Kinetic Schemes for Saint-Venant Equations with Source Terms on Unstructured Grids

(1)

HAL Id: inria-00072657

https://hal.inria.fr/inria-00072657

Submitted on 24 May 2006

HAL is a multi-disciplinary open access archive for the deposit and dissemination of sci- entific research documents, whether they are pub-

L’archive ouverte pluridisciplinaire HAL, est destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non,

Kinetic Schemes for Saint-Venant Equations with Source Terms on Unstructured Grids

Emmanuel Audusse, Marie-Odile Bristeau, Benoît Perthame

To cite this version:

Emmanuel Audusse, Marie-Odile Bristeau, Benoît Perthame. Kinetic Schemes for Saint-Venant Equa-

tions with Source Terms on Unstructured Grids. [Research Report] RR-3989, INRIA. 2000. �inria-

00072657�

(2)

ISRN INRIA/RR--3989--FR+ENG

a p p o r t

d e r e c h e r c h e

% &('*),+-&/.10-.324)5 )6087:9<;>=@?A&B'C+DFEG)'H?'@+I)6JLKM?N+-&(9<'M0

O

&P+Q2 09<K4;R.S)T+-);-5 0U9<' K*'H0+Q;VKH.W+-K*;-)6X Y<;-&:X*0

Z-[<[]\/^_`acbS_d_Fefeg`hji*\/kflm`ngo!dlmam`QpRkflegqg`P\/_]h/pR`^rclsq-tV`kfqvuF\/[<`

w-xzy({(|({

}_lsamam`~qccc

6z

(3)

(4)

+.,%/++%01 %+23+20

4&56587:9-;+< =?>@;BA;BCCD<FE GBH#7:IJK<MLN0AJK=O<PIJQC*R<S7S;(TUGBP< 9 VWJKRYXZ< IR[B7S56<]\

^&_/`acbedfhgWija@k/ljm n*ixSopbUnx:q/n*ijaisrm n*ixSo

t brDuvr*nD`abUr&wfxvaq/lsbUxvbUr

yZz x|{DbUn~} y(w

m q/q"xWzn

t

b"zbUw_/bzw_/bo"/y({(|({f:k/ilslsbUn"SS:fdSd]q-mSbUr

UM- 6b]wfxSo/rDit bznD_/bcg/mioWnbUo-m oWn]mxvzgW_-m lslx 8m n*bzDr*uWr*n*ba_/isw_8isr"m o

k/r*k-mlax t bUlBn x t bUr*wzib"n*_/b "x r&iso8zis¡Sbzrxvz&wfxWm r*nm l+mSzbm rM¢£^&_/isre_WuWq-bzxSlsiw@r*uWr*n*ba

x ¤wfxSo/r*bz¡|m n*ixSo¥ljmMrisrr xSls¡Sb t x:o¦k/o/r*nzk/wUn*k-zb t abUrD_/bUr~Wu¦m]§Siso/bUn*isw@r*w_/bacbe-m rDb t xSo

m@¨-o/isn*b@¡(xSlsk-ab"m q/q-z xvm w_B¢©o¦ijaq"xWznm oWn&q-z xSq-bznªux ¤n*_/isrr*w_/babisrn*_/bq-zbUr*bz¡ m n*ixSo

x ¤n*_/bm n*bz«_/bUisS_vn¬q"xSr*isn*is¡Wisnªu¦¬_/bUom q/q/lsiwmn*ixSo/risn*_

t

zuFmSzbmr®m:zbwfxSo/r*i

t

bzb

t

¢£¯/x:l°

lx iso/]r xvacb_vuWq"xSnD_/bUr*isr®mo¦bUovnz xSqvuiso/bU±Wk-m lsinu¦isrq-zx ¡Sb t ¢

^&_/berDnm o t mSz t §:iso/bUn*iswr*w_/bacbeisrax t i°¨-b t n x t bm lisn*_F¡ mSzuWiso/0b t rDlx:q-b"m o t q-mSz²

n*iswUk/ljmSzlsun xq-zbUr*bz¡SbZbU±vk/islsis-zisk-ar*nmn*bUrrDk/w_m rrDn*isll/m nDbzU¢B}xvzbfx|¡:bz+n*_/b£r xSk-zwUb£nDbz*ar

t k/bn x@n*_/bmSz/isnz*mSzu""x:n*n xva³n x:q"xSvzDm q/_vu@_-mM¡Sbn x@-b t isr*wzbUn*is´Ub t iso¥r*k/w_¦m@mu@n x-m lµ

m o/wUbnD_/b -k/xFvzDm t isbUovn*rZ¤xvzZn*_/bUr*bbU±Wk/islis-zisk-arM¢

¶bislslsk/r*nz*mn*bn*_/bq-zxSq-bzn*isbUrx¤n*_/br*w_/babRxSo t iµ· bzbUoWn&n*bUr*n&wm r*bUrZ¤xvzZ_/isw_¦bUx/m wUn

r x:lk/nDix:o/remSzb"mM¡|miljm /lsb"m o

t

x:oaxvzbzbm lsisr*n*isw0mq/q/liswm nDix:o/r¢

¸º¹v»+¼½¾-|¿MÀg/m isoWnbUo-m oWn&r*uvrDn*ba¦ÁSrD_-m lslx )m nDbzr*uWr*n*ba¦Á:k/o/r*nzk/wUnDk-zb t Wzit ÁS¨-o/isn*b

¡(x:lk-acb"abUn*_cx t ÁÂ§:iso/bUn*isw"r*w_/babUrÁWr x:k-zwUbenDbz*arÁWbU±Wk/ilsis-zisk-aÃr*w_/babS¢

ÄÆÅ ÇÈÉÊ£ÇËÌÍÈªÎKÏÐ

ÑÒÓÔ

Õ

ÒÓÔ£Ö ×-Ø&ÙDÚÜÛ Ý-ÒÙDÞQÚ°ß²àÆáâUÚÜÛ°ßãä-ÞQÚ°åQæQßÙDçvè1ÚÜé|ÞQÚ°ÙMãêsÞ

ë

ÒÓÔ£Ö ×-Ø&ÙDÚÜÛ Ý-ä+ß²é|ìDÚ°æãí/ß²ÞæKî ÙØ&ß*èß²é|åãêsÞ

(5)

e +.-%+ %)2 ¬ ¬3#,/++%01 %

Â 8owfxSo/r*i

t

`zblsbr*uWr*n*`ab

t

bUrU±vk-m nDix:o/r

t

b"g/m isovnbUo-m ovn¬±vk/ia!x

t

Ulsisr*b"lsbUr

UwfxSk/lsbabUoWn*rbUobm k/x¥q-bUkq-z x ¤x:o t bUr¢&br*uWr*n*`ab_vuWq-bz"xSlsis±vk/b t blxSisr t b"wfxSo/r*bz¡|mM

n*ixSo¥bUr*n£zUr xSlskFr*k-z t bUra]milsljm SbUrZocxSo]r*nzk/wUn*k-zUrq-mSz«k/o]r*w_a]mwUisoUn*is±Wk/be±Wk/i-r°m q/q/k/isb

r*k-zk/o/bm q/q-z xWw_/b&¡(xSlsk-abUr¨-o/isr¢o/b&q-z xSq-zi Un!~ijaqxvznmovn*b

t

bwUbr*w_a]mbUr*n

t

µm r*r*k-zbz

ljm@q"xSrDinDi¡Wisn" t bljm _-m k/n*bUk-z t bm k t m o/r£lsbUr£wmr x$#lsb t xWa]m iso/b t bwm lswUk/lio/wUlsk/r t bUr£´fx:o/bUr

r*`Uw_/bUr¢£g/x:k/rwUbznm iso/bUr&_WuvqxSn*_/`Ur*bUrÁ/xSo¦qbUk/neaxSoWnzbz«k/o/bisoUvm lsisn" t ÜbUoWnz x:q/ibS¢

