How to solve PDEs using MATHEMATIA and MATLAB

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G. Y. Park, S. H. Lee and J.K. LeeDepartment of Electronic and Electrical Engineering,

POSTECH

2006. 5. 17

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ContentsContents

• Solving PDEs using MATHEMATICA

- FTCS method

- Lax method

- Crank Nicolson method

- Jacobi’s method

- Simultaneous-over-relaxation (SOR) method

• Solving PDEs using MATLAB

- Examples of PDEs

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ReferencesReferences

• Textbook

- ‘Numerical and Analytical Methods for Scientists and Engineers Using Mathematica’, Daniel Dubin, Wiley, 2003

- ‘Applied numerical methods in C’, S. Nakamura

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PDE (Partial Differential Equation)PDE (Partial Differential Equation) PDEs are used as mathematical models for phenomena in

all branches of engineering and science.

- Three Types of PDEs:

1) Elliptic: Steady heat transfer, flow and diffusion

2) Parabolic: Transient heat transfer, flow and diffusion

3) Hyperbolic: Transient wave equation

fauuc )(

fauuct

ud

)(

fauuct

ud

)(2

2

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FTCS method for the heat equationFTCS method for the heat equation

FTCS ( Forward Euler in Time and Central difference in Space )

) :),(),(),(),0((

),(),(),(

21

2

2

ydiffusivitthermaltTtLTtTtT

txStxTx

txTt

Heat equation in a slab

),(1

t

TTtxT

t

nj

nj

nj

2

11

2

2 2),(

x

TTTtxT

x

nj

nj

nj

nj

)2( 1121 n

jnj

nj

nj

nj

nj TTT

x

ttSTT

njnj TtxT ),(Let

FTCS method for the heat equationFTCS method for the heat equation

Initial conditions

Plot

FTCS

01.0t

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Stability of FTCS and CTCSStability of FTCS and CTCSFTCS is first-order accuracy in time and second-order accuracy in space.So small time steps are required to achieve reasonable accuracy.

05.0t Courant condition for FTCS

2

12

x

t

)2(2

2 11211 n

jnj

nj

nj

nj

nj TTT

x

ttSTT

CTCS method for heat equation(Both the time and space derivatives are center-differenced.)

However, CTCS method is unstable for any time step size.

( unstable )

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Lax methodLax methodSimple modification to the CTCS method

In the differenced time derivative,

)2(2

2)( 1121

11

1211 n

jnj

nj

nj

nj

nj

nj TTT

x

ttSTTT

t

TTT

t

TT nj

nj

nj

nj

nj

2

)]([

2

)( 11

112

1111

Replacement by average value from surrounding grid points

The resulting difference equation is

Courant condition for Lax method

4

12

x

t( Second-order accuracy in both time and space )

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Crank Nicolson Algorithm ( Implicit Method )Crank Nicolson Algorithm ( Implicit Method )

BTCS ( Backward time, centered space ) method for heat equation

)2( 111

2

nj

nj

njx

tnj

nj

nj TTTtSTT

( This is stable for any choice of time steps, however it is first-order accurate in time. )

Crank-Nicolson scheme for heat equation

taking the average between time steps n-1 and n,

)22(

)(1

111

111

1211

2

nj

nj

nj

nj

nj

njx

t

nj

nj

nj

nj

TTTTTT

SStTT

( This is stable for any choice of time steps and second-order accurate in time. )

a set of coupled linear equations for njT

Crank Nicolson AlgorithmCrank Nicolson Algorithm

Initial conditions

Plot

Crank-Nicolson scheme

)/sin(),(22 / LxetxT Lt

Exact solution

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Crank Nicolson AlgorithmCrank Nicolson Algorithm

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Multiple Spatial DimensionsMultiple Spatial Dimensions

),,(),,(),,(2

2

2

2

tyxStyxTyx

tyxTt

FTCS for 2D heat equation

)2()2( 1121121 n

jknjk

njk

nkj

njk

nkj

njk

njk

njk TTT

y

tTTT

x

ttSTT

Courant condition for this scheme

22

22

2

1

yx

yxt

( Other schemes such as CTCS and Lax can be easily extended to multiple dimensions. )

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Wave equation with nonuniform wave speedWave equation with nonuniform wave speed

),,(),(),,(2

2

2

22

2

2

tyxzyx

yxctyxzt

2D wave equation

0

),()0,,(),,()0,,(

00

00

vz

yxvyxzyxzyxz

txxtxz

txztyztyz

2sin)1(),0,(

,0),1,(),,1(),,0(

Initial condition :

Boundary condition :

])5.0(5)5.0(2exp[7.01),( 22101 yxyxcWave speed :

)2()2(2 112

22

112

2211 n

jknjk

njk

jknkj

njk

nkj

jknjk

njk

njk TTz

y

tczzz

x

tczzz

CTCS method for the wave equation :

) ( 2

12

22

yxfortc

Courant condition :

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Wave equation with nonuniform wave speedWave equation with nonuniform wave speed

Since evaluation of the nth timestep refers back to the n-2nd step, for the first step, a trick is employed.

