PARTIAL DIFFERENTIAL PARTIAL DIFFERENTIAL EQUATION (PDE)MTK II – JURUSAN TEKNIK KIMIA FT UGM

How to solve PDE..??

Several methods can be used:Several methods can be used:a. FDA : Explicit (Forward)b. FDA : Implicit (Backward)p ( )c. FDA : Crank-Nicolson (CN)d. Lines Method

Finite Difference Approximation (FDA)pp ( )

Terdapat tiga jenis finite difference approximation:Terdapat tiga jenis finite difference approximation:

xyy

dxdy ii

Δ−

≈ +1Forward : Turunan kedua:

dd ⎞⎛⎞⎛xdx Δ

xyy

dxdy ii

Δ−

≈ −1Backward : 12

2

xdxdy

dxdy

dxdy

dxd

dxyd ii

Δ

⎟⎠⎞

⎜⎝⎛−⎟

⎠⎞

⎜⎝⎛

≈⎟⎠⎞

⎜⎝⎛= +

xdx Δ

xyy

dxdy ii

Δ−

≈ −+

211Central : 11

xyy

xyy

xdxdxdx

iiii

Δ−

−Δ−

≈

Δ⎠⎝

−+

xdx Δ.2

( )211 .2 yyy

xiii

Δ+−

≈

Δ≈

+−

( )2xΔ

FDA - METHOD

1 Discretize the domain into a grid of evenly spaces points 1. Discretize the domain into a grid of evenly spaces points (nodes)

2. Express the derivatives in terms of Finite Difference 2 2Approximations of O(h2) and O(Δt) [or order O(Δt2)]

2 Fi i 2 T∂2

2T

x∂∂

Finite Differences

2

2T

y∂∂

Tt

∂∂

3. Choose h = Δx = Δy, and Δt and use the I.C.'s and B.C.'s to solve the problem by systematically moving ahead in time.

Discretize the solution domain in space d i i h h Δ d k Δ

9

10

and time with h = Δx and k = Δt

t7

8

9t

4

5

6

1

2

3

Time0

1

0 1 2 3 4 5 6 7 8 9 10

(j index)

space xspace (i index)

x

FDA  : Crank‐Nicolson (CN)

Provides 2nd‐order accuracy in both space and time.Average the 2nd‐derivative in space for tj+1 and tj. 

tm+1 RightLeft

xi-1 xi xi+1

tmBndry50°C

Bndry100°C

tm+1/2

Initial temperature0 °C

)(1du)(1

,1, jiji uukdt

−= +Avarage FDA time level j+1 and j

⎤⎡2

FDA space

6⎥⎦⎤

⎢⎣⎡ +−++−= ++++−+− )2()2(

21

1,11,1,12,1,,122

2

jijijijijiji uuuhcuuu

hc

dxud

Example: Heat EquationExample: Heat Equationp qp q

Heat transfer in a one-dimensional rod

x = 0x = a

g (t) g (t)

uu 2∂∂

g1(t) g2(t)

ax0f(x)u(x 0)

Tt0 a,x0 ;x uc

tu

2

<<

≤≤<<∂∂

=∂∂

Tt0,)t(gt)u(0,

ax0 f(x),u(x,0)

1 ≤≤⎨⎧ =

<<=

Tt0 , )t(g)t,a(u

2

≤≤⎩⎨ =

Discretize the solution domain in space d i i h h Δ d k Δ

9

10

and time with h = Δx and k = Δt

t7

8

9t

4

5

6

1

2

3

Time

0

1

0 1 2 3 4 5 6 7 8 9 10

(j index)

space xspace (i index)

x

FDA : Crank-Nicolson (CN)

Crank-Nicolson method for heat equation

( )

