1. Modeling Debye Dispersive Media Using Efficient ADI-FDTD Method
Ding Yu Heh and Eng Leong Tan
School of EEE, Nanyang Technological University, Singapore
June 2009

2. ADI Scheme
ADI scheme originated from Peaceman and Rachford (and Douglas), calls for splitting formulae

3. Simplifying ADI Scheme
Introducing auxiliary variables to denote the right-hand sides of implicit equations, we can rewrite the original algorithm as
where the v’s serve as the auxiliary variables.

4. Fundamental ADI Scheme
By exploiting the auxiliary variables, we can obtain the most efficient ADI scheme having the simplest right-hand sides without any explicit matrix operators.
where
E. L. Tan, “Fundamental schemes for efficient unconditionally stable implicit finite-difference time-domain methods," IEEE Trans. Antennas Propag., Vol. 56, No. 1, , 2008.
E. L. Tan, “Efficient algorithm for the unconditionally stable 3-D ADI-FDTD method," IEEE Microw. Wireless Compon. Lett., Vol. 17, No. 1, 7-9, 2007.

5. COMPARISONS OF UNCONDITIONALLY STABLE FDTD METHODS
E. L. Tan, “Fundamental schemes for efficient unconditionally stable implicit finite-difference time-domain methods," IEEE Trans. Antennas Propag., Vol. 56, No. 1, , 2008.
Note: CPU time gain = 1.77

6. Comparison of original and fundamental algorithms of CNDS and CNCSU
Note: CPU time gain = 1.58 – 2.97
CNCSU is not unconditionally stable
E. L. Tan, “Efficient algorithms for Crank-Nicolson-based finite-difference time-domain methods," IEEE Trans. Microw. Theory Tech., Vol. 56, No. 2, , 2008.

7. ADI-FDTD for Debye Media
Time domain Maxwell’s equation and the constitutive relation
where
: static relative permittivity
: relative permittivity at infinite frequency
: relaxation time

8. ADI-FDTD for Debye Media(contd’)
Solving the previous equations using the ADI-FDTD method
where

9. Efficient Formulation
By introducing some auxiliary variables, we can obtain the most efficient ADI scheme having the simplest right-hand sides without any explicit matrix operators:
Matrix-operator-free RHS (less flops count and memory indexing)
Yield exactly the same computation as the previous conventional ADI-FDTD
Main fields u can be retrieved by
E. L. Tan, “Fundamental schemes for efficient unconditionally stable implicit finite-difference time-domain methods," IEEE Trans. Antennas Propag., Vol. 56, No. 1, , 2008.

10. Actual Algorithm Implementation
Define
First procedure from n to n+1/2

11. Actual Algorithm Implementation (contd’)
Still on first procedure from n to n+1/2
where

12. Actual Algorithm Implementation (contd’)
The update equations for all other fields can be written down by cyclically switching the subscripts.
The update equations for second procedure from n+1/2 to n+1 can also be derived in a simple manner.
Same memory storage requirement as conventional ADI-FDTD by appropriate reusing of field arrays.

13. Numerical Results
Plot of electric fields recorded in rectangular cavity filled with water (Debye medium) for Yee’s FDTD, efficient and conventional ADI-FDTD (CFLN=10).
All results are in good agreement to each other
Efficient and conventional ADI-FDTD have exactly the same results (same computation)

14. Numerical Results (contd’)
For preliminary efficiency comparison, the CPU time incurred for a run of 1600 time steps by both conventional and efficient ADI-FDTD method are found to be around 163s and 145s respectively (efficiency gain ~ 1.12).
Simulation is run using a Matlab program on a platform of Intel Duo Core 2.66GHz processor, 1.98 GB RAM.
The efficiency may be improved further by optimization and pre-compilation of the codes.
New timing using C++: conventional ADI-FDTD ~ 62s, efficient ADI-FDTD ~ 37s (efficiency gain ~ 1.68)

15. Conclusion
Efficient implementation of 3-D ADI-FDTD method for Debye dispersive media.
The formulation results in matrix-operator-free RHS.
The efficient ADI-FDTD method yields exactly the same computation as the conventional ADI-FDTD method, but with improved efficiency achieved.
It represents the fundamental scheme which is the simplest and most concise implicit scheme.
Can be further extended to higher order dispersive media, such as Lorentz, Drude, etc.

16. Thank You!