

(1+ 4A) f

i , j

n+1 − A fi+1, jn+1 + fi−1, j

n+1 + fi, j−1n+1 + fi, j+1

n+1( ) = fi , j

n

This gives a set of linear equations for the new temperatures:

Known source term

Recall forward in time method

Computational Fluid Dynamics

�

fi , j

n+1 = 11+ 4A

A fi+1, jn+1 + f i−1, j

n+1 + f i, j−1n+1 + f i, j+1

n+1( ) + fi , j

n( )

Isolate the new fi,j and solve by iteration

The implicit method is unconditionally stable, but it is necessary to solve a system of linear equations at each time step. Often, the time step must be taken to be small due to accuracy requirements and an explicit method is competitive.

Implicit Methods

Computational Fluid Dynamics

Second order accuracy in time can be obtained by using the Crank-Nicolson method

n

n+1

i i+1

i-1

j+1

j-1

j

Implicit MethodsComputational Fluid Dynamics

The matrix equation is expensive to solve

Crank-Nicolson

Crank-Nicolson Method for 2-D Heat Equation

⎟⎟⎠

⎞⎜⎜⎝

⎛∂∂+

∂∂+

∂∂+

∂∂=

Δ− +++

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++

−++

+−

++

+ −+++Δ+= nji

nji

nji

nji

nji

nji

nji fffff

htff α

hyx =Δ=ΔIf

( )njinji

nji

nji

nji fffff

ht

,1,1,,1,12 42

−+++Δ+ −+−+α

Computational Fluid Dynamics

Expensive to solve matrix equations.

Can larger time-step be achieved without having solve the matrix equation resulting from the two-dimensional system?

The break through came with the Alternation-Direction-Implicit (ADI) method (Peaceman & Rachford-mid1950’s)

ADI consists of first treating one row implicitly with backward Euler and then reversing roles and treating the other by backwards Euler.

Peaceman, D., and Rachford, M. (1955). The numerical solution of parabolic and elliptic differential equations, J. SIAM 3, 28-41

Computational Fluid Dynamics

The ADI Method

Computational Fluid Dynamics

Fractional Step:

Alternating Direction Implicit (ADI)

�

f n+1/ 2 − f n = αΔt2h2

fi+1, jn+1/ 2 − 2 f i, j

n+1/ 2 + f i−1, jn+1/ 2( ) + f i, j+1

n − 2 f i, jn + f i, j−1

n( )[ ]

( )hyx =Δ=Δ

�

f n+1 − f n+1/ 2 = αΔt2h2

fi+1, jn+1/ 2 − 2 f i, j

n+1/ 2 + f i−1, jn+1/ 2( ) + f i, j+1

n+1 − 2 f i, jn+1 + f i, j−1

n+1( )[ ]

Step 1:

Step 2:

⎥⎦

⎤⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛∂∂+

∂∂+

∂∂Δ=−

+++

2

2

2

12

2

2/121

21

2 yf

yf

xft

ffnnn

nn α

Combining the two becomes equivalent to:

Midpoint Trapisodial

Computational Fluid Dynamics

Computational Molecules for the ADI Method

n

n+1/2

n+1

i i+1

i-1

j+1

j-1

j

Computational Fluid Dynamics

Instead of solving one set of linear equations for the two-dimensional system, solve 1D equations for each grid line.

