One-Dimensional Diffusion Equation - CEProfs · 2020-03-17 · HC Chen 3/17/2020 Chapter 7 1D...

HC Chen 3/17/2020

Chapter 7 1D Diffusion Equation 1

Chapter 7

One-Dimensional Diffusion Equation

Diffusion Equation Diffusion equation contains the dissipation

mechanism for fluid flow with significant viscous or heat conduction effects

Provide guidance for choosing numerical algorithms for viscous fluid flow

)( , ,,

)( , ,,

)( , ,,

1jj1j2jxx2xx

1j1jjxx

1jj1jjxx

nj

1nj

1nj

2

2

TT2Tx

1TL121

x

1L

TTx2

1TL101

x2

1L

TT21TTM21M

t

TT

t

T

x

T

t

T

HC Chen 3/17/2020


7.1 Explicit Methods 7.1.1 FTCS scheme (Two-level scheme) Explicit time-marching

njxx

1nj

2

n1j

nj

n1j

nj

1nj

2

2

TLt

T

x

TT2T

t

TT

x

T

t

T

7.1 Explicit Methods 7.1.2 Richardson and DuFort-Frankel schemes

(Three-level schemes) Richardson

DuFort-Frankel

njxx

1nj

1nj

nj

1nj TL

t2

TT

t

T

t

T

2

1

njxx

nj

1nj

2

n1j

1nj

1nj

n1j

1nj

1nj

TLt

Ts

2

1

t

Ts

2

1

x

TTTT

t2

TT

)()(

)(

HC Chen 3/17/2020


7.1 Explicit Methods 7.1.3 General Three-level schemes

Numerical Consistency

1njxx

njxx

1nj

nj

1nj TeLTdLcTbTaT

n 1 nj j n n 1

xx j xx j

T T1 1 L T L T

t t1 1 2

i e a b c d 1 e t t t

( ) ( )

. ., , , , ( ),

Time derivative Spatial derivative

7.2 Implicit Methods 7.2.1 Fully-Implicit Scheme

Need to solve a system of coupled algebraic equations at time level (n+1)

njxx

nj

1njxx

1nj

2

1n1j

1nj

1n1j

nj

1nj

2

2

TLt

T

TLt

T

x

TT2T

t

TT

x

T

t

T

HC Chen 3/17/2020


7.2 Implicit Methods 7.2.2 Crank-Nicolson Scheme

7.2.3 Generalized Three-Level Scheme

0T 50T.5 0Lt

TTL

TL 50TL.5 0t

T

1nj

njxx

1nj

a

1njxx

njxx

1nj

.)(

.

0TT1Lt

T

t

T1TL

TLTL1t

T

t

T1

1nj

njxx

nj

1nj

a

1njxx

njxx

nj

1nj

)()()(

)()(

7.2 Implicit Methods 7.2.4 Higher-Order Schemes Hybrid FD-FE three-level scheme

Truncation error

0TT1L

t

TM

t

TM1TL

1nj

njxx

nj

x

1nj

xa

)(

)()(

n

jxxxx2

anj T

s12

1

2

1xsTLE )()(

HC Chen 3/17/2020


Higher-Order Scheme Truncation error

Fourth-order scheme

n

jxxxx2

anj T

s12

1

2

1xsTLE )()(

s12

1

2

1

2

112

1

6

5

12

121M

12

1x

then

),,() , ,( If

for any combination of (s, , )

General Three-Level Scheme Hybrid finite-difference/finite-element

j1 j j+1

n+1

n

n1

1

1

Implicit 1+

Explicit ( = 0)

12

HC Chen 3/17/2020


Active nodes for diffusion equation

General Three-Level Scheme Mass and difference operators

t

T

t

T21

t

T

t

TM

21 M

x

TT2TTL

t

TT

t

T

n1j

nj

n1j

nj

x

x

2

n1j

nj

n1jn

jxx

nj

1nj

1nj

)(

,,

HC Chen 3/17/2020


Explicit Three-Level Scheme Explicit Method ( = 0)

(1) FTCS Scheme

(2) Richardson Scheme

(3) DuFort-Frankel Scheme

(4) Fourth-order scheme

1nj

njxx

21n

jnj

1nj

1njxx

njxx

nj

1nj

a

TT1L1

xsT

1T

1

21T

0TLTL1t

T

t

T1TL

)(

)()()(

s12

1

2

1

s2

10

2

10

00

,

,

,

Explicit Three-Level Scheme Explicit Method ( = 0)

