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EEE 431 Computational Methods in Electrodynamics

EEE 431 Computational Methods in Electrodynamics. Lecture 6 By Dr. Rasime Uyguroglu Rasime.uyguroglu@emu.edu.tr. FINITE DIFFERENCE METHODS (cont). FINITE DIFFERENCE METHODS (cont). Finite Differencing of Hyperbolic PDE’s Consider the wave equation:. FINITE DIFFERENCE METHODS (cont).

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EEE 431 Computational Methods in Electrodynamics

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  1. EEE 431Computational Methods in Electrodynamics Lecture 6 By Dr. Rasime Uyguroglu Rasime.uyguroglu@emu.edu.tr

  2. FINITE DIFFERENCE METHODS (cont).

  3. FINITE DIFFERENCE METHODS (cont). • Finite Differencing of Hyperbolic PDE’s • Consider the wave equation:

  4. FINITE DIFFERENCE METHODS (cont). • Using central difference formula the wave equation may be approximated as:

  5. FINITE DIFFERENCE METHODS (cont). • Substituting • Let

  6. FINITE DIFFERENCE METHODS (cont). • Example: Solve the wave equation • Subject to the boundary conditions, • And the initial conditions:

  7. Finite Difference Method • Take r=1, • For j=0,

  8. Finite Difference Method • Substitute to get the starting formula:

  9. Finite Difference Method • Since u=1, r=1, chose, • Solve the problem for since it is symmetric. See the C code.

  10. Finite Difference Method • Finite Differencing of Elliptic PDE’s. Consider the two dimensional Poisson’s Equation:

  11. Finite Difference Method • Central difference approximation for the partial derivatives:

  12. Finite Difference Method • Where, • Assume: • FD approximation of the Poisson’s equation after simplification:

  13. Finite Difference Method • Gives: • Or

  14. 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:

  15. 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.

  16. 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:

  17. 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:

  18. 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.

  19. Accuracy and Stability FD Solutions • Unavoidable errors in numerical solution of physical problems: • modeling errors, • truncation (or discretization) errors, • round-off errors

  20. 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.

  21. 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.

  22. 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.

  23. Accuracy and Stability FD Solutions • Round-off Errors, are due to finite precision of computers. • May be reduced by using double precision.

  24. Accuracy and Stability FD SolutionsError as a function of a mesh size

  25. 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.

  26. Accuracy and Stability FD Solutions • For the stability of the difference scheme it is required that the above equation satisfies: • or

  27. 2D Potential Distribution in a Discrete Inhomogeneous Dielectric • The relevant equation is:

  28. 2D Potential Distribution in a Discrete Inhomogeneous Dielectric • Divide the domain into a grid.

  29. 2D Potential Distribution in a Disceat Inhomogeneous Dielectric • And:

  30. 2D Potential Distribution in a Discrete Inhomogeneous Dielectric • So,

  31. 2D Potential Distribution in a Discreat Inhomogeneous Dielectric • Similarly:

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