%+br*w_am]wUisoUn*is±Wk/b]r*nm o t m:z t bUr*n"ax t iµ¨&0qxSk-z¬nz*m isn*bz¬lsbUr¤xSo t r¡|m:zijm /lsbUrbUnÁ+bUo

q-mSzn*iswUk/lisbzUÁ:q"xSk-z1±Wk/b£lsb&r*w_amq-zUrDbz¡Sb t bUrUnm nDr t 'U±Wk/ilsis-zbnDbUlr±Wk/b£l(bmkcmk0zbUqxSr¢

)bq/lsk/rÁcxSo

t

¨-o/isnljm

t

isr*wzUn*isrm nDix:o

t

bUrnDbz*abUr£r xSk-zwUbUr

t

k/r+*@ljm"n xSq"xSWz*m q/_/isb

t

k¤xSo

t

qxSk-z&q-zUr*bz¡SbzZwUbUn,U±vk/islsis-zbS¢

oislslk/rDnzblsbUreq-z x:q-zi-Un"Ur t k6r*w_a]mr*k-z t iµ·zbUovnDreq-z x:/l`acbUren*bUrDn*r&q"xSk-z¬lsbUr*±vk/bUlsr

xSo¦wfx:o/o-m/.snljmr xSlsk/n*ixSom o-mluWn*is±Wk/bbUnr*k-z t bUrYmq/q/liswm nDix:o/rq/lsk/rYzmlisr*nDbUr¢

0 ¾-¼*/12¦ gWuWr*n*`ab t b~g/m isoWnbUo-m oWnÁUbm k/xqbUk"q-z x¤ xSo t bUrMÁa]m islsljm SbUrocxSo"r*nzk/wUn*k-zUrÁ

¡(x:lk-acbUr&¨-o/isrÁ/r*w_a]m r&wUisoUn*is±vk/bUrMÁ-n*bzDabUr&r xSk-zwUbUrÁ:r*w_a]m r

t

'U±Wk/ilsis-zbS¢

(6)

& '

++20%1

6bwfxSo/r*it bz¬n*_/b0g/m isoWnbUo-m oWnerDuvr*nDba¦Á/m]q-mSzn*iswUk/lÍm:zwm rDbWx ¤£r*_-m lslx &m n*bzZbU±vk-mn*ixSo/rÁ

_/isw_Fisrm o]k/rDk-m l ax

t

bUl-n x t

bUr*wzisbSÁS¤xvziso/r*nmo/wUbSÁvnD_/b& "x r£iso¦zis¡Sbzr xWzwfxvm r*nml-mSzbm r¢

^&_/isrrDuvr*nDbawm o¥-b

t

bzis¡Sb

t

¤z xvanD_/ben*_-zbUb

t

ijabUo/r*ixSo-ml iso/wfxvacq-zbUr*r*is/lsb(k/lsbzZr*uWr*n*ba

isn*_!m¶¤ zbUb¥r*k-z²¤m wUbSÁk/rDio/r*bU¡Sbz*m l&m r*r*k-acq/n*ixSo/rKra]m lsl&_/bUisS_Wn x¤Ym n*bzýÁ_Wu t z xSr*nmn*isw

q-zbUrDr*k-zbSÁS¡:bzn*iswm l/_cxvax:SbUo/bUisnªuWx¤?_cxWzis´fxSovnm lv¡:bUlxWwUisn*isbUr*)))U¢,+n~m lslx r£n x t bUrDwzis-bn*_/b

"x mn£n*ijab-!.0/1vm o t m nZn*_/bq"x:ioWn3254768:91:n*_-z xSk/:_0nD_/be_/bUisS_Wnx ¤B®mn*bz,;=<-1>2@?A.B/1

m o t isn*r¬¡SbUlxvwUisnuDC<->12?A4568 9 1/Wu]n*_/b_WuWq-bzxSlsiw@r*uWr*n*ba

E ;

E

-GFIHJLK

< ;@C?,MN/1

PO:¢QO

E ;C

E - FRHJLK

< ;@CSIC?

FIT

<VUWX;

9 ?

FYU

;

TZ MN/1

PO:¢°S

_/bzb

U t bUocxSn*bUrBn*_/bWz*mM¡visnuisovnDbUo/r*isnªum o t Z

< 2@?+n*_/bZ"xSn*n xva_/bUi:_vnÁ m o t n*_/bzb¤ xvzb[;

FZ

isr"nD_/b0lsbU¡SbUlx¤£®mn*bzrDk-z²¤ mwUb Kr*bUb¯Bis-¢KPOS¢QODU¢¶b t bUocxSn*b0mlr x5\<]->1>2@?3M^;<->12?_C<->12?UÁ

n*_/b

t

isr*w_-mSz:bS¢

`baPcd e afcd

g

I< <FCD;+I

g

7$hM<

¯BisSk-zbO:¢QOji

r*k-mllsuSÁ r*bU¡:bz*m l(xSnD_/bz+n*bz*armSzb~m t/t b t n xn*_/isrr*uWr*n*ba.isoxvz t bz+n xenm:§ bisovn xm wUwfxSk/oWn

¤ziswUn*ixSo/rzxSo!n*_/bp"xSnDn xva,m o t n*_/brDk-z²¤mwUbm o t xSnD_/bzq/_vuWr*iswm l¬¤Æbmn*k-zbUr/k/nF_/bzb&b

o/bUSlsbUwUnm lsl+n*_/bUr*bq/_/bUocxvabUo-m ¢

(7)

^&_/b]k/o/i

t

ijabUo/r*ixSo-m lZ¡Sbzr*ixSo<x ¤g/m isovnbUo-movnbU±Wk-m n*ixSo/r@ir@&bUll£m

t

m q/n*b t ¤xWzi

t

bml

zbUwUnm o/Sk/ljmSzzis¡Sbzr"m o t &bk/r*b]isnn x¦isovnz x t k/wUbxSk-zacbUn*_cx t r-b¤xvzbbUxWn*bUo t iso/isnisovn x

nx t iÍacbUo/r*ixSo/rr*isn*k-m nDix:o/r¢,+Qnisrezisn*n*bUo

E ;

E - F E

;C

E 2 MN/1

PO:¢yS

E

;C

E - F E

E 2

< ;C 9 F U ; 9 W^?

