Uniform Treatment of
Numerical Time-Integrations
of the Maxwell Equations
R. Horváth
TU/e, Philips Research Eindhoven
Scientific Computing in Electrical Engineering
Eindhoven, The Netherlands
June, 23rd - 28th, 2002
Eindhoven, 27th June, 2002
TU/e
Outline
• Introduction, Yee-method, problems
• Operator splitting methods
• Splitting of the semi-discretized Maxwell
equations (Yee, NZCZ, KFR)
• Proof of the unconditional stability of the
NZCZ-method
• Comparison of the methods
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Maxwell Equations
 : magnetic permeability
 : electric permittivi ty
H
1
    E,
t

H  ( H x (t , x, y, z ), H y (t , x, y, z ), H z (t , x, y, z ))
E 1
   H,
t 
magnetic field
E   0,
  H   0
E  ( Ex (t , x, y, z), E y (t , x, y, z), Ez (t , x, y, z))
electric field
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Space Discretization in the Yee-method
Second order spatial
discretization using
staggered grids.
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Time Integration in the Yee-method
So-called leapfrog
scheme is used:
E0 H1/2 E1 H3/2 E2 H5/2 E3 H7/2...
Stability condition:
t 
1
c 1 / x   1 / y   1 / z 
2
2
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Operator Splitting Methods I.
d
 A,  (0) is given
dt
 (t )  exp(tA) (0)
A  A1  A 2
d
 A 1
dt
d
 A 2
dt
Sequential splitting (S-splitting):
 (02) (0)   (0)
d
 A 1 (k1) , t  (t k 1 , t k ],  (k1) (t k 1 )   (k2)1 (t k 1 ),
k  1,2, 
dt
t k 1  ( k  1),
d (k2 )
( 2)
( 2)
(1)
 A 2  k , t  (t k 1 , t k ],  k (t k 1 )   k (t k ),
dt
  0,
(1)
k
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Operator Splitting Methods II.
Splitting error: Errspl.  ψ ()  ψ 1(s) () s: number of the
sub-systems
Lemma:
Errspl.  0  [ A1 , A 2 ] : A1 A 2  A 2 A1  0
For the S-splitting method (1st order):
ErrS  spl .  (exp( ( A1  A 2 ))  exp( A 2 ) exp( A1 ))ψ (0) 
2
 [ A1 , A 2 ]ψ (0)  (3 )
2
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Operator Splitting Methods III.
S-splitting: exp( (A1  A 2 ))  exp( A 2 ) exp( A1 )
Other splittings:
Strang-splitting (second order):
exp( ( A1  A 2 ))  S  : exp((  / 2) A1 ) exp( A 2 ) exp((  / 2) A1 )
Fourth order splitting:
exp( ( A1  A 2 ))  S S(12)  S ,   (2  3 2 ) 1
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Semi-discretization of the Maxwell Equations
Applying the staggered grid spatial discretization, the
Maxwell equations can be written in the form:
d
 A,  (0) is given
dt
A  IR 6 N 6 N:skew-symmetric, sparse matrix, at most four
elements per row. The elements have the form 1    . .
ψ  IR 6 N : consists of the field components in the form
H .,.,., E.,.,.
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Splitting I. (Yee-scheme, 1966)
A  A1Y  A 2Y
A1TY  -A 2Y
We zero the rows of the electric
field variables
We zero the rows of the magnetic field variables
Lemma: exp( A1Y )  I  A1Y , exp( A 2Y )  I  A 2Y
Ψ k 1  exp( A 2Y ) exp( A1Y )Ψ k  (I  A 2Y )(I  A1Y )Ψ k ,
k  0,1,...
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2D example for splitting
1
1 1 1 


1 1 1
 1


 1



1

dψ
1
 Aψ  
ψ
1
7
6
dt


1


9 2 5
1
8 1
Ez 


3
4
H
y
1

Hx

 1 1

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2D example - Yee-method
1
1 1 1 


1 1 1
 1


 1



1

dψ
1
 Aψ  
ψ
1
dt


1


1



1

 1 1

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Splitting II. (KFR-scheme, 2001)
A  A1K  A 2 K    A pK
The matrices
A1K , A 2 K ,  , A sK are skew-symmetric
matrices, where the exponentials exp(A.K) can be computed
easily using the identity: exp   0
   t 

