1. Implementation of Mur First Order Absorbing Boundary Condition in Efficient 3-D ADI-FDTD
Wei Choon Tay 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. Incorporating Mur ABC into 3-D ADI-FDTD
To incorporate Mur1 ABC into ADI-FDTD, we have adopted a consistent implementation:

8. Compact Matrix Form
By discretizing the equations in time using central difference and time averaging, we can formulate the Mur1 ABC for all E boundary fields into compact matrix form:

9. Compact Matrix Form
where

10. Incorporating Mur ABC into 3-D ADI-FDTD
It can be seen that the compact matrix form conforms to the ADI generalized splitting formulate:
where

11. Efficient Formulation
By introducing some auxiliary variables, we can obtain the most efficient ADI scheme having the simplest right-hand sides:
E. L. Tan, “Fundamental schemes for efficient unconditionally stable implicit finite-difference time-domain methods," IEEE Trans. Antennas Propag., Vol. 56, No. 1, , 2008.

12. Efficient Formulation
where
The algorithm constitutes the fundamental scheme that has its right-hand sides free from matrix operators and .
It reduces the arithmetic operations for computations, resulting in the simplest and most concise update equations.

13. Implementation and Comparison
Implementing Mur1 ABC in conventional 3-D ADI-FDTD algorithm (illustrated here for variable at boundary ):
Implementing Mur1 ABC in efficient 3-D ADI-FDTD algorithm:
Notice that RHS expression has been changed to auxiliary variables in the efficient 3-D ADI-FDTD

14. Numerical Results
Plot of reflection coefficients for Mur1 ABC in conventional and efficient 3-D ADI-FDTD with various CFLN in a free space medium:
Efficient and conventional ADI-FDTD with Mur1 ABC have exactly the same results (same computation)
Efficiency gain* of efficient over conventional ADI-FDTD with Mur1 ABC: 1.75
*Intel Core 2 CPU 2.66GHz, 1.98GB RAM, Microsoft Windows XP Pro SP3

15. Conclusion
Implementation of Mur first order ABC in efficient 3-D ADI-FDTD method.
The Mur1 ABC has been formulated in compact matrix form by conforming to the ADI generalized splitting formulae.
The computation for both conventional and efficient ADI-FDTD with Mur1 ABC yields exactly the same results, but with greater efficiency and simplicity for the latter.

16. Thank You!