⎩⎨⎧

======

jkt ,m/Tt kihx ,n/ax h

letj

i

ΔΔ

⎩

)uu2u(h2c)uu(

k1

j,1ij,ij,1i2j,i1j,i +−+ +−=−

)uu2u(h2c

)(h2

)(k

1j,1i1j,i1j,1i2

j,1ij,ij,1i2j,i1j,i

++++−

++

+−+

22 xtc

hckr

ΔΔ

==h2

Rearrange

j,1ij,ij,1i1j,1i1j,i1j,1i u2ru)r1(u

2ru

2ru)r1(u

2r

+−++++− +−+=−++−

Example: CrankExample: Crank‐‐Nicolson MethodNicolson Method• Heat Equation (Parabolic PDE)

1x0 ;2

2

≤≤=d

udcddu

4002u(x,0)

;2

⎨⎧ += x

dxdt

• c = 0 5 h = 0 25 k = 0 1

60),1( ,20t)u(0, 2⎩⎨

== −− tt etue

c   0.5, h   0.25, k   0.1

60e -2t20e -t1

60e 20e -t

1 2 3 4020 + 40 x

0

ExampleExample• Crank‐Nicolson method

22 8.0)250(

)10.0)(5.0(hckr ===

j1,iji,j1,i1j1,i1ji,1j1,i u2rr)u(1u

2ru

2rr)u(1u

2r

)25.0(h

+−++++− +−+=−++−

• Tridiagonal matrix (r = 0.8), dimulai dari j=0 dan i=1

j1,iji,j1,i1j1,i1ji,1j1,i 0.4u0.2u0.4u0.4u1.8u0.4u2222

+−++++− ++=−+−

Tridiagonal matrix (r   0.8), dimulai dari j 0 dan i 1

⎪⎫

⎪⎧ ++−+

⎫⎧⎥⎤

⎢⎡ −+ 10020100 ururu)r1(ur0rr1

⎪⎪

⎪⎪⎪

⎬

⎪⎪

⎪⎪⎪

⎨ +−+

+++

=⎪⎭

⎪⎬

⎫

⎪⎩

⎪⎨

⎧

⎥⎥⎥⎥⎥

⎢⎢⎢⎢⎢

−+−

+

0,30,20,1

1,00,20,10,0

1,2

1,1

u2ru)r1(u

2r

u2

u2

u)r1(u2

uuu

2rr1

2r

02

r1

⎪⎪⎪

⎭⎪⎪⎪

⎩++−+

⎪⎭

⎪⎩⎥⎥⎦⎢

⎢⎣

+− 1,40,40,30,21,3 u

2ru

2ru)r1(u

2ru

r12r0

• Solve the tridiagonal matrix

⎪

⎪⎬

⎫

⎪

⎪⎨

⎧

+++++

=⎪

⎪⎬

⎫

⎪

⎪⎨

⎧

⎥⎥⎥⎤

⎢⎢⎢⎡

−−− −

)50(4.0)40(2.0)30(4.0)e20(4.0)40(4.0)30(2.0)20(4.0

uu

4.08.14.004.08.1 1.0

1,2

1,1

⎪⎬

⎫⎪⎨

⎧=⎪⎬

⎫⎪⎨

⎧⇒⎪

⎬

⎫⎪⎨

⎧=

⎪⎭

⎪⎩ +++⎪

⎭⎪⎩⎥⎦⎢⎣ − −

299758553942144598.29

uu

4023869934.37

)e60(4.0)60(4.0)50(2.0)40(4.0u8.14.00

1,1

2.01,3

⎪⎭

⎬⎪⎩

⎨=⎪⎭

⎬⎪⎩

⎨⇒⎪⎭

⎬⎪⎩

⎨=42746748.4729975855.39

uu

64953807.6940

1,3

1,2

1 29.42 39.30 47.43

60e -2t20e -t

1 2 3 4020 + 40 x

0

Method of Lines

Converting Partial Differential Eqs into set of Converting Partial Differential Eqs into set of Ordinary Differential Eqs Discretizing only the spatial derivatives using FDA Discretizing only the spatial derivatives using FDA and leaving the time derivatives unchanged.

