Theoretical consideration of charge simulation method for numerical conformal mappings in a ring domain

(1)

Internat. J. Math. & Math. Sci. VOL. 21 NO. 2 (1998) 289-298

289

THEORETICAL CONSIDERATION

OF

CHARGE SIMULATION

METHOD

FOR NUMERICAL

CONFORMAL MAPPINGS

IN

A

RING DOMAIN

TETSUOINOUE

Department

of Applied Mathematics,Kobe Mercantile Marine

College

Higashinada,Fukae-Minami5-1-1, Kobe

658,

Japan

(Received August

16, 1996andinrevisedform March 17,

1997)

ABSTRACT.

The chargesimulationmethod has been appliedto solve a lot of problems in electrical engineering.

However,

theprinciple ofthe methodisnot knownenougheven now.

This paperisdevoted to giving the theoretical and mathematical baseforthechargesimulation method of numericalconformal mappings in ring domains. Thereforeforexample,the uniform ,convergence ofapproximations, the theoretical distribution of charge points, and the charges will be mathematically discussed.

An

example is shown to helpunderstanding oftheoretical considerations.

KEYWORDS AND PHRASES. Charge

Simulation

method,

Weightedasymptotic

theorem,

Potentialtheory, Hyperbolic polynomial, Conformalmapping, Uniformconvergence

1991

AMS SUBJECT

CLASSIFICATION

CODE.

30El0,

41A10,

65E05.

1.

INTRODUCTION

Thechargesimulationmethodisveryuseful to solve partial differential equationsinelectrical engineering, and has been studied and developed by alot of researchers. The method iseasy to

understand,

and canbeapplied only by solvinga system ofsimultaneous linear equations.

Many

examples show that the method makespossible toget rather precise solutions [or the boundaryvalueproblemswithrespectto domains boundedbysmoothcurves.

However,

manypartsof themethod dependonthe experimentfromthe examles.

For

instance.

the best distributionof chargepointsisnotknown.

In

this paperwewill give the theoretical and mathematical base for the chargesimulation

methodofnumericalconformalmappingsinring domains.

Then,

thepotential theory

[10,16],

especially the asymptotic theorems

[5-9,11-14]

onextremal weightedhyperbolicpolynomials play fundamental roles. The theoremsdependonthe notion

introduced in 1992 by Mhaskar andSaff

[13].

We

notethat numericalconformalmappings not dependingonthechargesimulationmethod have been shownin

[4,15].

Theresults established in this paperareasfollows:

(la)

A

newschemeforapproximations isproposed forthe numerical conformalmappingof

aringdomain.

(lb)

The ditributionof chargepointsischaracterized bytheweighted extremal points.

(lc)

It

istheoretically shown that the numbersof charge points in thecomplements ofaring

(2)

290 T. NOUE

(ld)

If theouter

boundary

is theunit circle with acenter at the origin, the chargepoints exteriorto it are represented by

( 1/,,,

}=t,

where

{z,,, }=

are the chargepoints interior to theinnerboundary.

(le)

The approximations convergeuniformlyin the domainandontheboundary,if thecharge pointsaretheoretically distributed.

The outlineofthis paperis asfollows.

In

sections2and

3,

the definitionof weighted hyperbolic capacity

(shortly

wh-capacity)

and asymptotic lemmas on extremal wh-polynomials will be introduced, respectively.

In

section4thelemmasareappliedtoestablishing algorithms ofa new

chargesimulation methodfornumericalconformalmappingsinring domainswhoseeachouter

boundaryistheunitcircle withacenteratthe origin.

In

section5 the theoretical distribution of chargepoinsis discussed.

In

section6 asimpleexampleis shown tohelp understanding of theoretical considerations forthenewchargesimulatinmethod.

2.

DEFINITIONS

In

approximationtheory,the asymptotic behaviorof extremal polynomials hasbeen studied byalotof researchers.