Truncation error

Fourth-order scheme

0TT1Lt

T

t

T1TL 1n

jnjxx

nj

1nj

a

)()()(

n

jxxxxxx2

42

n

jxxxx2

anj

T2s12s360

1

6

1xs

Ts12

1

2

1xsTLE

)(

)()(

2

1

s12

1

HC Chen 3/17/2020


Von Neumann Analysis Explicit Three-level scheme

Amplification factor

0TT1Lt

T

t

T1TL 1n

jnjxx

nj

1nj

a

)()()(

01s211s221GG1G11s2G21G1

eGeG11s2eGeG21eG1

1eGx

2

e2eeGx

12

x

1L

2

2

ji1njin

ji1njinji1n

jin2

iijin

2

n1j

nj

n1j2

njxx

)](cos[)])(cos([)(]))[((cos)()(

]))[((cos )()(

))(cos(

))(()(

Numerical Stability Explicit Three-level scheme

For 4th-order scheme

01s211s221GG1 2 )](cos[)])(cos([)(

2

1

s12

14th-order scheme

HC Chen 3/17/2020


Implicit Three-Level Scheme

Truncation error

Fourth-order scheme

0TT1Lt

TM

t

TM1TL 1n

jnjxx

nj

x

1nj

xa

)( )()(

n

jxxxxxx22

42

n

jxxxx2

anj

T2s12s360

1

6

1

s2s2s12xs

Ts12

1

s2

1xsTLE

)(

)()(

ss12

1

2

1

Implicit Three-Level Schemes

(1) Fully-Implicit

(2) Crank-Nicolson

(3) Linear FEM/Imp

(4) Linear FEM/C-N

(5) 3-level Fully Implicit

(6) 4th-order FDM

(7) 4th-order Composite

2

1

12

1s12

1

2

10

12

10

2

10

6

1

106

12

100

100

,

,

, ,

, ,

, ,

, ,

, ,

0TT1Lt

TM

t

TM1TL 1n

jnjxx

nj

x

1nj

xa

)( )()(

HC Chen 3/17/2020


Truncation Error Analysis Implicit three-level scheme

(a) Difference operator

0TLTL1t

TM

t

TM1 1n

jxxnjxx

nj

x

1nj

x

)( )(

n

jxxxxxx

4n

jxxxx

2n

jxx

n

jxxxxxx

6n

jxxxx

4n

jxx

2

2

n1j

nj

n1j2

njxx

T360

xT

12

xT

T6

xT

4

xT

2

x

x

2

TT2Tx

1TL

!!!

)(

Difference Operator

Therefore,

n

jxxxxxxxxn

jxxxxxx1n

jxxxxxx

n

jxxxxxxxx

22n

jxxxxxxn

jxxxx1n

jxxxx

n

jxxxxxx

22n

jxxxxn

jxx

n

jxxtt

2n

jxxtn

jxx1n

jxx

1n

jxxxxxx

41n

jxxxx

21n

jxx

1n1j

1nj

1n1j2

1njxx

TtTT

T2

tTtTT

T2

tTtT

T2

tTtTT

T360

xT

12

xT

TT2Tx

1TL

!

!

!

)(

n

jxxxxxx242n

jxxxx2n

jxx

n

jxxxx

22n

jxx

njxx

1njxx

Ts360

1

s122xsT

s12

1xsT

T12

x1

12

xtT 1

TL1TL

)(

)()(

)(

HC Chen 3/17/2020


Truncation Error Analysis (b) Mass operator

n

jxxxxxx

23n

jxxxx

2n

jt1n

jnj

nj

n

jxxxxxx

23n

jxxxx

2n

jt

n

jttt

2n

jttn

jt

n

jttt

3n

jtt

2n

jtnj

1nj

1nj

T6

tT

2

tTTT

t

1

t

T

T6

tT

2

tT

T3

tT

2

tT

T3

tT

2

tTt

t

1TT

t

1

t

T

)(

!!

!!)(

n

jxxxxxx

42n

jxxxx

2n

jt

n

jtxxxx

4n

jtxx

2n

jtn

jt

n

1jtn

1jtn

jt

n

1jtn

jtn

1jtn

jtx

T12

xT

2

xT

T4

xT

2

xT2T21

TTT21

TT21TTM

!

!!)(

)(

)(

Mass Operator

Time derivative

n

jxxxxxxxx

2n

jxxxxxxn

jxxxxxxx

n

jxxxxxx

2n

jxxxxn

jxxxxx

T2

x2TTM

T2

x2TTM

!