2dd2

2

=i

dxud

dtdu α

( )112 2 −+ +−Δ

= iiii uuu

xdtdu α

The complete set of differential equations for 0 <= i <=N ld bwould be:

( )1010 2 +−= uuudu α

(BC 1)Direction of Integration

( )

( )01221

1012

2

2 −

+−Δ

=

+Δ

=

uuuxdt

du

uuuxdtα

(BC 1)

t=21

t= j

( )2

....

+

Δ

idu

xdt

αt=0t=1

ii-2 i-1 i+1 i+2ddd d d( )112

.....

2 −+ +−Δ

= iiii uuu

xdt dtdu i

dtdu i 1−

dtdu i 2−

dtdu i 1+

dtdu i 2+

( )112 2 −+ +−Δ

= NNNN uuu

xdtdu α

(BC 2)

Simultaneous ODE

Example: Method of Lines• Heat Equation (Parabolic PDE)

1x0 ;2

2

≤≤=d

udcddu

4002u(x,0)

;2

⎨⎧ += x

dxdt

• c = 0 5 h=Δx = 0 25 k = Δt = 0 1

60),1( ,20t)u(0, 2⎩⎨

== −− tt etue

c   0.5, h Δx   0.25, k  Δt   0.1

60e -2t20e -t1

60e 20e -t

1 2 3 4020 + 40 x

0

ExampleExample

( )10120 2 TTT

ddT

+−= −α

⎪⎨⎧ == − 02458849.1920 05.0

10 eT( )

( )01221

1012

2 TTTxdt

dTxdt

+−Δ

=

Δ −

α ⎪⎩

⎪⎨

== − 29024508.5460 10.01,4

1,0

eT

( )12322 2 TTT

xdtdT

xdt

+−Δ

=

Δα

Dicari T1,1; T2,1; T3,1 dengan

( )

( )

23423 2

dT

TTTxdt

dT+−

Δ=

α

α menyelesaikan persamaan disampingsecara simultan.

( )34524 2 TTT

xdtdT

+−Δ

=α

Menuju ke “t “berikutnya….j y

Stability Analysis

Explicit Euler method for heat equation

y y

⎩⎨⎧

======

jkt ,m/Tt kihx ,n/ax h

letj

i

ΔΔ

)2()(1,1,,12,1,2

2

jijijijiji uuuhcuu

kdxudc

dtdu

+−+ +−=−⇒=

Rearrange

j1ijij1iji1ji )uu2u(ckuu +−+=

j,1ij,ij,1i

j,1ij,ij,1i2j,i1j,i

ruu)r21(ru

)uu2u(h

uu

+−

+−+

+−+=

++

22 xtc

hckr

ΔΔ

==

5.0r0 ≤<Stability:

j,1ij,ij,1i1j,i ruu)r21(ruu +−+ +−+=

60e -2t20e -t1

• Stable010980010010 ++⇒

1 2 3 4020 + 40 x

0

j,1ij,ij,1i1j,i

j,1ij,ij,1i1j,i

40204040

u 1.0u 8.0u 1.0u 1.0r

u01.0u98.0u01.0u 01.0r

+−+

+−+

++⇒

++=⇒=

++=⇒=

U bl ( i ffi i )j,1ij,1i1j,i

j,1ij,ij,1i1j,i

u 5.0u 5.0u 5.0r

u4.0u2.0u4.0u 4.0r

+−+

+−+

+=⇒=

++=⇒=

• Unstable (negative coefficients)

⎪⎨

⎧ +−=⇒= +−+ j,1ij,ij,1i1j,i

10191010u u u u 1r

⎪⎩

⎪⎨

+−=⇒=+−=⇒=

+−+

+−+

j,1ij,ij,1i1j,i

j,1ij,ij,1i1j,i

u100u 199u100u 100ru10u19u10u 10r

Heat Heat EquationEquation: Time: Time--dependent BCsdependent BCs

r = 0.4

Explicit Euler Method: StabilityExplicit Euler Method: Stability

r = 1

Unstable !!