In

thissection wedescribethenotionsofweighted hyperbolicpolynomials

(shortly wh-polynomials)

and weighted hyperboliccapacity

(wh-capacity)

introduced by the author

[5]

andMhaskar-Saff

[12,13],

respectively.

Let D

bearingdomainwhose outer andinnerboundariesaretheunitcircle_9’0 withacenter at the origin and a Jordan curve ", respectively.

For

the numerical conformalmapping of a

ring domainweestablishanewchargesimulationmethoddependingonasymptotic lemmason

extremalweighted hyperbolic polynomials. Therethe distancebetweenz, andzj isdefinedby

[lZ’-,z,

z,

I(

z,

)(

=,

),

where

w(z)

is aweight function.

All ofthedefinitions forthe usualweighted polynomialsholdanalogouslyforhyperbolicones.

Let

w

w(z)

be an arbitrary, continuous, positive functiondefinedonthe complex plane

C.

For

each integern

_>

1, welet

P,,

denote the class of all polynomials of theform

,(z)

(1-

,z

)()(,,)]

(2.1)

whichwe call wh-polynomialsof degreen.

Let

M{’)

denotetheclass ofall positive unitBorelmeasures whosesupport is

-.

We

define wh-capacity by

where

and

caph(w,

"),)

exp(V,),

(2.2)

/

z-lw(z)w(t.)]da(z)da(t)

(2.3)

I,(a)

log

[It

z

V,

V(w,

"y)

sup

I(a).

(2.4)

Throughoutthe remainder of this paper,we assumethat every

p,.,.,(z)

hasall of thezeroson".

(3)

ICAL CONFORMAL MAPPINGS IN A RING DOMAIN 291

Let

# 6

M(7)

bean extremalmeasuresuch that

I(tt)---

V,.

(2.5)

Theexistenceand the uniquenessof# wereshown in

[13,Theorem 3.1(b)].

We

assume that

S

7, where

S

isthesupport of t.

For

polynomials

p,,(z)

of degree n,the discreteunitmeasuredefinedoncompactsets in the complex plane

C

with mass

1In

ateach zeroof

p,,,(z)

will bedenoted

by/,,

t(p,.).

It

willbecalled the normalized countingmeasure onthezerosof

p,,,(z).

If

p,,,(z)

hasmultiple zeros,the obvious modification will be considered.

The weakconvergence of

v

tov asn x willbedefinedby

lira

/fdv,,--/fdv

(2.7)

foreverycontinuousfunction in thecomplex plane

C

withcompactsupport. 3.

LEMMAS

We

presentthefundamentallemmasonextrmalwh-polynomialsthataredevoted to mathe-matical considerationsof the newchargesimulation method.

Sinceallofthelemmas establishedfor usual weightedpolynomialshold analogously for hy-perbolic ones,weomittheproofs

(see

[5-9,11-14]).

Under the assumption mentioned in section

2,

we state the main lemma which has been verifiedin

[6].

Lemma

3.1.

Let

-

....

(z)(,,)]

be

wh-polynomJals

of degreensatis@ingthe conditions

(3a)

and

(35)

below:

(3a)

limsup,,_.oo

Ilp,,.(z)ll/"

<

caph(w,

(35)

liminf,,._.oo

’],,

log

w(z,,.,) > flogw(t)dt(t).

Then,

there holds theequality

lim

ip,,(z)l

’/’

exp{g(z)}

(3.1)

uniformlyoneverycompactsubset of

D,

where

(z)

log[I

1

lw()w(t)]d,,(t)..

(a.2)

We

notethatif

(a.1)

holds uniformlyonevery

compact

subsetof

D,

the

property

(ab)

follows withoutinfandwihequality.

It

iseasilyshown byletting tendto 1 in

(3.1).

Now,

weshowthe converseofLaminaa.1.

Lemma

3.2.

Let

lit(

v.,(z)

.