FYU

; E Z

E 2 MN/)

PO:¢Üdv

wVxSn*iswUb¥_cx|&bU¡SbznD_-m n"n*_/isrr*uWr*n*ba wfxvz*zbUr*qxSo

t

r"n x¶mpr*ijaq/lsbFwmr*bF/k/n]mpaxvzb]:bUo/bz*m l

bU±Wk-m n*ixSo¦wm o¦br*nm nDb

t

¤xvz&zis¡Sbzr¬inD_¦¡|mSzijm /lsbr*bUwUnDix:o/r¢

^&_/b t iFwUk/lsnun x¥k/o t bzr*nm o t n*_/b"wfx:ov¡SbzSbUo/wUbSx ¤oWk-abziswm lr*w_/bacbUreisr"zbUljm n*b t n x

n*_/b t bUbUqa]m nD_/ba]m n*iswm l/r*nzk/wUn*k-zbx ¤rDk/w_¥_vuWq-bz"xSlsiswr*uWr*n*ba n*_/b¬¨ zr*n&bUxWisr*n*bUo/wUbq-z xjx¤

x ¤&bmS§6r x:lk/nDix:o/rFmM¤Qn*bzr*_cxWwU§Sr]ison*_/bFljmSzSb¦isr

t

k/bn xQO |iso O {({/¢R+Qn]isr]-m rDb

t

xSo

n*_/b§Siso/bUn*iswisovnDbzq-zbUnm nDix:ox ¤nD_/b"r*uWr*n*ba ¬_/iw_6isrYm lsr x¦m¥abUn*_cx t n x t bzi¡:b"oWk-abziswm l

r*w_/babUr¬inD_¦(xjx t q-z xSq-bzn*isbUr¢

©#wUlÍmr*r*iswm l-m q/q-z xWm w_¤xvz1r xSls¡Wio/_Wuvqbz"xSlsiswr*uWr*n*bacrBwfxSo/r*isr*n*risok/r*iso/¨-o/inDb~¡BxSlsk-ab

r*w_/babUrÁ1r*bUbRx t lsbUr§:i®mo t mM¡vijmSznQOMÁ %+b²bU±vk/bQO ª¢¥^&_/isrcacbUn*_cx t wm o-b¥k/r*b t

xSo(SbUo/bz*mlnzijm o/Sk/ljmSz vzi

t

r"isn*_#m¥¨-o/isn*b¦bUlsbabUoWn

t

mnm¦r*nzk/wUn*k-zbk/r*iso/8mpq-mSzn*iswUk/ljmSz

wfxSoWnz x:l£¡(xSlsk-ab¥_/isw_#ir¥n*_/b¦ab

t

ijm o/b¦-m r*b t#t

k-ml~wUbUlslemr0isn]m risoWnz x t k/wUb

t

r*bU¡Sbz*m l

uSbm:zrem (xc¤xvz¬m bz x t uvo-mSaciwUrq-z xS/lsbarKr*bUbµ¤xvz&iso/r*nm o/wUbý¢+%+bUnk/rYzbUwm lslBn*_-m n&¨-o/isn*b

¡(x:lk-acbabUn*_cx t rczbU±vk/ijzb¥n x6wfxvaq/k/n*b -k/xWbUrFm nn*_/b¦wfxSoWnz xSl¡BxSlsk-abUr]isovnDbz²¤ mwUbUrÁm o t

n*_/bx ¡SbzDm lsl£r*nm/ilsisnu]x ¤enD_/b abUnD_cx t Qm o t axvzb¥q-zbUwUisr*bUlsuN _"!$#%!_"!&# @zbU±Wk/iÍzbUr

r xWabk/qWiso t io/io¦nD_/bisovn*bzq"xSljm n*ixSozx ¤Bn*_/b¬ -k/xvbUrMÁSn*_/isr&isrk/r*k-m lslsu]wm lslsb t n*_/b('°/k/islt iso/

/lxWw § xvz)'m q/q-z x xWiÍam n*b"&isba]m o/o6r x:l¡:bz jÁr*bUbQO"¤xvzm o8m q/q/lsiwmn*ixSo6n xFrD_-m lslx m n*bz

bU±Wk-m n*ixSo/r¢

*£k/nocxSo/bx ¤ÂnD_/bUr*b~m q/q-z x|xWija]m n*b&isba]m o/or x:l¡:bzrBisrm/lb&n x¬q-z x ¡vit b~mllWn*_/b t bUr*ijz*m/lb

q-z x:q-bzn*isbUrRx ¤~m+'°Bxjx t /r*w_/babSÁ+m o t n*_/ba]m iso t iFwUk/lsn*isbUr"mSzirDio/¥r*qbUwUi°¨-wmllsu6isn*_pn*_/b

g/m isoWnbUo-m oWnr*uWr*n*ba isrn*_-m n&n*_/b@r*w_/bacbq-zbUr*bz¡SbUr&bU±Wk/islis-zisk-a r*nm n*bUr&lsij§ b"nD_/b"r*n*bm t u

r*nm n*b]Sis¡SbUo(Wu8r*n*islslm n*bz] ;

FNZ

M-,. -Á CIM /Sý¢8^&_/isr t iFwUk/lsnu8m r@q"xSisoWn*b t xSk/n

Wu rDbU¡Sbz*m lm k/n*_cxWzrÁr*bUb/zbUbUovbz m o t %+bz xSk/x0OUdªÁ %+b²bU±vk/b1QO2 ª¢#©q/q-z xvm w_/bUrn x

/k/isl

t

rDnm /lsbr*w_/bacbUre_/isw_8q-zbUr*bz¡Sb"bU±Wk/islis-zisk-a _-m¡:b0bUbUo6q-z xSq"x:r*b

t

zbUwUbUovnDluS¢@^&_/b

xWbr x:l¡:bz3°4ÁS_-m rbUbUoFax t i°¨-b t io6xvz t bz1n xq-zbUr*bz¡:b£bU±Wk/ilsis-zisk-ar&iso*Zbz*a@k t bU´m o t

£m rD±vk/bU´5Üd6ªÁ7*Zbz*a@k t bU´bUnmly mo t n*_/b t iFwUk/lsnun xem lslx ;@n x¡ m o/isr*_"isrnzbm n*b t iso+°S ª¢

