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Finite Difference Methods in Electrodynamics: Wave and Poisson's Equations

In this lecture, Dr. Rasime Uyguroglu explores the finite difference methods applied to hyperbolic and elliptic partial differential equations, focusing on wave and Poisson's equations. The wave equation is approximated using central difference formulas, demonstrating boundary and initial conditions. The lecture further delves into the application of finite difference methods for solving elliptic PDEs, addressing the computational challenges of large systems of algebraic equations. Topics on the accuracy and stability of numerical solutions, including modeling errors and truncation errors, are discussed to provide a comprehensive understanding of these computational techniques.

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Finite Difference Methods in Electrodynamics: Wave and Poisson's Equations

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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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