,z)(z)(z-,’)l

bewh-polynomials of degreensatisfyingtheequa/ity

im

Iv.,(z)l

’"

x{(z)}

(3.)

uniformlyoneverycompact subset o[D.

Then, there hold the relations

(a)

nm

up,_

IIV,,(z)ll/"

<

(,

).

(4)

292 T. INOUE /.,emma3.3.

Let

n

Z-Z,,,

,()

[()()w(,,)]

andbt,

t(p,,,)

bewh-polynomials of degreenand the normalized countingmeasureofp, respectively,satisfyingtheequality

lim

i,(z)l

/

exp{g(z)}

(3.4)

uniformlyoneverycompactsubset of

D. Then,

thereholds

(A);

g,converges weaklyto

In

fact, Lemma

3.3 has beenproved for the case_{when 7 lies}on the real axisand for the w-polynomials

=HI(

_:

.

,)()1.

(3.5)

Sincetheprooffor thecaseof

Lemma

3.3issimilar,wewillomit it.

Thefollowingtheoremhas recentlyestablishedby theauthor.

Lemma

3.4.

Let

#,,converge weaklytoIzasn oo.

Then,

wehave

limsup

IIr,,,.,o(z)ll’/"

<_

,ph(o,,z).

(a.6)

From Lemmas

3.3 and 3.4 we have the following lemma under the assumption mentioned above.

Lemma

3.5.

_m

Ir,,,.,()l

’/"

exp{g(z

}

holds uni[ormlyoneverycompactsubsetof

D

if andonlyif

(A)

holds.

(3.7)

We

transform

(3.3)

totheform

li_m.

1-[

(lZ--

,%’

)(,,)1,/

=exp{

logtl

_tzlw(t)]dti(t)}.

(3.8)

It

isknownthattheequality

z-t

exp{

log[I

1

Zz

Iw(z)w(t)]dg.(t)}

caph(w,

7)

(3.9)

holds quasi-everywhere

(q.e.)

on7

[13];

that is, with the possible exception ofa set having capacityzero.

Therefore,

weobtainthe equalities

p{/og[l(

t)/(1

Zz)lw(t)ldg(t)}

exp{f

log[l(z

t)/(1

z)lw(z)w(t)]d(t)}

1

=w(z)

(3.10)

(5)

NUMERICAL CONFORMAL MAPPINGS IN A RING DOMAIN 293

Lemma

3.6.

2i-={I-[

l(i-

,

)I(’’)}’

=P{I

lg[l

I

-z

(3.11)

=l

holds uforyon every

comp

sue

of

D

if d onlyif

(A)

hM. rhermore, ifit is

saisHed, he

p(

om[l(

0/(1

)()].()}

1

(3.)

hMdsq.e. on %

t

D’

bearing domawhoouterand inner boundariesehe unitccle withacenter

atthe

orion

daJordancurve

’,

respectively.

e

sumethat he function w

f(z)

maps

D

U₀ onto

D’U

,

corresponding Thehnction

f(z)

isuniquelydeterminedunderthe condition

f(1)

I. Whenthe domain

D

given, the modulus

o

D

uniquelydetermined

[I].

The ymptotic formula

or

the nth root of

p,(z)

can be obtainedfrom oneof

]p,(z)[.

Th followsfrom he fact that

og(m,()),

-og

im,()l,

+

arg(p,(z))

’/

(3.13)

isanalytic in

D

andCauchy-Riemannrelationholdsbetweenthe real andimaginal parts

[14].

Thus,

using

Lemma

3.6with

w(z)

I/If(z)l

the asymptotic formulafor

f(z)

canbe obtained fromthenth rootofthe weighted polynomial

s--I

and

caph(w,

"y).

Lemma

3.7. Underthecondit/on

(A),

the approximation

f(z)

of

f(z)

can be obtainedby

/()

’o{1-[

’’

/

(;)

fora sull]ciently largenumbern. Then

f,(z)

converges to

f(z)

uniformlyon everycompact subset

of D.