©lsr x-ÁSmzbUwUbUovnm q/q-z xWm w_Wu %+b58U±vk/b9QO62/isr-m r*b

t

xSo"k/r*iso/n*_/b3Rx

t

k/ocx ¡r*w_/bab«¤xvz1m o

m q/q-z xSq-zijm n*bUlsu"bUxWn*bUo t b t

r*uWr*n*ba¦Á m o

t

m o]m q/q-z xWm w_@-m r*b

t

xSowUbUoWnzb

t

r*w_/babUrisrZk/r*b

t

io

°¢¯/xvz1n*_/b£rDiÍacq/lbzwmr*b x ¤-m¬r*wm ljmSz1wfxSo/r*bz¡ m n*ixSoljmM"Á|m§:io/bUnDiw&r*w_/babZn*_-m nBq-zbUrDbz¡SbUr

bU±Wk/islis-zisk-a m o t _/isw_8isrq-zx ¡Sb t n x¥-bwfxSov¡:bzSbUoWnÁ/isr"q-zbUrDbUovn*b t iso:*xSn*w_cxvzisr*_W¡vislsi

(8)

Ávm o

t

mocxSn*_/bzm q/q-z xWm w_@-m r*b

t

x:o]ocxSoWwfxSo/r*bz¡|m n*is¡Sb&q-z x t

k/wUn*rZ_-m r£bUbUo

t

bU¡SbUlxSq-b

t

Wu:RxSr*rDb m o t %+bz x:k/x QO ªÁ RxSr*rDb QO O ¢8¯Biso-m lslsuSÁ&&b zb¤Qbzn x Rx t io-mk t bUn]m l {&¤xvz

m¥o/bUm q/q-z xvm w__/bzbn*_/bocxSnDix:o$x¤~&isba]m o/o8r x:l¡:bzisr0zbUq/lÍmwUb t Wu¶m¦a!xvzb":bUo/bz*m l

wfxSo/wUbUq/nM¢

+Qo(n*_/isr"q-m qbz&b t bUr*wzisb mºq"xSr*rDi/lsb¦zbUr xSlsk/n*ixSo x ¤n*_/b]r*uWr*n*ba,PO:¢QOUÁ£PO:¢°S¬k/r*iso/

m b "m o t q-zbUrDbUovnm¦¡|m:zijm ovn@_/isw_(_-m r@n*_/bFq-z xSq-bznupn xq-zbUrDbz¡Sbn*_/b¥r*n*islsl

m n*bzbU±Wk/islsi-zik-aº¢F©lsr xisn@q-zbUr*bz¡SbUrnD_/b]¡Sbzupr*ijaq/lsiwUisnu]x ¤&n*_/b]k/r*k-ml£§Siso/bUn*isw¥r xSls¡Sbz

m o t isn*rr*nm /islsisnu¦q-z xSq-bzn*isbUr¢3(r*qbUwUijm lsluSÁ1isneisr&bUlsl§Socx opn x]bcm/lb@n x]nzbm n¡|mwUk/k-a

;YM /¥_/bzbSÁBisnwfxvzDzbUr*q"x:o t rn x t zur xSislrMÁBn*_/bUo8nD_/b0rDuvr*nDbaÃlx:r*r*bUr_Wuvqbz"xSlsiswUisnuÂ@m o t

rmn*isr¨-bUrq-zbUwUisr*bisoWwUbUllZbUovnz xSqWuFiso/bU±Wk-m lsinDibUrM¢"6bFzb¤Qbz¬n x ybzn*_-mSab °:4¤xvzm¥r*k-z¡:bUu

x ¤n*_/bn*_/bfxWzbUn*iswm l+q-z xSqbzn*isbUrx ¤nD_/bUr*br*w_/babUrÁ/ybzn*_-mSacbYm o t isk1°(y41¤xvzZn*_/b"mwUn*k-m l

ijaq/lsbabUoWnm n*ixSoionªx t ijabUo/r*ixSo/r~mo t *xSk/w_Wk/n -¤xvzm:bUo/bz*m l/wfxSo/r*nzk/wUn*ixSo¤xvz x/

r*uWr*n*bacr¢^&_/bUijz&a]m iso

t

zDm¬-m w § isr~nD_/bljm w §,x ¤m wUwUk-zDm wUuxSo]r*n*bm

t

u"wfxSoWnm wUn

t

isr*wfxSoWn*isoWk/i°

n*isbUr0Æn*_/bUuFmSzbwfxvaq/k/nDb t bUx/m wUn*lsu]WuRx t k/ocx|¡+Ár*_/bz²f(«o/S±vk/isr*nxvz&xWbr xSls¡SbzrZ_/isw_

a]m:§ bUrn*_/ba k/o-m/lb@n x]wfxvacq/k/n*b"xSk/o t m:zuFljmMuSbzrVxvz&r*_/bmSz~ "x|¬r~iso6wUljm r*r*iswm l? -k/it6t u:

o-mSaciwUrq-z xS/lsbarM¢e¯/xvzg/m isovnbUo-m ovn¬bU±vk-m nDix:o/ren*_/isrq-z xS/lsba t xvbUrocxSnbUxWisr*ne_cx &bU¡SbzUÁ

r*iso/wUbn*_/b,xSo/lsu¦q"xSr*rDi/lsb

t

irDwfxSovnDioWk/isn*isbUr"mSzbr*_cxWwU§SrÁm o

t

n*_/bzb¤xvzben*_/b§Siso/bUn*isw@r*w_/bacbUr

aisS_Wnb"m-bUn*nDbz£wfxWaq-z xvaisr*b-bUn&bUbUom wUwUk-z*mwUuSÁSr*nm/ilsisnum o t b4FwUisbUo/wUuS¢

©eocxSnD_/bz0m t ¡|m oWnm Sb6x ¤ §:iso/bUn*isw¦r*w_/babUr]isr]n*_-m nisnFm lslx r]n x t bzis¡Sbmpr*w_/bab¥¤xvz

n*_/b"m t ¡SbUwUn*ixSox ¤n*baq-bzDm n*k-zb G<->12?mxvzZm"wfx:o/wUbUovnzDm n*ixSox ¤q"xSlslsk/nm oWnZ_/isw_¥isr"x:n*_

ax:ocxSn xSo/bisom o t wfxSo/r*bz¡ m n*is¡Sbiso ;"¢^&_/isrYm t/t isn*ixSo-mlÆÁ+m o tpt bUwfxSk/q/lsb t bU±Wk-m n*ixSopisr

ezinDn*bUo¦isomwfxSo/r*bz¡ m n*is¡Sb¬¤xWz*a m r

E

;

E - FIHJLK

< ;@C:?,MN/ 1

PO:¢S

/k/n &#ir-bUnDn*bzZr*bUbUoxSo¦n*_/b t bU¡SbUlxSqb t ¤ xWz*a

E

E - F C

T

BMN/)

PO:¢S

6b t xocxSn&wfxSo/r*it bz&n*_-m n&qxSisovnisopn*_/isrq-m q-bz®/k/nem om o-m lsuWr*isrwm o¦b¤xSk/o t iso12 ª¢

©¤Qn*bz1q-zbUr*bUoWn*iso/n*_/be§:io/bUnDiw®¤xvzDa"k/ljm nDix:oSx ¤+n*_/b&r*_-m lslx|m n*bz+r*uWr*n*ba³isoFgWbUwUnDix:oF/Á

