Partial differential equations

Function depends on two or more independent variables

0

y

u

x

u

This is a very simple one - there are many more complicated ones

1252

2

2

3

yu

x

ux

yx

u

Order of the PDE is given by its highest derivative

xyx

u

y

u

yx

u4

2

2

2

2

22

4

is 2nd order

xyx

u

yx

u

y

u42

2

22

2

2

is 4th order

Linear PDE is linear in dependent variable, and all coefficients depend on independent variables only

222

2

3

3

4 yxx

uy

y

ux

Nonlinear PDEs violate these rules

14 222

2

2

2

3

3

uyxx

uy

y

uxu

PDE that often appears in engineering is second order, linear PDE

General form:

02

22

2

2

Dy

uC

yx

uB

x

uA

A, B, C are functions of x and y

D is function of x,y,u and andx

u

y

u

Can use values of coefficients A,B,C to characterize the PDE

B2-4AC Type Example

<0 Elliptic Laplaceequation

=0 Parabolic Heatconduction

>0 Hyperbolic Waveequation

02

2

2

2

y

T

x

T

2

2

x

Tk

t

T

01

2

2

22

2

t

y

cx

y

Why categorize?

Different methods to solve different types

Different types describe different engineering problems

• Elliptic - steady state

• Parabolic - propagation

• Hyperbolic - vibrations

Analytic solutions - there aren’t many

Often can use analytic tools to get idea of behavior of a PDE, especially as parameters are changed

Important for limiting cases

Elliptic PDEs

Steady-state two-dimensional heat conduction equation is prototypical elliptic PDE

02

2

2

2

y

T

x

T

This is the Laplace equation

yxfy

T

x

T,

2

2

2

2

This is the Poisson equation

Think of a small box

qx, in

qy, out

qx, out

qy, in

At steady state, net change in heat is 0, so

0,,,, outyinyoutxinx qqqq

Shrink to differential size

0

y

q

x

q

Fourier’s law of heat conduction

a

TCkq pa

Substitute

02

2

2

2

y

T

x

T

0

y

TCk

yx

TCk

x pp

We will solve with finite differences

Discretize PDE so that we have a mesh of grid points with boundary conditions

0

1

2

3

4

5

6

7

8

9

10

0 1 2 3 4 5 6 7 8 9 10

Index for x is i

Index for y is j

Use finite differences for the derivatives

2

,1,,1

2

2 2

x

TTT

x

T jijiji

0

1

2

3

4

5

6

7

8

9

10

0 1 2 3 4 5 6 7 8 9 10

Ti-1,j

Ti,j

Ti+1,j

Now the y derivative

2

1,,1,

2

2 2

y

TTT

y

T jijiji

0

1

2

3

4

5

6

7

8

9

10

0 1 2 3 4 5 6 7 8 9 10

Ti,j+1

Ti,j

Ti,j-1

Substitute these expressions back into original elliptic PDE

022

2

1,,1,

2

,1,,1

y

TTT

x

TTT jijijijijiji

Assume x=y. Can rearrange to get

04 ,1,1,,1,1 jijijijiji TTTTT

True for all interior points

0

1

2

3

4

5

6

7

8

9

10

0 1 2 3 4 5 6 7 8 9 10

Need to define values on ALL boundaries - Dirichlet boundary condition

(Neumann BC fix flux at boundary)

Each interior point has an equation - for 9 x 9 interior points - 81 equations

Adds up quickly

Example: 4 x 4 grid - 2 x 2 interior points

0

1

2

3

0 1 2 3

75

20

10

02550

60 45 30

75

75

75

1354

046075

04

1,12,11,2

1,12,11,2

1,10,12,11,01,2

TTT

TTT

TTTTT

0

1

2

3

0 1 2 3

75

20

10

02550

60 45 30

75

75

75

Let i=1, j=1

Fill in the matrix

35

65

125

135

411

141

141

114

2,2

1,2

2,1

1,1

T

T

T

T

Generally, we get a sparse matrix (big, too)

Technique most often used is Gauss-Seidel or some variation of it - matrix is always diagonally dominant - also called Liebmann’s rule

Apply these steps iteratively until T converges

41,1,,1,1

,

jijijijiji

TTTTT

Rewrite equation in Gauss-Seidel form

04 ,1,1,,1,1 jijijijiji TTTTT

Use overrelaxation (if desired)

oldji

newji

newji TTT ,,, 1

Solving our example - the four equations are

0

1

2

3

0 1 2 3

75

20

10

02550

60 45 30

75

75

75

0654

01354

2,12,21,1

1,11,22,1

TTT

TTT

0354

01254

2,21,22,1

1,22,21,1

TTT

TTT

Rewrite them in Gauss Seidel form

4

354

1254

654

135

1,22,12,2

2,21,11,2

2,21,12,1

1,22,11,1

TTT

TTT

TTT

TTT

and assume initial values for T

75

75

75

75

2,2

1,2

2,1

1,1

T

T

T

T

Run without overrelaxation

iterationstart 1 2 3 4 5 6 7 8 9T11 75 71.25 64.38 60.31 58.59 57.58 57.15 56.89 56.79 56.72T12 75 53.75 45.63 42.19 40.16 39.3 38.79 38.57 38.45 38.39T21 75 68.75 60.63 57.19 55.16 54.3 53.79 53.57 53.45 53.39T22 75 46.25 39.38 35.31 33.59 32.58 32.15 31.89 31.79 31.72ea1 0.053 0.107 0.067 0.029 0.018 0.008 0.004 0.002 0.001ea2 0.395 0.178 0.081 0.051 0.022 0.013 0.006 0.003 0.001ea3 0.091 0.134 0.06 0.037 0.016 0.009 0.004 0.002 0.001ea4 0.622 0.175 0.115 0.051 0.031 0.013 0.008 0.003 0.002

0

10

20

30

40

50

60

70

80

0 2 4 6 8 10 12

T11

T12

T21

T22

0

1

2

3

0 1 2 3

75

20

10

02550

60 45 30

75

75

75

End result

56.72 38.39

31.7953.39

What about derivative (flux) boundary conditions

I.E. if we insulate one side of the plate, is 0 there

x

T

Create an imaginary point outside boundary

T0,j+1

T-1,j T1,jT0,j

T0,j-1

Equation becomes

04 ,0,1,11,01,0 jjjjj TTTTT

Now consider finite difference for derivative at 0

0,1,1

,1,1

0

2

2

t

TxTT

x

TT

t

T

jj

jj

Substitute

042 ,00

,1,11,01,0

jjjjj Tt

TxTTTT

Derivative BC now included in equation

Irregular domains (funny shapes)

What do you do with a domain like?

Your book uses , to scale the x, y

Different x, y

0

1

2

3

4

5

0 1 2 3 4 5

1 x

2 x

1 y

2 y

Can develop equations for edge points

02

2

212

,1,

211

,1,

2

212

,,1

211

,,

2

jijijiji

jijijijii

TTTT

y

TTTT

x

Now use a Gauss-Seidel or other matrix approach