0

isdeterminedbythe condition

f,(1)

i.

4.

ALGORITHMS

Usingtheterminologiesof thechargesimulationmethod,

Lemma

3.7ismentionedaafollows for the numerical conformal mapping ofaringdomain.

Algorithm 4.1. The approximation

f(z)

of

f(z)

is obtainedasfollows:

(4a)

The chargepoints

{z,,},__

satisfyingthe condition

(A)

arechosenon %

(4b)

The chargeat everychargepointisassumed to be

1/n.

(4c)

Theapproximation

f(z)

of

f(z)

can be obtainedby

Z-Zr,,t

for asufl]ciently large number n. Then

f,(z)

converges to

f(z)

uniformlyon everycompact subsetof

D.

is determinedbythe condition

f

(1)

1.

(6)

24 T. INOUE

Algorithm 4.2. Theapproximation

f

z)

of

f

z)

isobtainedasfollows:

(4d)

The chargepoints

{z, }=

are’theoretically’chosen interior to".

(4e)

Thechargeat everychargepointisassumed to be

1/n.

(4f)

Theapproximation

f(z)

isrepresented by

f,(z)

e’"{H

1

,,z

}1/’

(O;real)

(4.2)

t=l

for astdticientlylarge numbern. Then

f(z)

converges to

f(z)

uniformlyon everycompact subset of

D

U".

O,

isdetermined bythe condition

f

(1)

1.

Algorithm4.2suggestsusthe following

f,(z)

of

f(z)

(49)

The charge points

{

z,,

}

[’=,

andthe collocation points

{

(,,

}’:,

are’appropriately’chosen inteior to 7 andon7, respectively.

(4h)

Whena,

(i

1,

2,...,

n)

arethesolutionsofasystem ofsimultaneous linear equations a,log

li

-_

":,,-,k

log

If(,,,k)l

(k

1,2,...,

n),

(4.3)

a

+

32+".

+

an

1,

(4.4)

thechargestit

{,r,,

=1

tiregiwen by

{l,

}..1

(4i)

Theapproximation

A

(z)

isrepresented by

2I

z

)"’

(0,,;

real)

(4.,5)

f(z)--e

’"

(1

,z-..z,,.,

g,

forasutticiently largenumbern. 0,,isdeterminedbythe condition

f,

(1)

1.

Algorithm 4.3requires n- 1 values

I(,)1

within k

1,2,...,n

to be given.

However,

all of the values arenotrequiredwhen the image domain

D’

is adisk with acenter at the origin

(o=ly

thevalue

II(C,)I-

(k

1,2,...,,t)

is =ot

k=ow).

From

Algorithm 4.3 for this case,

wehave thefollowing

f(z)

off(z)

(4j)

Thechargepoints

{z,,,}l’=

and the collocation points

{,,,

},=,

are appropriatelychosen inteior to/andon %respectively.

(4k)

Whena,

(i

0, 1,

2,...,

n)

arethesolutionsofasystem ofsimultaneous linear equations

’., _,o

(

,

2,...,,),

(4.)

cq

+

32

+-"

+

c,, 1,

(4.7)

(4/)

The approximation

f,(z)

isrepresented by

,=

forascJently largenumber n. 8, Jsdeternedbythe

contJon

f

(1)

1. The numerical experiments show that the approximations

1

a0logp, a,-

(i=l,2,...,n)

n

(4.8)

(7)

NUMERICAL CONFORMAL MAPPINGS IN A RING DOMAIN 295

with

p

cph(1,

"),,,)

(4.10)

hold, when the charge points and the collocation points are theoretically distributed. The method introduced in this section hasthe following advantages compared tothe standardone:

(4m)

Thechargepoints

1

exterior to70 isuniquelydeterminedbyones

{z.,,

(4n)

The numberof charge pointsexteriorto % isequaltooneof theminterior to -yp.