jkt

jkjkjk atvzz 2001 2

)2()2( 112

20

100

12

2njk

njk

njk

jkkjjkkj

jkjk TTz

y

czzz

x

ca

Since initial velocity and value,

0

01

00

jk

jkjk

z

zv

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Wave equation with nonuniform wave speedWave equation with nonuniform wave speed

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Wave equation with nonuniform wave speedWave equation with nonuniform wave speed

2D Poisson’s equation2D Poisson’s equation

),(),()( 2

2

2

2

yxyxyx

Poisson’s equation

Direct Solution for Poisson’s equation

jkjkjkjkkjjkkj

yx

2

11

2

11 22

Centered-difference the spatial derivatives

Jacobi’s method ( Relaxation method )Jacobi’s method ( Relaxation method )Direct solution can be difficult to program efficiently.

Relaxation methods are relatively simple to code, however, they are not as fast as the direct methods.

Idea :

I. Poisson’s equation can be thought of as the equilibrium solution to the heat equation with source.

II. Starting with any initial condition, the heat equation solution will eventually relax to a solution of Poisson’s equation.

)2(

1)2(

1

2

111211222

221 n

jknjk

njk

nkj

njk

nkjjk

njk

njk yxyx

yx

)(1

)(1

2

111211222

22njk

njk

nkj

nkjjk yxyx

yx

),(),(),,(t

),(),( 22 yxyxtyxyxyx

) 2

1 (

22

22

yx

yxt

FTCS

(Maximum time step satisfying Courant condition)

Jacobi methodJacobi method

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Simultaneous OverRelaxation (SOR)Simultaneous OverRelaxation (SOR)

The convergence of the Jacobi method is quite slow.

Furthermore, the larger the system, the slower the convergence.

Simultaneous OverRelaxation (SOR) :

the Jacobi method is modified in two ways,

)(1

)(1

2)1(

1112

1112

22

221

njk

njk

nkj

nkjjk

njk

njk

yx

yx

yx

1. Improved values are used as soon as they become available.

2. Relaxation parameter ω tries to overshoot for going to the final result. ( 1<ω<2)

Simultaneous OverRelaxation (SOR)Simultaneous OverRelaxation (SOR)

O.V. Manuilenko

MATLAB The Language of Technical Computing

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MATLAB PDE

Run: dftcs.m >> dftcs

dftcs - Program to solve the diffusion equation using the Forward Time Centered Space scheme.

Enter time step: 0.0001Enter the number of grid points: 51Solution is expected to be stable

2

2 ),(),(

x

txT

t

txT

O.V. Manuilenko

MATLAB The Language of Technical Computing

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MATLAB PDE

Run: dftcs.m >> dftcs

dftcs - Program to solve the diffusion equation using the Forward Time Centered Space scheme.

Enter time step: 0.00015Enter the number of grid points: 61WARNING: Solution is expected to be unstable

O.V. Manuilenko

MATLAB The Language of Technical Computing

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MATLAB PDE

Run: neutrn.m >> neutrn Program to solve the neutron diffusion equation using the FTCS.Enter time step: 0.0005Enter the number of grid points: 61Enter system length: 2 => System length is subcriticalSolution is expected to be stableEnter number of time steps: 12000

),(),(),(

2

2

txnx

txn

t

txn

O.V. Manuilenko

MATLAB The Language of Technical Computing

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MATLAB PDE

Run: neutrn.m >> neutrn Program to solve the neutron diffusion equation using the FTCS.Enter time step: 0.0005Enter the number of grid points: 61Enter system length: 4 => System length is supercriticalSolution is expected to be stableEnter number of time steps: 12000

O.V. Manuilenko

MATLAB The Language of Technical Computing

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MATLAB PDE

Run: advect.m >> advect

advect - Program to solve the advection equation using the various hyperbolic PDE schemes: FTCS, Lax, Lax-Wendorf

Enter number of grid points: 50Time for wave to move one grid spacing is 0.02Enter time step: 0.002Wave circles system in 500 stepsEnter number of steps: 500

FTCS FTCS

0),(),(

x

txu

t

txu

O.V. Manuilenko

MATLAB The Language of Technical Computing

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MATLAB PDE

Run: advect.m >> advect

advect - Program to solve the advection equation using the various hyperbolic PDE schemes: FTCS, Lax, Lax-Wendorf

Enter number of grid points: 50Time for wave to move one grid spacing is 0.02Enter time step: 0.02Wave circles system in 50 stepsEnter number of steps: 50

Lax Lax

O.V. Manuilenko

MATLAB The Language of Technical Computing

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MATLAB PDE

Run: relax.m >> relax

relax - Program to solve the Laplace equation using Jacobi, Gauss-Seidel and SOR methods on a square grid

Enter number of grid points on a side: 50Theoretical optimum omega = 1.88184 Enter desired omega: 1.8Potential at y=L equals 1 Potential is zero on all other boundariesDesired fractional change = 0.0001

0),(),(

2

2

2

2

y

yx

x

yx