!

n

jxxxxxx2

42

n

jxxxx2n

jt

n

jxxxx22n

jt

n

jxxxxxx

3n

jxxxx2n

jtx

nj

x

1nj

x

T6

1

s2

1

s12xs

Ts2

1xsT

Tt2

1xT

T6

tTt

2

21TM

t

TM

t

TM1

)(

HC Chen 3/17/2020


Diffusion Equation Combine time and spatial derivatives

Numerical consistency

n

jxxxxxx22

42

n

jxxxx2n

j

nj

n

jxxt

Ts360

1

s1226

1

s2s2s12xs

Ts12

1

s2

1xsE

ETTPDE

0tx0E nj , as

Summary Implicit three-level scheme

Explicit three-level scheme ( = 0)

Fourth-order schemes

Composite scheme

n

jxxxx2n

j Ts12

1

s2

1xsE

n

jxxxx2n

j Ts12

1

2

1xsE

ss12

1

2

1s12

1

2

1

:Implicit

:Explicit

2

1

12

1 choose

(Independent of s)

HC Chen 3/17/2020


TnewTold

HC Chen 3/17/2020


Explicit Schemes - DIFEX

3L-4th scheme

Told Tnew

HC Chen 3/17/2020


3-level 4th-order scheme

HC Chen 3/17/2020


Exact Solution

Explicit schemes - DIFEX

s12

1

2

1

289012

1s

6

1s

:4th3L

. :FrankelDuFort

:FTCS

(s < 0.41 for = 1)

HC Chen 3/17/2020


Implicit Schemes - DIFIM

HC Chen 3/17/2020


Tridiagonalmatrix system

HC Chen 3/17/2020


HC Chen 3/17/2020


Explicit Scheme - DIFEX

Implicit Scheme - DIFIM

Numerical Accuracy

Implicit Schemes – DIFIM

(s = 1)

FDM-2nd

FEM-2nd

FDM-4th

FEM-4th

Composite

FDM-2nd

FEM-2nd

FDM-4th

FEM-4th

Composite

correct

HC Chen 3/17/2020


Numerical Accuracy

S = 0.41: stability for explicit FDM-4th

7.3 Boundary and Initial Conditions

Neumann boundary conditions

HC Chen 3/17/2020


1st

2nd

2nd

2nd

2nd

2nd

4th

4th

Rate of convergence depends on the boundary conditions, not just the accuracy of discretization scheme

Explicit treatment of Neumann BCs

Implicit treatment of Neumann BCs

1st

2nd

2nd

2nd

It is important to maintain the same order of accuracy for discretization scheme and boundary conditions

HC Chen 3/17/2020


7.4 Method of Lines (Semi-Discretization)

Discretize the spatial term first

Reduced to ODE in time

Solve a system of ODEs by Runge-Kutta or multistep methods

fuAdt

ud

0uu2ut

s

dt

du

TT2Txdt

dT

TT

1jj1jj

1jj1j2

j

xxt

)(or

)(

Time Integration Linear multistep method (for time integration)

One-step integration (m = 1)

Euler scheme (1= 0 = 0 = 1, 1 = 0)

n

m

0

nm

0

fut

1 Weighted-average of time-derivative and

slope f

)( 1n1

n0

1n1

n0 fftuu Two-level

scheme

njxx

n1j

nj

n1j

n

nn1n

uLuu2ut

sf

ftuu

)(

FTCS

scheme

HC Chen 3/17/2020


Crank-Nicolson Scheme One-step integration (m = 1)

Trapezoid scheme (1= 0 = 1, 0 = 1 = 1/2)

All two-level schemes are special cases of the one-step method

)( 1n1

n0

1n1

n0 fftuu

njxx

n1j

nj

n1j

n

1nnn1n

uLuu2ut

sf

f50f5.0tuu

)(

).(

Two-Step Integration Two-step integration (m = 2)

General three-level schemes

Explicit three-level scheme (2 = 0)

Implicit three-level scheme (2 = 1)

e.g, Three-level fully implicit scheme

)( 2n2

1n1

n0

2n2

1n1

n0 ffftuuu

, , ,

. , ,.

001

50251

012

012

3LFI

HC Chen 3/17/2020


Classical 4th-order Runge-Kutta Method

Most widely used one-step methodn

xxnnn1n uLfff2f2f

6

tuu

; **)****(

weighted-average slope

**)****( ,

**)*,(*** ,** ***

*)*,(** ,* .**

*),(* , .*

ff2f2f6

1fftuu

utffftuu

utffft50uu

utffft50uu

nn1n

1nn

2

1n

n

2

1n

nn

Error in eq.(7-53), p. 244

Classical 4th-orderRunge-Kutta Method

t n t n+1/2

f n

f *

f **

f ***

**)****( ff2f2f6

1f n

f

t n+1

One-Dimensional Diffusion Equation - CEProfs · 2020-03-17 · HC Chen 3/17/2020 Chapter 7 1D...

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