eee 431 computational methods in electrodynamics
DESCRIPTION
EEE 431 Computational Methods in Electrodynamics. Lecture 6 By Dr. Rasime Uyguroglu [email protected]. FINITE DIFFERENCE METHODS (cont). FINITE DIFFERENCE METHODS (cont). Finite Differencing of Hyperbolic PDE’s Consider the wave equation:. FINITE DIFFERENCE METHODS (cont). - PowerPoint PPT PresentationTRANSCRIPT
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FINITE DIFFERENCE METHODS (cont).
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FINITE DIFFERENCE METHODS (cont). Finite Differencing of Hyperbolic PDE’s Consider the wave equation:
2 2
22 2
ux t
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FINITE DIFFERENCE METHODS (cont).
Using central difference formula the wave equation may be approximated as:
2
2
2 2
2 2
( 1, ) 2 ( , ) ( 1, )
( )
( , 1) 2 ( , ) ( , 1)
( )
i j i j i j
x x
i j i j i j
t t
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FINITE DIFFERENCE METHODS (cont).
Substituting
Let
22 2
( 1, ) 2 ( , ) ( 1) ( , 1) 2 ( , ) ( , 1)
( ) ( )
i j i j i i j i j i ju
x t
22 2
2
( )u t u tr
xx
( 1, ) 2 ( , ) ( 1, ) ( , 1) 2 ( , ) ( , 1)
( , 1) 2(1 ) ( , ) ( 1, ) ( 1, ) ( , 1)
r i j i j i j i j i j i j
i j r i j r i j i j i j
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FINITE DIFFERENCE METHODS (cont).
Example: Solve the wave equation
Subject to the boundary conditions,
And the initial conditions:
0 1, 0tt xx x t
(0, ) 0 (1, ) 0t t t
( ,0) sin , 0 1,
( ,0) 0, 0 1t
x x x
x x
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Finite Difference Method
Take r=1,
For j=0,
( , 1) 2(1 ) ( , ) ( 1, ) ( 1, ) ( , 1)
( , 1) ( 1, ) ( 1, ) ( , 1), 1
i j r i j r i j i j i j
i j i j i j i j j
( ,0) ( ,1) ( , 1)0
2( ,1) ( , 1)
t
x i i
t ti i
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Finite Difference Method
Substitute to get the starting formula:
( ,1) ( 1,0) ( 1,0) ( , 1)
( ,1) ( 1,0) ( 1,0) ( ,1)
1( ,1) ( 1,0) ( 1,0)
2
i i i i
i i i i
i i i
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Finite Difference Method
Since u=1, r=1, chose,
Solve the problem for since it is symmetric. See the C code.
,
0.1
x t
x t
0 0.5x
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Finite Difference Method
Finite Differencing of Elliptic PDE’s. Consider the two dimensional Poisson’s Equation:
2 2
2 2( , )g x y
x y
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Finite Difference Method
Central difference approximation for the partial derivatives:
2
2
22 2
22 2
( 1, ) 2 ( , ) ( 1, )( )
( )
( , 1) 2 ( , ) ( , 1)( )
( )
i j i j i jO x
x x
i j i j i jO y
y y
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Finite Difference Method
Where,
Assume:
FD approximation of the Poisson’s equation after simplification:
, , 0,1,2,3...x i x y j y and i j
x y h
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Finite Difference Method
Gives:
Or
2( 1, ) ( 1, ) ( , 1) ( , 1) 4 ( , ) ( , )i j i j i j i j i j h g i j
21( , ) ( 1, ) ( 1, ) ( , 1) ( , 1) ( , )
4i j i j i j i j i j h g i j
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Finite Difference Method
When the source term vanishes, the Poisson’s equation leads to the Laplace’s equation. Thus for the same mesh size h:
1( , ) ( 1, ) ( 1, ) ( , 1) ( , 1)
4i j i j i j i j i j
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Finite Difference Method
The application of the finite difference method to elliptic PDEs often leads to a large system of algebraic equations to be solved.
Solution of such equations is a major problem. Band matrix and iterative methods are commonly used to solve the system of equations.
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Finite Difference Method
Band Matrix Method Notice that only nearest neighboring
nodes affect the value of at each node.
Application of the FD equations results in a set of equation such that:
A X B
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Finite Difference Method
Where is a sparse matrix (it has many zeros) , is the column matrix consisting of the unknown values, and
is the column matrix containing the known values of . So:
A X
B
1X A B
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Accuracy and Stability FD Solutions
Accuracy is the closeness of the approximate solution to the exact solutions.
Stability is the requirement that the scheme does not increase the magnitude of the solution with increase in time.
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Accuracy and Stability FD Solutions
Unavoidable errors in numerical solution of physical problems:
modeling errors, truncation (or discretization) errors, round-off errors
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Accuracy and Stability FD Solutions
Modeling errors: Several assumptions are made for obtaining the mathematical model. i.e. nonlinear system may be represented by a liner PDE.
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Accuracy and Stability FD Solutions
Truncation errors, arise from the fact that in numerical analysis we can deal only with finite number of terms of a series.
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Accuracy and Stability FD Solutions
Truncation errors may be reduced: By using finer meshes. i.e. smaller time
and space step sizes and more number of points.
By using a large number of terms in the series expansion of derivatives.
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Accuracy and Stability FD Solutions Round-off Errors, are due to finite
precision of computers. May be reduced by using double
precision.
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Accuracy and Stability FD SolutionsError as a function of a mesh size
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Accuracy and Stability FD Solutions
To determine whether the FD scheme is stable, define an error, , which occurs at time step n, assuming a single independent variable. Define the amplification of this error at time step n+1 as:
Where is known as amplification factor.
n
1n ng g
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Accuracy and Stability FD Solutions
For the stability of the difference scheme it is required that the above equation satisfies:
or
1n n
1g
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2D Potential Distribution in a Discrete Inhomogeneous Dielectric
The relevant equation is:
( ) ( ) 0x x y y
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2D Potential Distribution in a Discrete Inhomogeneous Dielectric
Divide the domain into a grid.
( 1, ) ( 1, )
2
i j i j
x x
1/ 2, 1/ 2,( ) i j i jx x
x x x
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2D Potential Distribution in a Disceat Inhomogeneous Dielectric
And:
1, ,
1/ 2
, 1,
1/ 2
i j i j
i
i j i j
i
x x
x x
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2D Potential Distribution in a Discrete Inhomogeneous Dielectric
So,
1/ 2, 1, , 1/ 2, , 1,
2
1/ 2, 1, 1/ 2, 1/ 2, , 1/ 2, 1,
2
( ) ( )( )
( )
( )
( )
i j i j i j i j i j i j
i j i j i j i j i j i j i j
x x x
x
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2D Potential Distribution in a Discreat Inhomogeneous Dielectric
Similarly:
, 1/ 2 , 1 , , 1/ 2 , , 1
2
, 1/ 2 , 1 , 1/ 2 , 1/ 2 , , 1/ 2 , 1
2
( ) ( )( )
( )
( )
( )
i j i j i j i j i j i j
i j i j i j i j i j i j i j
y y x
x