EEE 431 Computational Methods in Electrodynamics

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EEE 431Computational Methods in Electrodynamics

• Lecture 6
• By
• Dr. Rasime Uyguroglu
• Rasime.uyguroglu_at_emu.edu.tr

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FINITE DIFFERENCE METHODS (cont).
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FINITE DIFFERENCE METHODS (cont).

• Finite Differencing of Hyperbolic PDEs
• Consider the wave equation

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FINITE DIFFERENCE METHODS (cont).

• Using central difference formula the wave
  equation may be approximated as

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FINITE DIFFERENCE METHODS (cont).

• Substituting
• Let

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FINITE DIFFERENCE METHODS (cont).

• Example Solve the wave equation
• Subject to the boundary conditions,
• And the initial conditions

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Finite Difference Method

• Take r1,
• For j0,

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Finite Difference Method

• Substitute to get the starting formula

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Finite Difference Method

• Since u1, r1, chose,
• Solve the problem for since it is
  symmetric. See the C code.

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Finite Difference Method

• Finite Differencing of Elliptic PDEs. Consider
  the two dimensional Poissons Equation

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Finite Difference Method

• Central difference approximation for the partial
  derivatives

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Finite Difference Method

• Where,
• Assume
• FD approximation of the Poissons equation after
  simplification

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Finite Difference Method

• Gives
• Or

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Finite Difference Method

• When the source term vanishes, the Poissons
  equation leads to the Laplaces equation. Thus
  for the same mesh size h

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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

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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

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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 n1 as
• Where is known as amplification factor.

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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

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2D Potential Distribution in a Discrete
Inhomogeneous Dielectric

• The relevant equation is

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2D Potential Distribution in a Discrete
Inhomogeneous Dielectric

• Divide the domain into a grid.

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2D Potential Distribution in a Disceat
Inhomogeneous Dielectric

• And

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2D Potential Distribution in a Discrete
Inhomogeneous Dielectric

• So,

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2D Potential Distribution in a Discreat
Inhomogeneous Dielectric

• Similarly