This document discusses unconditionally stable finite-difference time-domain (FDTD) methods for solving Maxwell's equations numerically. It outlines FDTD algorithms such as Yee's method from 1966 which discretize the equations on a staggered grid. It also discusses the von Neumann stability analysis and compares implicit Crank-Nicolson and alternating-direction implicit methods to conventional explicit FDTD methods. The document notes the advantages of unconditionally stable methods but also mentions potential disadvantages.

1. Unconditionally Stable
FDTD Methods
SUBHASH YADAV
RF & MICROWAVE ENGG
ROLL-NO-11EC63R10
21 MARCH 2012 IIT KHARAGPUR

2. Outlines…………..
1.Some computational methods for
maxwell,s equations.
2.First fdtd algorithm YEE 1966
discritizations.
3.Some problems and condition in
conventional fdtd.
4.Unconditionally stable fdtd use to
solve heat equation
21 march 2012 2

3. Outlines….
5.Imclicit Cranknicolson ADI
Methods
6-Von neumann stability
7.Advantages
8.Disadvantages
9.Conclusion
10.References
21 march 2012 3

4. Maxwell equations:
µ : magnetic permeability
∂H 1
= − ∇ × E, ε : electric permittivity
∂t µ
∂E 1 H = ( H x (t , x, y, z ), H y (t , x, y, z ), H z (t , x, y, z ))
= ∇ × H,
∂t ε magnetic field
∇( εE ) = 0,
E = ( E x (t , x, y, z ), E y (t , x, y, z ), Ez (t , x, y, z ))
∇ ( µH ) = 0
electric field

5. Computational Electromagnetics
Finite-difference Transmission line
time-domain matrix (TLM)
(FDTD)
Finite element
method (FEM) Method of Moments
(MoM)
Finite-difference Fast multipole
frequency-domain method (FMM)
(FDFD)

6. Computational Electromagnetics
Finite-difference Transmission line
time-domain matrix (TLM)
(FDTD)
Finite element
method (FEM) Method of Moments
(MoM)
Finite-difference Fast multipole
frequency-domain method (FMM)
(FDFD)
Frequency

7. FDTD Overview – Updating Equations
 Three scalar equations can be obtained from one vector curl equation.
∂Ex ∂H z ∂H y
εx = −
∂t ∂y ∂z
∂E ∂E y ∂H x ∂H z
ε = ∇× H εy = −
∂t ∂t ∂z ∂x
∂Ez ∂H y ∂H x
εz = −
∂H x ∂E y ∂Ez ∂t ∂x ∂y
µx = −
∂t ∂z ∂y
∂H y ∂Ez ∂Ex ∂H
µy = − µ = −∇ × E
∂t ∂x ∂z ∂t
∂H z ∂Ex ∂E y
µz = −
∂t ∂y ∂x 7/60

8. Finite Difference
 Taylor’s series

9. II. Finite Difference
 Eror
Taylor’s
series

10. Finite Difference Time Domain
Method
• Divide the interval x into sub-intervals,
each of width h
• Divide the interval t into sub-intervals,
each of width k t
• A grid of points is used for
the finite difference solution
• Ti,j represents T(xi, tj)
• Replace the derivates by x
finite-difference formulas
10

11. The Yee Discretization (1966)
(i, j+1)
(i+1, j+1, k+1)
Hz
Ey
(i, j, k+1) Hz
Hx
Ez Hy (i, j) (i+1, j)
(i+1, j+1, k)
Ex
Ey
(i, j, k) (i+1, j, k)
Ex
Staggered grid in space:
— every field component is stored on a different grid

12. The Yee Discretization (1966)
∂
(i, j+1)
H 1
= ∇ E ⇒L
− ×
∂t µ Ey
Hz
∂Hz 1 ∂Ey ∂Ex 
=−  − 
∂t 1
i+ , j+
1 µ  ∂x ∂y  (i, j) (i+1, j)
2 2 Ex
 1 1 1 1 
E (i +1 j + )−Ey (i, j + ) Ex (i + , j +1)−Ex (i + , j) 
1 y
,
≈−  2 2 − 2 2 
µ ∆x ∆y 
 
+ O(∆x2) + O(∆y2)
all derivatives become center differences…

13. FDTD Overview – Updating Equations
∂Ex ∂H z ∂H y
εx = −
∂t ∂y ∂z
Exn +1 (i, j , k ) − Exn (i, j , k )
ε x (i, j , k ) =
∆t
n + 0.5
H zn + 0.5 (i, j , k ) − H zn + 0.5 (i, j − 1, k ) H y (i, j , k ) − H y + 0.5 (i, j , k − 1)
n
−
∆y ∆z