&bbUxWq/ljm isoFn*_/bGO )r*nmo t mSz t §Siso/bUn*iswr*w_/babem o t &bbUrDnm /lsisr*_Fisn*rZq-z xSqbzn*isbUr£iso¦gWbUwUn*ixSo

y/Á&bq-zbUr*bUoWn"n*_/b¥m r*r xWwUijm n*b t ovk-acbziswm l£zbUr*k/lsn*r@io#gWbUwUn*ixSo8d¢ ^®_/bUoBÁ&bisovnz x t k/wUb¥m

¡ mSzijm oWn_/isw_pisrem t m q/n*b t n x]n*_/br*n*islslBm n*bzZbU±Wk/islis-zisk-aÃiso¶gWbUwUnDix:o ]m o t &bnzbm n&inDr

bUxWn*bUo/r*ixSo¥¤xvz&$) q-z xS/lsbar&iso¶gWbUwUn*ixSo//¢^&_/b"gWbUwUnDix:o/2"islslsk/r*nz*mn*bUrn*_/bqxSr*r*is/islsinDibUrWx¤

n*_/b"acbUn*_cx t Wu]oWk-abziswm lzbUr*k/lsn*r¢

(9)

+,F+.&. !"."/+ 20+³¬ %"

®

6b zbUwmll_/bzbpr xvabp~bUlsl§Socx o#q-z xSq-bznDibUrzx ¤"n*_/bpr*_-m lslx ®mn*bz@r*uvrDn*ba¦Ám o t vu

m o-mlx:Suisn*_R(k/lsbzbU±Wk-m n*ixSo/r¤xvzwfxvaq-zbUr*r*is/lsb]vm rÁ+&b]lsiso-§6n*_/b]a]mwz xSr*wfxSq/isw]g/m isoWn

bUo-m oWn&r*uWr*n*ban xcmaiswz x:r*wfxSq/isw

t

bUrDwzisq/n*ixSox¤n*_/b¬ -k/i

t

¬_/iw_¦isr&n*_/b

t

bUq-m:zn*iso/q"xSisoWn

x ¤n*_/b"abUnD_cx t r t bU¡SbUlxSqb t iso¦n*_/isreq-mq-bzU¢

! "#$

6b]zbUwm lslBnxFa]m isopq-z xSq-bzn*isbUr-x ¤n*_/b"r*_-mllx )m n*bz®r*uvrDn*ba¦¢&¯Bijzr*nRx ¤£m lslQÁ-n*_/b@r*uWr*n*ba

m t acinDrcmo isoW¡|mSzijm oWn0zbUSixSo ;<->12? . //Á£m o t n*_/b¥®mn*bz_/bUisS_vn;(wm o(iso t bUb t ¡|m o/isr*_

"x/x t io/@´fxSo/bUrÁ t zu0zbU:ix:o/rÁ n*it m l/ mn*r£U¢^®_/irZisr®zbUlÍmn*b t n x"mn*_/bfxWzbUn*iswm l/m o t oWk-abziµ

wm l t iFwUk/lsnªubUwm k/r*bn*_/b¬r*uvrDn*ba³lxSr*r*bUrZ_vuWq-bzxSlsiswUinuFmn ; MN/ ¢©ocxSn*_/bz1¤Qk/o t mSabUoWnm l

q-z x:q-bznu¥irn*_/b@bUovnz xSqvuq-z xSq-bznªuS¢Z6bwm lsl! _"!$# L#$ ] Fn x]n*_/b"g/m isoWnbUo-m oWnr*uWr

n*baºÁ£mp~bm:§6r xSlsk/n*ixSo#_/isw_#rm n*isr¨-bUrn*_/b¥bUoWnz xSqWu6io/bU±Wk-m lsisnu#Sis¡SbUo ison*_/b]¤xSlslx iso/

n*_/bfxWzba¦¢

%'&¹W¾-M¹ )(!*,+.- # > 0/21j%315476/81j%:9;4B</21j%>=?476/81j%A@B4B_ > ] # #"! ?CL]

;EDB/%F- # G! b > _"! #G/"H Q G #I#J4

K < ; 1>C 1 Z

?XMN;ML C L9

W F U ; 9

W

FRU

; Z 1

Q ¢QO

H >N!

E

E -

<OK ?

FRHJLKP C <#K

F U ; 9 W ?8QRB/)

Q ¢°S

¶b t x¥ocxSnq-z x ¡Sbn*_/isrn*_/bfxvzba _/isw_6zbUlsibUr,xSonD_/b"wUljm r*r*iswm ln*_/bfxvzux ¤£_WuWq-bzxSlsiw

bU±Wk-m n*ixSo/rFmo t r*ijaq/lsb¶m lsSbU-z*miw¦wmlwUk/ljm nDix:oBÁ&r*bUbgWbz*zb°2 ªÁ )mM¤Qbz*axSr+| ª¢#6b0{Dk/r*n

zbUwmllnD_-m n&¤xvzraxjx:n*_¦r xSlsk/n*ixSo/rn*_/b@iso/bU±vk-m lsisnu6io!Q ¢°S&isr"m o6bU±Wk-m lsinuS¢*+Qoq-mSzn*iswUk/ljmSz

n*_/brDuvr*nDba m t aisn*rem ¤m:aisluzx ¤r*n*bm t ur*nm n*bUr&Sis¡SbUo¦Wu]n*_/b"zbUlÍmn*ixSo/r

;:C M SUT1

Q ¢yS

LC L9

W F U ; F U Z M S 9 1

Q ¢Üdv

_/bzb9,VT®mo t , 9

m:zbnxmSz/isnz*m:zuwfxSo/r*nm oWn*r¢

(10)

O "5

6bocx bUxWq/ljm iso¶nD_/b0§:io/bUnDiw@¤xvz*a@k/ljm n*ixSox ¤£nD_/irr*uWr*n*ba¦¢¯/xWzezbmr xSox ¤£r*ijaq/lsiswUisnu&b

isoWnz x t

k/wUb"isn&¤xvzZn*_/bWxSo/b

t

ijabUo/r*ixSo-m l1q-z xS/lsba /k/nn*_/bwfx:o/r*nzk/wUn*ixSo¦isr¡:bzuFrDiÍaciljmSz®io

nx t iÍacbUo/r*ixSo/r0Kr*bUb"gWbUwS¢SU¢ %?bUnk/risovnz x t k/wUb"mzbm l¤Æk/o/wUnDix:o t b¨-o/b t xSo 68³_/isw_

_-m r¬n*_/b¤x:llx iso/]q-z x:q-bzn*isbUr

A< :? M!<?A.B/1

68

!<:?BMj1

68 9 !<:?BMj1

Q ¢S

m o t lsbUnk/rwfxSo/r*nzk/wUnenD_/b t bUo/rDinux ¤&q-mSzn*iswUlsbUr <->12X1 ? t b¨-o/b t vum¥r xwm lslb t #C C

"#$L#C#$

<-1>2 1 ?M < ;X1 C? M

;=<-1>2@?

A<

C=<-1>2?

