Uniform Treatment of Numerical Time-Integrations of the Maxwell Equations R. Horváth

Uniform Treatment of Numerical Time-Integrations

of the Maxwell EquationsR. Horváth

TU/e, Philips Research Eindhoven

TU/eEindhoven, 27th June, 2002

Scientific Computing in Electrical Engineering

Eindhoven, The Netherlands

June, 23rd - 28th, 2002

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


Maxwell Equations

0

,0

1

,1

H

E

H E

E H

,t

t typermittivi electric :

typermeabili magnetic :

)),,,(),,,,(),,,,((E zyxtEzyxtEzyxtE zyx

field magnetic

field electric

)),,,(),,,,(),,,,((H zyxtHzyxtHzyxtH zyx


Second order spatialdiscretization usingstaggered grids.

Space Discretization in the Yee-method


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:

222 /1/1/1

1

zyxct


Operator Splitting Methods I.

givenisdtd

)0(, A )0()exp()( Att

21 AAA 1A

dtd

2Adtd

Sequential splitting (S-splitting):

),()(],,(,

),()(],,(,

)1(1

)2(1

)2()2(

1)2(11

)1(1

)1()1(

kkkkkkkk

kkkkkkkk

tttttdt

d

tttttdt

d

2

1

A

A

,0

),1(

,2,1

1

kt

k

k

)0()0()2(0


Operator Splitting Methods II.

Splitting error: )()(. (s)1ψψsplErr

)()0(],[2

)0())exp()exp())((exp(

32

.

ψAA

ψAAAA

21

1221splSErr

For the S-splitting method (1st order):

Lemma:

0:],[0. 122121 AAAAAAsplErr

s: number of thesub-systems


Operator Splitting Methods III.

S-splitting: )exp()exp())(exp( 1221 AAAA

Other splittings:

))2/exp(()exp())2/exp((:))(exp( 12121 AAAAA S

Strang-splitting (second order):

Fourth order splitting:13

)21( )22(,))(exp( SSS21 AA


Semi-discretization of the Maxwell Equations

Applying the staggered grid spatial discretization, theMaxwell equations can be written in the form:

:skew-symmetric, sparse matrix, at most four elements per row. The elements have the form .

: consists of the field components in the form

.1

givenisdtd

)0(, A

.,.,..,.,., EH

NNRI 66 A

NRI 6ψ


Splitting I. (Yee-scheme, 1966)

YY 21 AAA

We zero the rows of the magnetic field variables

We zero the rows of the electric field variables

YTY 21 -AA

,...1,0

,))(()exp()exp( 12121

k

kYY

kYY

k ΨAIAIΨAAΨ

YYYY 2211 )exp(,)exp( AIAAIA Lemma:

ψAψψ

11

1

1

1

1

1

1

1111

1111

1dtd


2D example for splitting

1 2

3 4

7 6

58

H

z

x

E

yH

9

ψAψψ

11

1

1

1

1

1

1

1111

1111

1dtd


2D example - Yee-method


Splitting II. (KFR-scheme, 2001)

pKKK AAAA 21

sKKK AAA ,,, 21 The matrices are skew-symmetric

matrices, where the exponentials exp(A.K) can be computed

easily using the identity:

)cos()sin(

)sin()cos(

0

0exp

tt

tt

t

t

RIik

iKipi

k

,)exp(},,1{

1 ψAψ

Stability: the KFR-method is unconditionally stable by construction.

ψAψψ

11

1

1

1

1

1

1

1111

1111

1dtd


2D example - KFR-method


Splitting III. (NZCZ-scheme, 2000)

NN 21 AAA

Discretization of first terms in the curl operator

Discretization of second terms in the curl operator

NN 21 , AA skew-symmetric

The sub-systems with and cannot be solved exactly. Numerical time integrations are needed.

N1A N2A

kNNNN

kNNN

k

ΨAAAA

ΨAAAΨ

))2/exp(())2/exp(())2/exp(())2/exp((

))2/exp(()exp())2/exp((

2112

2121


Splitting III. (NZCZ-scheme, 2000)Solving the systems by the explicit, implicit, explicit and implicit Euler-methods, respectively, we obtain the iteration

Theorem: the NZCZ-method is unconditionally stable.

Proof: We show that if is fixed

( ),

then the relation

is true for all k with a constant K independent of k.

kNNNN

k ψAIAIAIAIψ )2

()2

)(2

()2

( 21

111

21

hcq /

),,min( zyxh 0ψψ Kk


Splitting III. (NZCZ-scheme, 2000)

02

12 )

2()

2( ψAIAIψ

NNk

202 1,1 qKqk ψψ

121 q

ψAψψ

11

1

1

1

1

1

1

1111

1111

1dtd


2D example - NZCZ-method

22

11

11

11

11

11

11

2111111

2111111

21

A


2D example - Real Schur Decomposition

11

11

11

11

11

11

111111

111111

21

21 AA

0],[ 21 AA No splitting error!


2D examples - Comparisonq YEE NZCZ KFR1 KFR2 SCHUR

1 6.70 1.12 2.05 0.62 1.1210-14

0.707 2.90 0.52 1.08 0.26 1.1910-14

0.5 1.29 0.21 0.70 0.11 4.5810-15

0.05 0.01 2.1210-4 7.6810-3 1.1110-4 6.7910-16

0.005 1.0010-4 2.0910-7 7.5210-5 1.1210-7 8.9210-16

0.0005 1.0010-6 2.0810-10 7.5010-7 1.1010-10 5.5610-16

/q

)exp()( AAU


Comparison of the methods

1,

010

100

001

010

1

xdtd

A



t YEE NZCZ KFR4

0.08 - 7.4610-2 0,04

-

0.008 - 7.8010-3 0,43

-

0.004 - 2.1310-4 0,88

-

0.002 2.0510-15 0,36

7.0910-6 1,83

5.4010-2 0,62

0.0005 2.3510-6 1,42

2.6610-6 7,34

2.4410-4 2,49

0.00005 2.3610-6 14,3

2.3710-6 72,15

2.3910-6 25,3

500/1x

)cos()cos(),(

)sin()sin(),(

txxtH

txxtE

y

z



302

1025.12

]ψA,[A 2Y1Y

What is the reason of the difference between the Yee and the KFR-method?Compare Yee and KFR1:

73.02

02

]ψA,[A 2K1K

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 time-step the method is faster than the others (with acceptable error).


Thank You for the Attention

Documents

Uniform Treatment of Numerical Time-Integrations of the Maxwell Equations R. Horváth