14. FDTD Overview – Updating Equations
∂H x ∂E y ∂Ez
µx = −
∂t ∂z ∂y
H xn + 0.5 (i, j , k ) − H xn −0.5 (i, j , k )
µ x (i, j , k ) =
∆t
E y (i, j , k + 1) − E y + 0.5 (i, j , k ) Ezn (i, j + 1, k ) − Ezn (i, j , k )
n n
−
∆z ∆y

15. FDTD Overview – Updating Equations
 Express the future components in terms of the past components
 E y (i, j , k + 1) − E y + 0.5 (i, j , k ) 
n n
 
∆t  ∆z 
H xn + 0.5 (i, j , k ) = H xn −0.5 (i, j , k ) +
µ x (i, j , k )  Ez (i, j + 1, k ) − Ez (i, j , k ) 
n n
− 
 ∆y 
 H zn + 0.5 (i, j , k ) − H zn + 0.5 (i, j − 1, k ) 
 
∆t  ∆y 
Exn +1 (i, j , k ) = Exn (i, j , k ) −
ε x (i, j , k )  H y + 0.5 (i, j , k ) − H y + 0.5 (i, j , k − 1) 
n n
− 
 ∆z 

16. Fundamentals of the FDTD method
Slides from 5-15 may be skipped because it was taught in the class
 Accuracy and stability
Accuracy ∆ ≤ λ / 10
1 ∆
Stability ∆ t ≤ ∆ t max = ∆ t ≤ ∆ t max =
1 1 1 c 3
c 2
+ 2+ 2
∆x ∆y ∆z
1
2D: ∆ t ≤
1 1
c + 2
∆x 2 ∆y
1D: ∆ t ≤
∆
Physically, this condition means that the time
c step should be smaller than the time for the
wave to propagate from one cell to the
neighbor one

17. III. Fundamentals of the FDTD method
 Dispersion relation
2
2 2 2 2 ω
In free space (ideal) k = kx + ky + kz =
c
In FDTD computation
(numerical)
~ 2
 1 ~
 k y ∆y 
2
 1 ~ 2 2
 1  k x ∆x    k z ∆z    1  ω ∆t  
 sin    +  sin    +  sin   =  sin  
 ∆x  2    ∆y  2    ∆z  2    c∆ t  2  
       
( )  ~ y ∆y   1
2
ω  ~ x ∆x   1  ~ z ∆z 
2 2
2  1 k   +  sin 
k   +  sin  k
v pnum = ~ = ~ Arc sin c∆ t u u =  sin 
∆x  2   ∆y  2   ∆z  2 


k k∆ t         
The numerical medium is dispersive : the propagation of the wave varies with
frequency and angle

18. III. Fundamentals of the FDTD method
 Dispersion relation

19. Limitations of FDTD method
1-Grid spacing should be ~λ/10.
2-According to Courant’s stability condition,
time step Δt becomes small when FDTDgrid
spacing becomes small.
3-In 3-D simulation, simulation time scales
like N^4, and required memory size scales
like N^3.
4-Application is restricted to relatively small
size.

20. Space Domain Discretization
• Heat Conduction Equation
∂T ( x, y, t ) ∂ 2T ( x, y, t ) ∂ 2T ( x, y, t ) g ( x, y, t ) κ
=α +α + α=
∂t ∂x 2
∂y 2
ρc p ρcc
• Central-Finite-Difference Approximation
∂ 2T Ti +1, j − 2Ti ,nj + Ti −1, j
n n
n
= + O(∆x) 2
∂2x
i, j
( ∆x ) 2
Ti +1, j − 2Ti ,nj + Ti −1, j
n n
δ x2T n
≈ =
( ∆x ) 2 ( ∆x ) 2
21 march 2012 20

21. Finite-Difference Formulation of the
Heat Conduction on a Chip
(0,J)
Ti,nj+ 1
…
∆y
(0,j)
Tin1,j
-
Ti,nj Tin 1,j
+ • Space Domain
Ti,nj− 1 • Time Domain
…
∆x
X
(0,0) … (i,0) … (I,0)
21 march 2012 21

