Understanding Finite Difference Time Domain (FDTD-ABC’s) Method in Electrodynamics

1. EEE 431Computational Methods in Electrodynamics Lecture 11 By Dr. Rasime Uyguroglu Rasime.uyguroglu@emu.edu.tr

2. FINITE DIFFERENCE TIME DOMAIN METHOD (FDTD-ABC’s) Absorbing Boundary Conditions

3. FINITE DIFFERENCE TIME DOMAIN METHOD (FDTD-ABC’s) • A simple absorbing boundary condition (ABC) was used to terminate the grid. • It is based on the fact that the fields were propagating in one dimension and the speed of propagation was such that the fields moved one spatial step for every time step (i.e., the Courant number was unity, Lecture 8 or Courant number was ½, Lecture 9) )

4. FINITE DIFFERENCE TIME DOMAIN METHOD (FDTD-ABC’s) • The node on the boundary was updated using the value of the adjacent interior node from the previous time step. • However, when a dielectric was introduced, and the local speed of propagation was not equal to c , this ABC failed to work properly. • One would also find that in high dimensions this simple ABC would not work even in free space.

5. FINITE DIFFERENCE TIME DOMAIN METHOD (FDTD-ABC’s) • One would also find that in high dimensions this simple ABC would not work even in free space because the courant number cannot be unity in free space.

6. FINITE DIFFERENCE TIME DOMAIN METHOD (FDTD-ABC’s) • The wave equation which governs the propagation of the electric field in one dimension is: • where the second form represents the equation in terms of an operator operating on Ex.

7. FINITE DIFFERENCE TIME DOMAIN METHOD (FDTD-ABC’s) • This operator can be factored as:

8. FINITE DIFFERENCE TIME DOMAIN METHOD (FDTD-ABC’s) • Their solutions: • i.e a wave traveling in the negative z direction. • i.e a wave traveling in the positive z direction.

9. FINITE DIFFERENCE TIME DOMAIN METHOD (FDTD-ABC’s) • Grid Termination: Let us now consider how the wave equation can be used to provide an update equation for a node at the end of the computational domain. • Consider field at z=0. • Remember that the interior nodes can be updated before the boundary node. • Assume that all the adjacent nodes in space-time are known.

10. FINITE DIFFERENCE TIME DOMAIN METHOD (FDTD-ABC’s) • i.e. are known. • At the left end of the grid, the fields should only be traveling to the left. • Thus the fields satisfy the equation given by (1).

11. FINITE DIFFERENCE TIME DOMAIN METHOD (FDTD-ABC’s) • The finite-difference approximation of this equation provides the necessary update equation, but the way to discretize the equation is not entirely obvious. • Equation (1) is expanded about the space-time point .

12. FINITE DIFFERENCE TIME DOMAIN METHOD (FDTD-ABC’s) • To obtain an approximation of; • Similarly:

13. FINITE DIFFERENCE TIME DOMAIN METHOD (FDTD-ABC’s) • Therefore the temporal derivative can be approximated by the following finite difference:

14. FINITE DIFFERENCE TIME DOMAIN METHOD (FDTD-ABC’s) • The spatial derivative can be approximated as:

15. FINITE DIFFERENCE TIME DOMAIN METHOD (FDTD-ABC’s) • Combination of two difference equations:

16. FINITE DIFFERENCE TIME DOMAIN METHOD (FDTD-ABC’s) • Letting and solving for : • Where s is the Courant number

17. FINITE DIFFERENCE TIME DOMAIN METHOD (FDTD-ABC’s) • This equation provides a first-order absorbing boundary condition which updates the field on the boundary using the values of past and interior fields. • Note that when is unity, which would be the case and a unity courant number, this equation reduces to the ABC used before (Taflove).

18. FINITE DIFFERENCE TIME DOMAIN METHOD (FDTD-ABC’s) • A nearly identical equation can be obtained for the other end of the grid. • Equation (2) would be expanded in the neighborhood of the last node of the grid. • Although (1) and (2) differ in the sign of one term, when (2) is applied it is “looking” in the negative z direction. • That effectively cancels the sign change.

19. FINITE DIFFERENCE TIME DOMAIN METHOD (FDTD-ABC’s) • Hence the update equation for the last node in the grid, which is identified here as