ψ
k 1


i{1,, p}
t    cos( t ) sin( t ) 
  

0    sin( t ) cos( t )
exp( i A iK ) ψ , i  IR
k
Stability: the KFR-method is unconditionally stable by
construction.
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2D example - KFR-method
1
1 1 1 


1 1 1
 1


 1



1

dψ
1
 Aψ  
ψ
1
dt


1


1



1

 1 1

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Splitting III. (NZCZ-scheme, 2000)
A  A1N  A 2 N
A1N , A 2 N
skew-symmetric
Discretization of second terms in
the curl operator
Discretization of first terms in the curl operator
Ψ k 1  exp((  / 2) A 2 N ) exp( A1N ) exp((  / 2) A 2 N )Ψ k 
 exp((  / 2) A 2 N ) exp((  / 2) A1N ) exp((  / 2) A1N ) exp((  / 2) A 2 N )Ψ k
The sub-systems with A1N and A 2 N cannot be solved exactly.
Numerical time integrations are needed.
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Splitting III. (NZCZ-scheme, 2000)
Solving the systems by the explicit, implicit, explicit and implicit
Euler-methods, respectively, we obtain the iteration
ψ
k 1




1
1
 (I  A 2 N ) (I  A1N )(I  A1N ) (I  A 2 N )ψ k
2
2
2
2
Theorem: the NZCZ-method is unconditionally stable.
Proof: We show that if q  c / h is fixed
( h  min( x, y, z ) ),
then the relation
ψk  K ψ0
is true for all k with a constant K independent of k.
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Splitting III. (NZCZ-scheme, 2000)


1
ψ  (I  A 2 N )  (I  A 2 N )  ψ 0
2
2
k
1
 1 q2
ψk  1 q2 ψ0 , K  1 q2
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2D example - NZCZ-method
1
1 1 1 


1 1 1
 1


 1



1

dψ
1
 Aψ  
ψ
1
dt


1


1



1

 1 1

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2D example - Real Schur Decomposition



 1
1
1
A  1
2 
 1
1

1
 2
1
1
1
1
1
1
2
1 1 1 1 1 1 2 


 1 1 1  1 1 1  2



 1

 1
 1 
   1
 2 

1

1



1


[A1 , A2 ]  0
1 1 1 1 1 1
1 1 1 1 1 1
1
1
1
1
1
1






  A1  A2






No splitting error!
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q   /
2D examples - Comparison
q
YEE
NZCZ
KFR1
KFR2
SCHUR
1
6.70
1.12
2.05
0.62
1.1210-14
0.707
2.90
0.52
1.08
0.26
1.1910-14
0.5
1.29
0.21
0.70
0.11
4.5810-15
0.05
0.01
2.1210-4
7.6810-3
1.1110-4
6.7910-16
0.005
1.0010-4
2.0910-7
7.5210-5
1.1210-7
8.9210-16
0.0005
1.0010-6
2.0810-10
7.5010-7
1.1010-10
5.5610-16
U (A)  exp( A) 
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Comparison of the methods
0

d
1  1
 A 
dt
x  0



0  0

 0 1
0  1 0 
1
0
,  1
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Comparison of the methods
t
YEE
NZCZ
KFR4
0.08
-
7.4610-2
0,04
-
0.008
-
7.8010-3
0,43
-
0.004
-
2.1310
0,88
-4
-
0.002
2.0510-15
0,36
7.0910-6
1,83
5.4010-2
0,62
0.0005
2.3510-6
1,42
2.6610-6
7,34
2.4410-4
2,49
0.00005
2.3610-6
14,3
2.3710-6
72,15
2.3910-6
25,3
x  1 / 500
E z (t , x)  sin( x) sin( t )
H y (t , x)   cos( x) cos( t )
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Comparison of the methods
What is the reason of the difference between the Yee and the KFRmethod?
Compare Yee and KFR1:
2
[A1Y , A 2Y ]ψ 0  1.25  10 3
2
2
[A1K , A 2K ]ψ 0  0.73
2
Yee: sufficiently accurate, strict stability condition
KFR: unconditionally stable, but an accurate solution requires
small time-step. In the long run it is slower than the Yee-method.
NZCZ: unconditionally stable. With a suitable choice of the timestep the method is faster than the others (with acceptable error).
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Thank You for the Attention
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