(40)

Whenthe charge pointsonthe analyticcurvewhose imageis acircle with center at the origin,typical examples showthat numericalresults ofhigh accuracy can be obtain.

5.

DISTRIBUTION

OF CHARGE

POINTS

In

this sectionweshowtwokindsofweightedextremal pointssatisfyingthe condition

(A).

For

each integer n

>_

1, let

Q.

beasetof wh-polynomials

where thezeros

{z.,

},=

lieontheboundary7 of

D.

Let

q,(z)

beapolynomiM such that

IIq2,(z)ll

inf

IIq.,(z)ll.

Theexistenceof

q,,,(z)

is easily proved by the usualmethod.

Then,

q,,(z)

is calleda

wh-Chebyshev

polynomial with zeroson7-

It

isknown thatitsatisfies the condition

(A) [5].

To

showanother polynomialsatisfyinghe condition

(A),

weintroducethe definitionof wh-transfinite diameter.

For

each integern

>

9., let

z,,,,-

z,,,,

lw(z,,.,)w(z,,,,)]}_/(,,(,,_,))

(5.3)

.(,)

sup

{

l-[

[I_,,:,,

(, )

im

.(, )

(.4)

iscalled wh-transfinite diameterof_7.

It

is saidthat

{z,,},"=,

arewh-Fekete points.

It

is then known thatwh-polynomials

z-k,.(z)

(.)

2=I

stisfythe condition

(A) [11].

B. NUMERICAL EXPERIMENTS

In

thissection,asimple exampleisshowntohelp understanding oftheoretical considerations

(8)

296 T. NOUE

Let

z--a

(0<a< ).

I()

-z This function mapsconformallyadomain

D

onto

(6.1)

{;

p

<

I1

<

1}

(6.2)

where

D

isaring domain whose outerboundaryistheunit circle.

Let

f(z)

map a ring domain

D

o bounded by % and a Jordan curve /p onto an annulus

{w;

1

>

Iwl

>

p}

bounded by%andaJordancurve

7’p,

_{corresponding %}to_"p withp

>

0.

The inverse function of

f(z)

is

w+a

The charge points

(z,,

},1

on

"r.

and the collocation points

{.,, },=1

on _{7 are} chosenas the imagesofthe points

pexp(2rj(i-

1)),

j

v/-Z

(6.4)

being distributedon

Iw]

p under the mapping z

h(w).

We

considerthecasewhere p 0.1 and 0.6.

It

isknown that these pointsareuniformly distributed

(which

meansthat these points satisfythe condition

(A))

onthe circles₇and7, respectively

[3].

For

n

10,

the

charge

points withp 0.1and the collocation points with p

0.6,

wesolve

asystemof simultaneous linear equations

(4.6)

and

(4.7)

andobtain the following results

(to

knowthe distributionprecisely,allofthechargesare

denoted)-Table 1:

Charges

by Algorithm4.4for n 10

ao

-0.51082563E+00

a3 0.99986970E-01

as

’10003428E+00

a

0.99997138Ffi01

a 0.10000704E+00

a

0.10002397EA:00

c7 0.99970734E-01

al0 0.99993097E-01

a

0.10000i23E+00

a

0.99968159E-01

as

0.10001742E+00

Thesolutionsatisfiestherelation

(4.9)

1

a0--log0.6

=-0.510825623...,

a,--

]-

(i

1,2,...,

10)

(6.5)

withhighaccuracy

(it

isremarkable that the number ofchargepointsisonly

10).

Using the charges

{a,}’__

obtained

above,

the approximations

(4.8)

are represented. The accuracy of the errorsareestimatedbythe maximumof

If.(.,,+,/)

(6.6)

where

,,+/2

isthe middle point between

,,,

and

,,,+1.

In

fact,

wemustestimatetheerrors

alsoat the collocation points.

