finite difference method for hyperbolic problems

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EEE 484Finite Differences Method

forHYPERBOLIC PROBLEMS

Presentationby

ÖZDEN PARSAK

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Content

Partial Differential Equations (PDE’s)

Finite Difference Method (FDM)

Hyperbolic PDE’s – Wave Equation

Applying FDM to the Wave Equation

References

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Partial Differential Equations

Physical problems that involve more than one variable are often expressed using equtions involving partial derivatives. And it is called Partial Differential Equation (PDE’s).

There are three types of partial differential equations

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Types of Partial Differential Equations (PDEs)

1) Elliptic:

2 2

2 2( , ) ( , ) ( , )u ux y x y f x y

x t

Poisson equation, Laplace equation for

steady state.

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2) Parabolic:

Transient heat transfer, flow and diffusion

equations

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2( , ) ( , ) 0u ux t x t

t x

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3) Hyperbolic:

Transient Wave Equation

2 22

2 2( , ) ( , ) 0u ux t x t

t x

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Content

Partial Differential Equations (Partial Differential Equations (PDE’sPDE’s))




References

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FINITE DIFFERENCE METHODS

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Introduction

For such complicated problems numerical methods must be employed.

Finite Difference Method (FDM) is one of the available numerical methods which can easily be applied to solve Partial Differential Equations (PDE’s) with such complexity.

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Introduction

Finite Difference techniques are based upon approximations of differential equations by finite difference equations.

Finite difference approximations:• have algebraic forms,

• Relate the value of the dependent variable at a point in the solution region to the values at some neighboring points.

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Introduction

Steps of finite difference solution: Divide the solution region into a grid of

nodes, Approximate the given differential

equation by finite difference equivalent, Solve the differential equations subject to

the boundary conditions and/or initial conditions.

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Introduction

Rectangular Grid Pattern

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Finite Difference Schemes

The derivative of a given function f(x) can be approximated as:

0 0 0

0 0 0

0 0 0

( ) ( ) ( )

( ) ( ) ( )

( ) ( ) ( )

2

df x f x x f xForward Difference Formula

dx xdf x f x f x x

Backword Difference Formuladx x

df x f x x f x xCentral Difference Formula

dx x

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0x x

f(x)

x

A

P B

0x x 0x

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Estimate of second derivative of f(x) at P by using Central Difference Formula;

2 3320 0 0

0 0 2 3

2 3320 0 0

0 0 2 3

( ) ( ) ( )1 ( )( ) ( ) ( ) ...

2! 3!

( ) ( ) ( )1 ( )( ) ( ) ( ) ...

2! 3!

f x f x f xxf x x f x x x

x x x

f x f x f xxf x x f x x x

x x x

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Adding these equations:

Assuming that higher order terms are neglected:

22 40

0 0 0 2

( )( ) ( ) 2 ( ) ( ) ( )

f xf x x f x x f x x O x

x

20 0 0

2 2

( ) 2 ( ) ( )

( )

f f x x f x f x x

x x

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To apply the difference method to find the solution of a function the solution region is divided into rectangles:

( , )x t

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Let the coordinates (x, t) of a typical grid point or a node be:

The value of at a point:


, 0,1,2,...

, 0,1,2,...

x i x i

t j t j

( , ) ( , )i j i x j t

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Using this notation, the central difference approximations of at (i, j) are: First derivative:

( , )

( , )

( 1, ) ( 1, )

2

( , 1) ( , 1)

2

i j

i j

i j i j

x x

i j i j

t t

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Second derivative:

2

2 2( , )

2

2 2( , )

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

( )

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

( )

i j

i j

i j i j i j

x x

i j i j i j

t t

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Content

Partial Differential Equations (Partial Differential Equations (PDE’sPDE’s))




References

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Hyperbolic PDE’s - Wave Equation

The wave equation which is an example of Hyperbolic Equation is given by the differential equation.

for , 0 x l 0t

2 22

2 2( , ) ( , ) 0u ux t x t

t x

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Boundary conditions subject to the wave equation are given.

, for

(0, ) ( , ) 0u t u l t 0t

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Initial conditions subject to the wave equation are given.

, for

( ,0) ( )u x f x

( ,0) ( )ux g x

t

0 x l

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Content

Partial Differential Equations (PDE’s)




References

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Select ; length step-size for and time-step size

The mesh point are defined by

0k 0m /h l m

( , )i jx t

ix ih

jt jk

0,1,...,i m

0,1,.....j

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At any interior mesh point , the wave equations becomes

( , )i jx t

2 22

2 2( , ) ( , ) 0i j i j

u ux t x t

t x

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The Difference method is applied to the second partial derivatives using the centred-difference formula, then we obtain the second order difference equations;

for some in ,

1 1( , )j jt t j

2 2 41 1

2 2 4

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

12i j i j i j

i j i j

u x t u x t u x tu k ux t x

t k t

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for some in .

i

1 1( , )i ix x

2 41 1

2 4

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

12i j i j i j

i j i j

u x t u x t u x t h ux t t

h t

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Substituting these equations into the wave equation, we obtain;

1 1

2

( , ) 2 ( , ) ( , )i j i j i ju x t u x t u x t

k

4 42 2 2

4 4

1( , ) ( , )

12 i j i j

u uk x h t

t x

1 122

( , ) 2 ( , ) ( , )i j i j i ju x t u x t u x t

h

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Neglecting the error term

Then the difference equation

4 42 2 2

4 4

1( , ) ( , )

12ij i j i j

u uk x h t

t x

, 1 , , 1 1, , 1,22 2

2 20i j i j i j i j i j i jw w w w w w

k h

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With , we can solve the difference

equation for , the most advanced time-step

approximation is to be obtained,

/k h

, 1i jw

2 2, 1 , 1, 1, , 12(1 ) ( )i j i j i j i j i jw w w w w

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2 2, 1 , 1, 1, , 12(1 ) ( )i j i j i j i j i jw w w w w

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This difference equation holds for, and

From boundary conditions;

for each

From initial conditions;

for each

0,1,.....j

0 0j mjw w 1,2,3.....j

0 ( )i iw f x 1,2,..., ( 1)i m

1,2..., ( 1)i m

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It is easy to solve that equation when writing this set of equations in matrix form;

2 2

1, 1 1, 12 2 2

2, 1 2, 1

2

2 2

1, 1 1, 1

2(1 ) 0 0

2(1 )

0 0. .

. .0 0 2(1 )

j j

j j

m j m j

w w

w w

w w

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References

Numerical Methods, J. Douglas FAIRES,

Richard L. BURDEN

Brooks Cole; Third edition (June 18, 2002)

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Thanks for your attention!!!

finite difference method for hyperbolic problems

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