?1

Q ¢S

isn*_

9 M U ; W )

Q ¢2S

!isn*_¦n*_/bUr*b t b¨-o/isn*ixSo/r~bezisn*bm§Siso/bUn*isw"zbUq-zbUr*bUovnm n*ixSox ¤n*_/b"g/m isovnbUo-m ovnZr*uWr

n*bam o t &be_-mM¡Sbn*_/b¤xSlslx iso/]nD_/bfxvzba i

%'&¹W¾-M¹ )(!*>( - # ] $ < ;X1>C@? _ H > #$ $ 5 34#

/81 %>=;4 6/81j%A@B4D # < ; 1 C? >N! "#

E

E -NF E

E 2 U E Z

E 2 E E

M! D<-1>2X1" ?1

Q ¢|S

$# > L>] &%' D<-1>2 1 ?UH ] Q:N!,j%% ->1A2

68

() M /1

68

() MN/)

Q ¢{S

*_AQ _"! #GL#$ ]7 ] #

68 9 () RB/ )

Q ¢QOS

(11)

M¾ ¾¾ %V&¹v¾¹ ( *(1¢ ^&_/b¦q-z xjx ¤zbUlsibUr6xSo!m6¡Sbzu xSv¡WixSk/r]wfxvacq/k/nm n*ixSoB¢(^&_/b

nx"g/m isovnbUo-m ovnbU±Wk-m n*ixSo/rMÁSm o t n*_/bbUoWnz x:qvuiso/bU±Wk-m lsinu¶/¢°SmSzb&bU±Wk/i¡ m lsbUovn&isn*_n*_/b

bU±Wk-m n*ixSo#Q/¢|S xSo/wUb"isoWn*bUvz*mn*b t iso m Wm iso/r*nSÁ m o t 9

¢^&_/b@lsb¤Æn*_-mo t r*it bUrwfxSiso/wUit b

bUx/m wUn*lsun*_-m o-§:rBn xn*_/b&zbUljm n*ixSo/r~Q ¢SBm o t n*_/b~zi:_vn*_-mo t r*it bUr¡ m o/isr*_xvzBmSzbZocxSo/q"x:r*isn*is¡Sb

m wUwfxWz t iso/]n xQ/¢{:£m o t Q/¢QOMSU¢^®_/bUr*bYm:zbwfxSo/r*bU±Wk/bUo/wUbUrx ¤nD_/bebm rDu zbUljm nDix:o/r t b t k/wUb t

¤z xvaQ/¢Si

;

;@C

9

F 9 M 68

9 < ;X1" C@? 1

Q ¢QOjO

KD< ;X1>C 1 Z

?XM

68 < 9

W

FRUZ

? < ;X1" RC?))

Q ¢QOS

^&_/isr0nD_/bfxvzbaÃ_-m rm¡:bzu6iÍacq"xvznm ovnwfx:o/r*bU±vk/bUo/wUbjinD_/b]ocxSoWlsiso/bmSz@r*_-m lslx| m n*bz

r*uWr*n*ba wm o#¡Sb¦¡WibU&b t m r¥m6r*iso/Slsblsiso/bmSzbU±vk-m nDix:oCxSo!m6ocxSo/lsiso/bmSz±vk-m oWn*isnu Á¤xvz

_/isw_FisnZir&bm rDibzn x¨-o

t m r*ijaq/lsbeoWk-abziswm l/r*w_/bab¬inD_](x/x

t

n*_/bfxvzbUn*iswm l/q-z xSqbzn*isbUr¢

w-xSnDiwUb_cx|&bU¡Sbz¬n*_-m nn*_/isr0^®_/bfxvzbaÃisr0m¥¡Sbzu&bmS§¦rDnm n*babUoWn¢G(r*qbUwUijm lsluion*_/b

] G# Q $Sx¤eisr*bUovnz xSq/isw¦vm r t uWo-mSaiswUrK_/isw_!m:zb a!xvzb¥SbUo/bz*m lZn*_-m o(n*_/b

abzb"g/m isovnbUo-movn&r*uWr*n*ba]ýÁvm lslBn*_/b¤mSaislsuzx ¤&bmS§bUoWnz xSqWu]iso/bU±Wk-m lsinDibUrmSzbrm n*isr¨-b t Á

ocxSn"{Dk/r*nxSo/bSÁm o t n*_/b(zisS_vnD_-m o t r*it b ,¡|mo/irD_/bUr¦¤xvz¥rax/xSn*_r xSlsk/n*ixSo/rÁocxSnxSo/lsu

nx a!xvabUoWn*rÁ/r*bUb QO {4¢0^&_/isr¤xvz*a]mlisra_-m rn*_/b t z*mM-mw §n x¦lsbm t n xaxvzb"wfxvaq/lsbUx

r*w_/babUr¢

^x(wfxSo/wUlsk t b8n*_/isr¥r*k//r*bUwUn*ixSoBÁ¬~bpq-zbUr*bUoWn m ¨ zr*n¥bUx/mSaq/lsb<x ¤³¤Æk/o/wUnDix:o/r¢^&_/b

r*ijaq/lsbUr*nq"x:r*r*is/lsb"w_cxSiswUbir

!<!?,M

T

9 ¤xvz L L

R

/ xSn*_/bz¬irDbS¢

Q ¢QO yS

©ocxSn*_/bzZo-m n*k-zDm l+w_cxSiswUbisrq-zbUrDbUovn*b t ljm nDbzU¢

(/ ¬02+2 +p-+8 & - !"]$%&

6bocx !q-m rDr£n x@n*_/b

t

bUr*wzisq/n*ixSozx ¤ ¤xvznD_/beg/m isovnbUo-m ovnZr*uWr*n*ba¦¢+Qo¦m

¨ zrDner*k//rDbUwUn*ixSo&b0zbUwm lslr xWab"ocxSnm n*ixSo/r&¤xvz¬n*_/b¨-o/isn*b¡(xSlsk-ababUn*_cx t xSo6m]:bUo/bz*m l

(12)

abUrD_ Kr*bUb O ÆU¢ +Qo mr*bUwfxSo

t

rDk//r*bUwUn*ixSo!&b¦k/r*bpn*_/b¶m"x ¡Sb§:io/bUnDiwº¤xWz*a@k/lÍmn*ixSo#n x

isoWnz x t k/wUbYm§:io/bUnDiwrDw_/babm o t &ber*_cx r xvab¬¤Æk/o t m:abUovnm l/q-z xSqbzn*isbUr&lij§ bqxSr*isn*is¡Winu

m o t isoWwUbUlslbUovnzxSqvuio/bU±Wk-m lsisnu¦io¦nD_/bo/bUxvnrDk//r*bUwUn*ixSo/r¢

# "

6bk/rDbm8abUr*_Cx ¤*68 ¬_/iw_!¡:bzn*iswUbUr]mSzb t bUocxSn*b t Á 4¢8^&_/b¥wfxSovnzxSl£¡BxSlsk-ab

KwUbUlsljZisr

t

b¨-o/b

t

m r, M P

1

Q+isn*_

M <

T F

?

