Finite Difference Method for Hyperbolic Problems
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Transcript of 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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Types of Partial Differential Equations (PDEs)
2) Parabolic:
Transient heat transfer, flow and diffusion
equations
22
2( , ) ( , ) 0u ux t x t
t x
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Types of Partial Differential Equations (PDEs)
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))
Finite Difference Method (FDM)
Hyperbolic PDE’s – Wave Equation
Applying FDM to the Wave Equation
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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Finite Difference Schemes
0x x
f(x)
x
A
P B
0x x 0x
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Finite Difference Schemes
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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Finite Difference Schemes
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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Finite Difference Schemes
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:
Finite Difference Schemes
, 0,1,2,...
, 0,1,2,...
x i x i
t j t j
( , ) ( , )i j i x j t
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Finite Difference Schemes
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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Finite Difference Schemes
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))
Finite Difference Method (FDM)
Hyperbolic PDE’s – Wave Equation
Applying FDM to the Wave Equation
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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Hyperbolic PDE’s - Wave Equation
Boundary conditions subject to the wave equation are given.
, for
(0, ) ( , ) 0u t u l t 0t
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Hyperbolic PDE’s - Wave Equation
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)
Finite Difference Method (FDM)
Hyperbolic PDE’s – Wave Equation
Applying FDM to the Wave Equation
References
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Applying FDM to the Wave Equation
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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Applying FDM to the Wave Equation
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Applying FDM to the Wave Equation
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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Applying FDM to the Wave Equation
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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Applying FDM to the Wave Equation
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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Applying FDM to the Wave Equation
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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Applying FDM to the Wave Equation
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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Applying FDM to the Wave Equation
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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Applying FDM to the Wave Equation
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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Applying FDM to the Wave Equation
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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Applying FDM to the Wave Equation
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!!!