The directions can be alternated to prevent any bias

Computational Fluid Dynamics

In matrix form, for each row

fi,1n+1/ 2

fi,2n+1/ 2

fi,N

n+1/ 2

⎡

⎣

⎢⎢⎢⎢⎢⎢

⎤

⎦

⎥⎥⎥⎥⎥⎥

=

fi,1n

fi,2n

fi,N

n

⎡

⎣

⎢⎢⎢⎢⎢⎢

⎤

⎦

⎥⎥⎥⎥⎥⎥

+αΔth2

−2 1 01 −2 10 1 −2

⎡

⎣

⎢⎢⎢⎢

⎤

⎦

⎥⎥⎥⎥

fi,1n+1/ 2

fi,2n+1/ 2

fi,N

n+1/ 2

⎡

⎣

⎢⎢⎢⎢⎢⎢

⎤

⎦

⎥⎥⎥⎥⎥⎥

+ source

This equation is easily solved by forward elimination and back-substitution

Computational Fluid Dynamics

Stability Analysis:

ADI Method is accurate

Similarly,

( )22 ,htO Δimyikxnn

j eeεε =

ε n+1/2 = ε n + αΔth2

ε n+1/2 eikh − 2 + e− ikh( ) + ε n eimh − 2 + e− imh( )⎡⎣ ⎤⎦

ε n+1/2

ε n=1− 2αΔt

h2sin2 kh

21+ 2αΔt

h2sin2 mh

2

ε n+1

ε n+1/2=1− 2αΔt

h2sin2 mh

21+ 2αΔt

h2sin2 kh

2

Computational Fluid Dynamics

Combining

Unconditionally stable!

1

2sin21

2sin21

2sin21

2sin21

22

22

22

221

<⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜

⎝

⎛

Δ+

Δ−

⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜

⎝

⎛

Δ+

Δ−=

+

khht

mhht

mhht

khht

n

n

α

α

α

α

εε

The 3-D version does not have the same desirable stability properties. However, it is possible to generate similar methods for 3D problems

Computational Fluid Dynamics

Implicit methods for parabolic equations• Allow much larger time step (but must be

balanced against accuracy!)• Preserve the parabolic nature of the equation

But, require the solution of a linear set of equations and are therefore much more expensive that explicit methods

ADI provides a way to convert multidimensional problems into a series of 1D problems

Computational Fluid Dynamics

Approximate factorization

Splitting

Computational Fluid DynamicsApproximate Factorization - 1

The Crank-Nicolson for heat equation becomes

Define δ xx ( ) = 1Δx2

( )i−1, j − 2( )i, j + ( )i+1, j⎡⎣ ⎤⎦

δ yy ( ) = 1Δy2

( )i, j−1 − 2( )i, j + ( )i, j+1⎡⎣ ⎤⎦

f n+1 − f n

Δt=α2δ xx f n+1 + f n( ) + α

2δ yy f n+1 + f n( ) +O(Δt 2 ,Δx2 ,Δy2 )

1− αΔt2

δ xx −αΔt2

δ yy⎛⎝⎜

⎞⎠⎟f n+1 = 1+ αΔt

2δ xx +

αΔt2

δ yy⎛⎝⎜

⎞⎠⎟f n

+ΔtO(Δt 2 ,Δx2 ,Δy2 )

which can be rewritten as

Computational Fluid DynamicsApproximate Factorization - 2

Factoring each side

+ΔtO(Δt 2 ,Δx2 ,Δy2 )

1− αΔt2

δ xx⎛⎝⎜

⎞⎠⎟1− αΔt

2δ yy

⎛⎝⎜

⎞⎠⎟f n+1 −

α 2Δt 2

4δ xxδ yy f

n+1 =

or

1+ αΔt2

δ xx⎛⎝⎜

⎞⎠⎟1+ αΔt

2δ yy

⎛⎝⎜

⎞⎠⎟f n −

α 2Δt 2

4δ xxδ yy f

n

1− αΔt2

δ xx⎛⎝⎜

⎞⎠⎟1− αΔt

2δ yy

⎛⎝⎜

⎞⎠⎟f n+1 = 1+ αΔt

2δ xx

⎛⎝⎜

⎞⎠⎟1+ αΔt

2δ yy

⎛⎝⎜

⎞⎠⎟f n

+α 2Δt 2

4δ xxδ yy f n+1 − f n( ) + ΔtO(Δt 2 ,Δx2 ,Δy2 )