W

¢¯/xvzrDiÍacq/liswUisnu]&b

r*k/q/qxSr*bn*_-mn&be_-mM¡Sbm]zbUSk/ljmSz&abUrD_m o t &b t bUocxSn*b2ºn*_/blsbUo/Sn*_x ¤mcwUbUll<, ¢«¶b

m lsr xk/r*b"mn*ijabUr*nDbUq G-Á-m o t r*bUn -"! M$#%G-¢

^xq-zbUr*bUoWnn*_/bSbUo/bz*ml-¨-o/isn*b@¡(xSlsk-ababUn*_cx t ÁW~bwfx:o/r*it bzm@r*uvrDn*ba x ¤¨ zr*nxvz t bz

wfxSo/rDbz¡|m nDix:o¦lÍmMr&isn*_cx:k/n~r xSk-zwUbn*bz*acri

E'&

E - F

E)( < & ?

E 2 M /)

y ¢QO

¯/xvz¬g/m isovnbUo-movnrDuvr*nDba POS¢yS+*?PO:¢ÜdSisn*_

Z M ,. -Á/&b_-m¡:bji

& M ;\ 1 ( < & ? M ,, \

\ 9

; F U ; 9

W --

)

6b t bUocx:n*bvu & !

n*_/bwUbUlslmM¡Sbz*m SbVx ¤nD_/bbUxÂm wUn&r x:lk/nDix:om n&n*ijab*-!

&.!

M

2

/10

&

<-

!

1>2@? 2 )

y ¢°S

6b"isoWn*bUvz*mn*benD_/bbU±vk-m nDix:o y ¢QOx:o¦n*_/br*bUn , '2

<-

!

1V-

! T

?UÁ/m o

t Á ioWn*bUvzDm n*iso/"WuFq-m:zn*r

n*_/b t is¡SbzSbUo/wUbn*bz*a¦ÁW&bRxS/nm iso

& ! T

& !

F G-

2 P( ! 3

( !

4

QMN/

y ¢yS

isn*_

( !

M

G- 6587:9

57 ( < &

<-1

?_? ->)

y ¢Üdv

^ xxS/nm iso m¥¨-o/isn*b]¡BxSlsk-ab]r*w_/bab]isn]zba]m iso/rn xº¨-o t m¦oWk-abziswm l¡ m lsk/b ( !

mr"m o

isoWn*bzq"x:lÍmn*ixSox ¤Bn*_/bwfxvaqxSo/bUoWn x ¤nD_/b -k/x ( < &

?xSo¦nD_/beisoWn*bz²¤m wUbYmn£n*_/bqxSisovn

¢

6bislslk/r*bn*_/b0§:iso/bUn*isw"¤xvz*a@k/ljm n*ixSo¦n x t b¨-o/b]m o¶m q/q-z x|xWija]m n*ixSox¤£n*_/isr -k/x K&b@k/r*b

n*_/brmSabocxSnm n*ixSo/rZ¤xvzZn*_/b"m q/q-z x xWija]m n*b t ±Wk-m oWn*isn*isbUrU¢

(13)

O ! "

^&_/b ] ¬isr"m ok/qWiso

t

ioWn*bzqxSljm n*ixSopio<xvz

t

bz&n x t

bUn*bzDaiso/b0nD_/b"oWk-abziswm l

-k/xWbUr ( !

iso y/¢ySUÁ/-m rDb

t

x:o¶nD_/bc§:iso/bUn*isw¤xvz*a@k/ljm n*ixSo¦isovnz x t k/wUb

t

iso gWbUwUn*ixSo8/¢ ¶b

m rDr*k-abn*_-m n&n*_/b"x:n*n xvan xSqxSvz*mq/_vu

Z

ir¬Sis¡SbUom nbm w_¥ocx

t

bS¢

¶bm r*rDk-abn*_-m nZ&be§Socx #n*_/br xSlsk/n*ixSo

& !

m n£nDiÍacb:- ! ¤xvz«bm w_ocx t bSÁW&b t b¨-o/bn*_/b

¤xSlslx|¬io/¥m ls(xvzisn*_-aisn*_¦n*_-zbUbr*n*bUq/ri

6b t b¨-o/b$!

M < ;1!

1 C1!

?&isn*_ < ;X1" C@? t b¨-o/b t Wu¶Q/¢SU¢

6b¦k/r*b¥n*_/b¦aiswz xSr*wfx:q/iw¥bU±Wk-m n*ixSo Q ¢|SU¢gWiso/wUb¦n*_/isr]bU±Wk-m n*ixSoirliso/bmSzc~bwm o

m q/q/lsu6m¥r*ijaq/lsb0k/qWiso

t

r*w_/bab@isn*_8mFabUnD_cx

t

x ¤~w_-mSz*m wUnDbzisr*n*iswUr&¤xvznD_/b¤xvzwUb"nDbz*a¦¢

+ nlsbm t ren x0nD_/b¤ xSlslx iso/ t isr*wzbUn*isrmn*ixSoF¬_/iw_ t b¨-o/bUrm t bUo/r*isnu]¤Qk/o/wUn*ixSo

! T

< ?i

! T

< U

G-

Z

? !

< ? F b-

2 P !

3

< ? !

< ?QMN/1

y ¢S

_/bzb

Z

M 09

0

9 m o

t

n*_/b -k/x$!

3 < ?Zisrwfxvacq/k/n*b

t

vu¦m o¥k/qv¬io

t ¤xWz*a@k/lÍm

!

< ?,M

$!

< ? ¤xvz' .B/1

!

T < ? ¤xvz RB/)

^&_/b t bUo/rDinu¤Æk/o/wUnDix:o[< ?®ir¬ocxSnem o¥bU±vk/islsis-zisk-a,Kr*bUb"zbamSz*§¦y/¢QOM*U¢

6b t b¨-o/b

& ! T

Wu

& ! T

M 68

! T

< ? ))

y ¢S

^&_/bUoBÁ ocxSn*iswUiso/]n*_-m n

68

! T

< U

G-

Z

? MN\

! T

FYU G-A;

! T

Z 1

&bWxS/nmio¥¤z xvaÃy/¢:UÁOy/¢SnD_/b"a]m wz xSrDwfxSq/iswr*w_/bab

& ! T

& !

FP ( !

( !

8

Q F

G-< & ! T

?,M /1

y ¢2S

isn*_

M b-

2 1

(14)

m o

t

¬_/bzbn*_/boWk-abziswm l -k/xvbUrmSzbSis¡SbUo¦vu¥n*_/b ]« -k/x¥r*q/lsisn*n*iso/

( !

M < & !