22. Time domain discretization
• Heat Conduction Equation
n +1
T −T  δ x2T ? δ y T ? 
n 2
1
=α + 2
+ g
∆t  (∆x) (∆y )  ρc p
2
 
δ x2T n = Ti −1, j − 2Ti ,nj + Ti +1, j

n n
 2 n
δ y T = Ti , j −1 − 2Ti , j + Ti , j +1
n n n

– Simple Explicit Method
– Simple Implicit Method
– Crank-Nicolson Method
21 march 2012 22

23. Can we check if a numerical
scheme is stable without
computation?
Von Neumann stability John von Neumann
• analysis
Analyze if (or for which conditions) a 1903-1957
numerical scheme is stable or unstable.
• Makes a local analysis, coefficients of PDE are
assumed to vary slowly (our example:
constant).
• How will unavoidable errors (say rounding
errors)
evolve in time? 23

24. Von Neumann stability
analysis
Ansatz:
Wave number k and amplification factor:
A numerical scheme is unstable if:
24

25. Simple Explicit Method
n +1
T −T  δ x2T n δ y T n 
n 2
1
=α + 2
+ g
∆t  (∆x) (∆y )  ρc p
2
 
δ x2T n = Ti −1, j − 2Ti ,nj + Ti +1, j

n n
 2 n
δ y T = Ti , j −1 − 2Ti , j + Ti , j +1
n n n

Accuracy: [ ( ∆t ) , ( ∆x ) , ( ∆y ) ]
2 2
•
• Stability Constraint:  1 1  1
γ = α∆t  + ≤
2 
 ( ∆x ) 2 ( ∆y )  2
• No matrix inversion but time steps are limited
by space discretization
21 march 2012 25

26. Simple Implicit Method
n +1
T −T  δ x2T n +1 δ y T n +1 
n 2
1
=α + 2 
+ g
∆t  (∆x)

2
(∆y )  ρc p

δ x2T n +1 = Ti −1,1j − 2Ti ,nj+1 + Ti +1,1j

n+ n+
 2 n +1
δ y T = Ti , j −1 − 2Ti , j + Ti , j +1
n +1 n +1 n +1

Accuracy: [ ( ∆t ) , ( ∆x ) , ( ∆y ) ]
2 2
Unconditionally Stable
No limits on time step but involves with large
scale matrix inversion
21 march 2012 26

27. Crank-Nicolson Method
n +1
T −T  δ x2T n +1 + δ x2T n δ y T n +1 + δ y T n 
n 2 2
1
=α + + g
∆t 2(∆x) 2(∆y )  ρc p
2 2

 
δ x2T n = Ti −1, j − 2Ti ,nj + Ti +1, j

n n
δ x2T n +1 = Ti −1,1j − 2Ti ,nj+1 + Ti +1,1j

n+ n+
 2 n  2 n +1
δ y T = Ti , j −1 − 2Ti , j + Ti , j +1 δ y T = Ti , j −1 − 2Ti , j + Ti , j +1
n n n n +1 n +1 n +1
 
Accuracy: [ ( ∆t ) , ( ∆x ) , ( ∆y ) ]
2 2 2
Unconditionally stable
No limits on time step but involves
with large scale matrix inversion
21 march 2012 27

28. Analysis of Crank-Nicolson
Method
e.x. m=4,n=4
Total node number N = mn *
*
*
* * *
* 

 
 * * * * 
 
 * * * * 
* * * * * 
 
n 

*
*
* * *
* * *
*
*


 
 * * * * * 
 * * * * * 
 
 * * * * * 
 
 * * * * * 
 * * * * *
 
 * * * * 
 * * * * 
 
m 

*
*
* *
*
*
16×16
*
 
Matrix size = NxN
21 march 2012 28

29. Alternating Direction Implicit Method
Solves higher dimension
problem by successive Lower
dimension methods
Accuracy: [ ( ∆t ) , ( ∆x ) , ( ∆y ) ]
2 2 2
Unconditionally stable
No limits on time step and no
large scale matrix inversion
21 march 2012 29

30. Alternating Direction Implicit Method
Step I:
x-direction implicit
y-direction explicit
n
Step II:
x-direction explicit
y-direction implicit
• Peaceman-Rachford Algorithm
• Douglas-Gunn Algorithm
21 march2012 30