Crank-Nicolson

Computational Fluid DynamicsApproximate Factorization - 3

The ADI method can be written as

1− αΔt2

δ xx⎛⎝⎜

⎞⎠⎟f n+1/2 = 1+ αΔt

2δ yy

⎛⎝⎜

⎞⎠⎟f n

1− αΔt2

δ yy⎛⎝⎜

⎞⎠⎟f n+1 = 1+ αΔt

2δ xx

⎛⎝⎜

⎞⎠⎟f n+1/2

Eliminating fn+1/2

1− αΔt2

δ yy⎛⎝⎜

⎞⎠⎟1− αΔt

2δ xx

⎛⎝⎜

⎞⎠⎟f n+1 = 1+ αΔt

2δ yy

⎛⎝⎜

⎞⎠⎟1+ αΔt

2δ xx

⎛⎝⎜

⎞⎠⎟f n+1

Computational Fluid Dynamics

�

αΔt2

⎛ ⎝ ⎜

⎞ ⎠ ⎟ 2

δyyδxx f n+1 − f n( ) ≈ α2

⎛ ⎝ ⎜

⎞ ⎠ ⎟ 2

Δt 3δxxδyy∂f∂t

= O(Δt 3)

Up to a factor:

ADI is an approximate factorization of the Crank-Nicolson method

Approximate Factorization - 4

Computational Fluid Dynamics

The Peaceman-Rachford methods does not have the same stability properties in 3D as in 2D. A method with similar properties is:

1− αΔt2

δ xx⎛⎝⎜

⎞⎠⎟f n+1* = 1+ αΔt

2δ xx + 2δ yy + 2δ zz( )⎡

⎣⎢⎤⎦⎥f n

1− αΔt2

δ yy⎛⎝⎜

⎞⎠⎟f n+1** = f n+1* − αΔt

2δ yy f

n

1− αΔt2

δ zz⎛⎝⎜

⎞⎠⎟f n+1 = f n+1** − αΔt

2δ zz f

n

Several other splitting methods have been proposed in the literature.

Approximate Factorization - 5Computational Fluid Dynamics

Similar ideas can be used for time splitting:

fi, jn+1/2 = 1+αΔtδ xx( ) fi, jn

fi, jn+1 = 1+αΔtδ yy( ) fi, jn+1/2

fi, jn+1 = 1+αΔtδ yy( ) 1+αΔtδ xx( ) fi, jn

Eliminate the half step:

Which is equivalent to the standard FTCS method, except for the

α 2Δt 2δ yyδ xx fi, jn

term, which is higher order.

Approximate Factorization - 6

Computational Fluid Dynamics

Why splitting?

1.  Stability limits of 1-D case apply.

2.  Different Δt can be used in x and y directions.

Computational Fluid Dynamics

Implicit time marching by fast

elliptic solvers

Computational Fluid Dynamics

fi, jn+1 − fi, j

n

Δt= α∇h

2 fi, jn+1

∇h2 fi, j

n+1 −fi, jn+1

αΔt= −

fi, jn

αΔt

∇h2 fi, j

n+1 − λ fi, jn+1 = S

Backward Euler

Rearrange

or

Can be solved by elliptic solvers

Computational Fluid Dynamics

One-Dimensional Problems•  Explicit, implicit, Crank-Nicolson•  Accuracy, stability•  Various schemes

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

Splitting

Outline

Solution Methods for Parabolic Equations

Computational Fluid Dynamics

TheAdvection-Diffusion

Equation

http://www.nd.edu/~gtryggva/CFD-Course/Computational Fluid Dynamics

�

∂u∂t

+ u∂u∂x

+ v∂u∂y

= −1ρ∂P∂x

+µρ

∂ 2u∂x 2

+∂ 2u∂y2

⎛ ⎝ ⎜ ⎞

⎠ ⎟

Navier-Stokes equations

�

∂u∂x

+∂v∂y

= 0

Parabolic part

Hyperbolic part

Elliptic equation

Summary

The Navier-Stokes equations contain three equation types that have their own characteristic behavior