However,

wecanobtainanlogousresults there.

Applying

reflexion principle and maximum principle, the error on the unit circle will be estimated

by

oneontheinner

boundary

%.

Then,

theaccuracymay decrease byorderof 10-1 at most.

(9)

NUMERICAL CONFORMAL MAPPINGS IN A RING DOMAIN 297

Table 2:

Errors

of

(4.8)

forn I0

4.4703483E-08

6.6fi40019E-08 1.2435162E-07

I.

1920929E-07

1.1920929E-07

8.4293693E-08

1.1920929E-07

9.2534291E-08 8.4293693E-08 5.9604644E-08 ******

Applying Algorithm 4.2 with n

I0,

usingthe above distribution of charge points and the theoretical values

1

(i

1:2,

10),

(6.7)

a0=log0.6,

,=-6

theerrors

IJ’

(z)-

f(z)l

ofthe approximation

(4.2)

areestimated. Thentheerrors are asfollows: Table3:

Errors

of

(4.2)

forn 10

1.2731556E-07 1.0745380E-07 8.2987058E-08

1.2287812E-07

7.4505805E-08 1.2731556E-07

1.4901161E-07

1.4790153E-07

1.999’2004E-07

6.6640019E-08 ***** *****

For

the aboveexamplewe haveusedtheparameterp 0.1 forthechargepoints. Ifthecharge pointsarenottooneartheboundary,analogousaccuracyisobtained.

The numerical caluculation has been performed by

Runfort-f77

(PC98-486AV)

and single precision.

7.

REMARKS

We

have shown that themethodintroduced in this papar has theadvantages coparedtothe standard one, when thechargepointsaretheoreticallydistributed.

Amano’s

method

[2]

for the conformalmapping from ageneral ring domain toa standard

oneisapplicable, evenwhen the theoretical distributionis notknown.

It

has also been shown that the approximations withhighaccuracycanbe obtained.

Algorithm 4.4is applicablewhen thecharge pointsand collocation pointsaredistributedby the way of

Amano.

We

wishthat this paper would contribute to presenting the theoretical base also for

Amano’s

method.

REFERENCES

[1]

AHLFORS

L.V.,

Complex

Analysis, 2nd.

ed., McGraw-Hill, New York,

1966.

[2]

AMANO

K.,

A

chargesimulation method for the numericalconformalmappingofinterior, exterioranddoubly-connected domains,

J. Comput.

App.

Math.,

53

(1994),

353-370.

[3]

GAIER

D.,

Lecture

on

Complex

Approximation, Birkhiiuser, 1987.

[4]

HENPJCI

P.,

Applied and Computational Complex Analysis

3,

John Wiley

&

Sons,

New

York,

1986.

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INOUE

T., A

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Chebyshev

polynomials,

Appl.

Math.

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INOUE

T.,

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’acad.

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29-34.

(10)

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INOUE

T.,

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151-155.

[9]

T.

INOUE,

Fundamentalpropertiesof weighted polynomials, Mathdrnatica,

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’acad. Rdpu.

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[10]

LANDKOF N.S.,

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Modern Potential Theory, Springer,

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MHASKAR

H.N.,

Finite-infiniterange inequalitiesin thecomplex plane, lnternat. J. Math.

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625-638.

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MHASKAR H.N.

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E.B.,

Wheredoesthesupnormofaweighted polynomiallive?

Constr.

Approx.,

1

(1985),

71-91.

[13]

MHASKAR H.N.

and

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E.B.,

Weightedanalogues ofcapacity, transfinite diameterand Chebyshev constant,

Constr.

Approx.,

8

(1992),

105-124.

[14]

STAHL

H.

and

TOTIK

V.,

Nthroot asymototic behaviorof

orthogonal

polynomials,

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Orthogonal Polynomials,

(Nevai P., eds.)

KluwerAcad. Pub.,

(1990),

395-417.