31. Peaceman-Rachford
Algorithm
rx 2 ry 2 n +1 rx 2 ry 2 n ∆t
(1 − δ x )(1 − δ y )T = (1 + δ x )(1 + δ y )T + g
2 2 2 2 ρc p
• Step I
rx 2 n + 1 ry 2 n ∆t
(1 − δ x )T 2
= (1 + δ y )T + g
2 2 2 ρc p
• Step II
ry rx 2 n + 1 ∆t
(1 − δ y )T = (1 + δ x )T
2 n +1 2
+ g
2 2 2 ρc p
21 march 2012 31

32. Douglas-Gunn Algorithm
rxδ ryδ y2 ∆t
(T ) (T )
2
T n +1 − T n = x n +1
+T n + n +1
+T n + g
2 2 ρc p
• Step I
n+
1
rxδ x2 n + 1 ∆t
T 2
−T =
n
(T 2
+ T ) + ryδ y T +
n 2 n
g
2 ρc p
• Step II
rxδ x2 n + 1 ryδ y n +1 ∆t
2
n +1
T −T =n
2
(T 2
+T ) +
n
2
T +T +n
ρc p
g ( )
21 march 2012 32

33. Illustration for ADI
Step I Step II
X-direction implicit Y-direction implicit
n n
Ti ,nj++11
n+ 1 n+ 1 n+ 1
T 2
i −1, j Ti, j
2
T 2
i +1, j
Ti ,nj+1
…
…
Ti ,nj+−11
2 2
j=1 j=1
i=1 2 … m 1 2 … m
21 march 2012 33

34. Analysis of ADI Method
X-direction implicit Tridiagonal Matrix
n * * 
* * * 
 
 * * * 
 
n+ 1 n+ 1 n+ 1   
T 2
i −1, j T
i, j
2
T 2
i +1, j  * * *
 
…

 * * m × m

2xnxm = 2nm =2N
2
2 steps n matrices tridaigonal matrix
j=1
i=1 2 … m Time complexity: O(N)
21 march 2012 34

35. ADVANTAGE over conditionally stable
Reduce simulation time
Good accuracy
Size of geometrical feature may be
typical of wavelengh
Good geometrical flexibility to allow with
corner or high curvature than fdtd.

36. Include examples of unconditionally stable fdtd and prove the advantages
 Advantages of the FDTD method over other methods
• It is conceptually simple.
• The algorithm does not require the formulation of integral equation, and relatively complex
scatters can be treated without inversion of large matrices.
• It is simple to implement for complicated, inhomogeneous conducting or dielectric structures
because constitutive parameters can be assigned to each lattice point.
• Its computer memory requirement is not prohibitive for many complex structures of interest.
• The algorithm make use of the memory in a simple sequential order.
• It is much easier to obtain frequency domain data from time domain results than the converse.
Thus, it is more convenient to obtain frequency domain results via time domain when many
frequencies are involved.

37.  Disadvantages
• Its implementation necessitates modeling object as well as its surroundings. Thus, the
required program execution time may be excessive.
• Its accuracy is at least one order of magnitude worse than that of the method of
moments, for example.
• Since the computational meshes are rectangular in shape, they do not conform the
scatterers with curved surfaces, as is the case of the cylindrical or spherical boundary.
Its computer memory requirement is not prohibitive for many complex structures of
interest.
• As in all finite difference algorithms, the field quantities are only known at grid nodes.

38. Applications of FDTD method
AElectromagnetic scattering & antenna
design
d EMC/EMI design
ESimulation of wave propagation problem
SSolving partial defferential equation
SWaveguide analysis
Stripline & microstrip line analysis…..
etc …

39. References………
1-Electromagnetic simulation using FDTD method
“Dennis M. Sillevan” IEEE press series on rf &
mivcrowave technology “Roger D.Pollard & Richard
Bootan”series
2.Understanding the FDTD method “John
B.Schneider” may 8 ,2011.
3. 3 D -ADI -Mehod-unconditionaly stable time
domain algorithom for solving maxwell equations
“IEEE Transaction on microwave theory &
techniques vol-48,No,10 october 2010. “Takefumi
IEEE.

40. 4.High order split step unconditionally stable
FDTD method &numerical analysis. “IEEE
Transactions on antinna & propagation,vol 59
no.9,sept2011” Yong-Dan Kong, & Qing xin chu
;senior member,IEEE
5.Genaral Finite DifferenceSchemes for Heat
equations “Indian J. Pure Apple.math
10(2);209-222,Febuary 1979.”by P.S Jain &D.N
Holla department of methemetics.
IITB 400076,