Depending on the governing parameters, one behavior can be dominant

The different equation types require different solution techniques

For inviscid compressible flows, only the hyperbolic part survives

Computational Fluid Dynamics

�

∂f∂t

+U ∂f∂x

= D∂ 2 f∂x 2

�

f jn+1 − f j

n

Δt+U

f j+1n − f j−1

n

2h= D

f j+1n − 2 f j

n + f j−1n

h2

1D Advection/diffusion equation

Forward in time/centered in space (FTCS)

Stability:

22

21111

1

hD

hU

t

nj

nj

nj

nj

nj

nj

nj −+−++ +−

=−

+Δ− εεεεεεε

jikxnnj eεε =

Write:ε j±1n = ε neikx j e±ikhand use

Computational Fluid Dynamics

Gives)2()(

21 2

1ikhikhikhikh

n

n

eehtDee

htU −−

+

+−Δ+−Δ−=εε

�

=1− 4 DΔth2

sin2 k h2− iUΔt

hsinkh α =

kh2

A = DΔth2

B =UΔth

G2 = 1− 4Asin2α( )2+B2 sin2 2α ≤1

For stability

1−8Asin2α +16A2 sin4α +B2 sin2 2α ≤1

8Asin2α 2Asin2α −1( )+ 4B2 cos2α sin2α ≤ 0AB2

2cos2α

2Asin2α −1( )+1≤ 0or

Must be <0

Must be <-1

A ≤ 12

2 AB2

≥1

ΔtDh2

≤12

U2ΔtD

≤ 2

Computational Fluid Dynamics

U 2Δt2D

≤ 1 & DΔth2 ≤

12

�

∂f∂t

+U ∂f∂x

= D∂ 2 f∂x 2

�

f jn+1 − f j

n

Δt+U

f j+1n − f j−1

n

2h= D

f j+1n − 2 f j

n + f j−1n

h2

1D Advection/diffusion equation

Forward in time/centered in space (FTCS)

Stability limitsR =

ULD

Δt = 2D

U 2 & Δt = h2

2D

�

Δt→ 0 For high and low D

Computational Fluid Dynamics

�

∂f∂t

+U ∂f∂x

= D∂ 2 f∂x 2

�

O Δt,h2( )FTCS

�

UΔt2D

≤1 & DΔth2

≤ 12

�

O Δt,h( )Upwind

�

UΔth

+ 2DΔth2

≤1

�

O Δt 2,h2( )L-W

�

UΔth

⎛ ⎝ ⎜

⎞ ⎠ ⎟ 2

≤ 2 DΔth2

⎛ ⎝ ⎜

⎞ ⎠ ⎟ ≤1

�

O Δt 2,h2( )C-N Unconditionally stable

Computational Fluid Dynamics

Steady state solution to the advection/diffusion equation

U∂f∂x

= D∂ 2 f∂x2

Exact solution

�

f =exp RL x /L( ) −1exp RL( ) −1

�

RL = ULD

�

f = 0

�

f =1

�

U

�

L

Computational Fluid Dynamics

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

RL=1RL=5RL=10RL=20

Computational Fluid Dynamics

The Cell Reynolds number

Computational Fluid Dynamics

�

U ∂f∂x

= D∂ 2 f∂x 2

�

Uf j+1 − f j−12h

= Df j+1 − 2 f j + f j−1

h2

Centered difference approximation

Numerical solution of:

�

Uf j − f j−1h

= Df j+1 − 2 f j + f j−1

h2

Upwind

Computational Fluid Dynamics

Ufj+1 − f j−12h

= Df j+1 − 2 f j + f j−1

h2

Uh2

2hDfj+1 − fj−1( ) = fj+1 − 2 fj + fj−1

R =UhD

Where:

Rearrange:

Rearrange:

R− 2( ) fj+1 + 4 fj − R + 2( ) fj−1 = 0

Centered difference approximation

Computational Fluid Dynamics

fj = qj

Divide by

Solution

Substitute:

R− 2( )qj+1 + 4qj − R+ 2( )qj−1 = 0

R− 2( )q2 + 4q1 − R+ 2( ) = 0

q j−1

R− 2( ) fj+1 + 4 fj − R + 2( ) fj−1 = 0

Computational Fluid Dynamics

Solving for q gives two solutions:

q1 =1 and

The general solution is:

R− 2( )q2 + 4q1 − R+ 2( ) = 0

q2 =2 + R2 − R

fj = C1q1j + C2q2

j

fj = C1 + C22 + R2 − R

⎛ ⎝

⎞ ⎠

jor

Computational Fluid Dynamics

Apply the boundary conditions

f0 = C1 + C22 + R2 − R

⎛ ⎝

⎞ ⎠

0

= C1 + C2 = 0

fN = C1 + C22 + R2 − R

⎛ ⎝

⎞ ⎠

N

= 1

fj =

2 + R2 − R

⎛ ⎝

⎞ ⎠

j

−1

2 + R2 − R

⎛ ⎝

⎞ ⎠

N +1

−1

The final solution is:

Computational Fluid Dynamics

�

Uf j − f j−1h

= Df j+1 − 2 f j + f j−1

h2

�

R + 2( ) f j − R +1( ) f j−1 − f j+1 = 0R =

UhD

�

f j = q j

�

q2 − R + 2( )q1 + R +1( ) = 0

�

f j =1− 1+ R( ) j

1− 1+ R( )NSolution

Upwind

Try solutions

giving

or

Computational Fluid Dynamics

fj =

2 + R2 − R

⎛ ⎝

⎞ ⎠

j

−1

2 + R2 − R

⎛ ⎝

⎞ ⎠

N +1

−1

�

f =exp RL x /L( ) −1exp RL( ) −1

�

RL = ULD

R =UhD

Exact solution

Centered differences

Upwind

�

f j =1− 1+ R( ) j

1− 1+ R( )N

Computational Fluid Dynamics

UpwindExact

Centered

Computational Fluid Dynamics

UpwindExact

Centered

Computational Fluid Dynamics

UpwindExact

Centered

Computational Fluid Dynamics

When centered differencing is used for the advection/diffusion equation, oscillations may appear when the Cell Reynolds number is higher than 2. For upwinding, no oscillations appear. In most cases the oscillations are small and the cell Reynolds number is frequently allowed to be higher than 2 with relatively minor effects on the result.

�

R = UhD

< 2

Computational Fluid Dynamics

2D example

�

∂f∂t

+U ∂f∂x

+V ∂f∂y

= D ∂ 2 f∂x 2

+ ∂ 2 f∂y 2

⎛

⎝ ⎜

⎞

⎠ ⎟

�

f =1Flow

�

f = 0

�

f = 0

Computations using centered differences on a 32 by 32 grid

Computational Fluid Dynamics

D=0.02t=1.5088

Recell=3.2258

Computational Fluid Dynamics

D=0.01t=1.5088

Recell=6.4516

Computational Fluid Dynamics

D=0.005t=1.5088

Recell=12.9032

Computational Fluid Dynamics

D=0.02t=1.50

0

5

10

15

0

5

10

150

0.5

1

1.5

05

1015

2025

30

05

1015

2025

300

0.5

1

1.5

Recell=3.2258 Recell=6.6716

Fine grid Coarser grid

Computational Fluid Dynamics

The Cell Reynolds number limitation becomes an issue when the advection terms are discretized using centered differences.

In many cases the oscillations only appear in isolated regions where diffusion and advection are of comparable magnitude

Upwind avoids the problem, but generally adds dissipation