1 & ! T

?1

y ¢|S

< &

1*?,M

( < & ? F (

< ?1

y ¢{S

( < & ? M

< ;X1 C?)1

y ¢QOS

( < & ? M

< ;X1 C?)1

y ¢QOjO

<

& ! T

?,M

/

U ; ! T

Z

)

y ¢QOS

bUo/wUb¤xvzn*_BÁ &b_-mM¡SbVxS/nm iso/b t mwfxSo/r*isr*nDbUovnZr*w_/bacb-bUwm k/rDb x ¤Bn*_/b¬q-z xSq-bznªuSÁSmvm iso

t b t k/wUb t ¤ z xWa /¢QOjOýÁ

( < & ? F ( < &

?,M

68

< ;X1" C? M ( < &

?)

^&_/bwfx:o/r*bz¡|mn*ixSox¤«a]mr*rm o

t

axvabUoWn*k-a ¤xvz

Z

MN/@isrYm lsr x6xSv¡WixSk/r¢

¹ (/!*,+ -X3 !_ G! = A#j#X3LQ> G ! H _ Qj

0 #C C "#$ #C# I Q4 G> ] b /"=$%J4j% * 3>

> ! H _ ! Qj> ! T

< ?C# ! T

< ?FH ]D H # )! Q => L>] $

G 3

CBC>" # LC >#$ H,#$! !# I %

" #$ B! !"

6b"ocx .bUr*nm/lisr*_pn*_/br*nm/ilsisnuFq-z xSq-bznux ¤n*_/b§Siso/bUn*isw"rDw_/babS¢

%'&¹W¾-M¹ !*,+ _>#$ b ] # # ] /"9 %J4 0> !j

># !!>A # 4 68G1 >#D !<:? MN/ L L . )

y ¢QO yS

- !#$j* "!"# > 0/ j ! j3 %$'&4

G-)(+*-,X<

LC ! L F

! ? R 2X1

y ¢QOUdv

] b /"= %/.476 /"= %,1 9;4 ! H bI ! > % % ; !

.^/R

# # %

(15)

w-xSnDiwUbnD_-m n&n*_/br xSk-zwUbnDbz*a ; Z

_-m rocx]isoW -k/bUo/wUb,xSo¦n*_/b ¯%8wfx:o

t

inDix:o-bUwmk/r*b

isnir¬nzbm n*b t ijaq/lsiswUinDbUluS¢

M¾ ¾]¾ %'&¹W¾-M¹ !*,+* gWk/q/qxSr*b&b_-m¡:b ; ! . /)4*)/¯/z xvan*_/b t b¨-o/isn*ixSo x¤

n*_/b¬¤Qk/o/wUn*ixSo& io /¢Sm o t n*_/beqxSr*isn*is¡Winux¤n*_/b&¤Qk/o/wUn*ixSo$£Á/&bija]ab t ijm n*bUlsu t b t k/wUb

!

< ?A.B/ )_)

6b"ocx .isovnz x t k/wUbnD_/b±vk-m oWn*isn*isbUr Á 6t b¨-o/b t Wu

M (+*-,X< /1" ? 1 M (+*-,X< /1 ? 1

m o

t r x]&bYzbUzisn*benD_/b"aiswz xSr*wfxSq/iswr*w_/babFy ¢Sisopn*_/b¤xvz*a

! T

< U

G-

Z ? M^<"

L L

?

!

< ? F !

T < ? F ! 3 T < ?)

y ¢QOS

^&_/bUoBÁÂ¤xvzZbm w_+ÁWn*_/b¡|m lsk/bWx ¤ ]isrbUisn*_/bz&r*k/w_¦n*_-m n

L C ! L

.!

!

m o

t

nD_/bUo¥¤z xWanD_/bwfxSo

t

isn*ixSo y/¢QO ySxSo(m o

t

n*_/b

t

b¨-o/isn*ixSox¤B¤Æk/o/wUnDix:o &b_-mM¡Sb

!

< ?,MN/1

xvz®n*_/be¡ m lsk/bSx ¤ ]isrr*k/w_¦n*_-m n

L C ! L R!

!

_/isw_8ijaq/lsisbUr0nD_-m n

L L R < LC ! L F !

?"m o t n*_/bUoBÁ1k/r*iso/n*_/b¯%#wfxSo t isn*ixSo.y/¢QOUdWUÁ

&bWxS/nmio

L L Rj)

^&_/bzb¤ xvzbSÁvisopn*_/b"zbUljm n*ixSo8y/¢QO6SUÁ ! T

< U

b-

Z

?&isrmwfxSoW¡SbUx]wfxva@/iso-m n*ixSozx ¤ocxSo/b

vmn*is¡Sbe±Wk-m oWn*isn*isbUrem o

t

n*_Wk/r

! T

< U

b-

Z

?A.0/)

inD_¶mr*ijaq/lsb@ioWn*bUvzDm n*ixSo]iso-Á/&bRxS/nm iso

; ! T

.B/)

(16)

0 ! "O$

©ocxSn*_/bzmo tt

bUbUq-bz¬r*nm/ilsisnu¦wfxSo

t

isn*ixSo6isren*_/b@isoWwUbUllbUoWnz x:qvu]iso/bU±Wk-m lsisnªupiQ¢ÜbS¢n*_/b

t

isr

wzbUnDb"m o-m lxSSx¤n*_/biso/bU±vk-mlisnu Q/¢µSU¢B^&_/isrisrqxSr*r*is/lsb¤xvz« m n&"xSnDn xvarZisn*_ mom q/q-z x

q-zijm nDbw_cxSiswUb¤xvzZn*_/b¤Qk/o/wUn*ixSo ¢

%'&¹W¾-M¹ !*>( $>&j ! > # QX#$ j N C#

!<:?,MA<

9

?

y ¢QOS

H,NM MIB j Q# 5 ]$E/"9 %J4 % _>#$b

Z M /

"!"# > /"=$%31$@74 %- 7 /"= %/.476/"=$%31 9?4 >N!

$ X _"!$#"#L&#

K ! T

0K

!

F <P

!

3

!

8

QRB/1

y ¢QO2S

H,D Q _G I# _"!$# #

K !

MN;

!

< C ! ? 9

W F U

< ; ! ? 9

W 1

! 3

M < & ? F

< & T ?1

< &

?,M

P < ;X1 C@?

F

& < ;X1" C@? Q )1

< &

?,M

P < ;X1 C@?

F

& < ;X1" C@?

Q )1

*£b¤xvzbq-z x|¡Wiso/FnD_/ir^&_/bfxvzba¦Á/&b"o/bUb

t

m opioWn*bzq-zbUnm n*ixSozx ¤£n*_/b ¤Æk/o/wUn*ixSo m rm

is//rbU±Wk/ilsis-zisk-a¦¢

«¹ !*,+.- # ] < ; 1 IC? C # L H,5b_"! # ] </"= %,1 J45

GG #I#

K < ?M

68 P 9

W [< ? F 9

,< ? Q 1

y ¢QO |S

#$j G> $> _

.B/1

68

[< ? ) MN; 1 68 ,< ?) M ; CX)

(17)

_ _> LIG G# Q /9 %3154DE/"9 %,11J4A G G#

K < ; 1 C?XM K < ; 1>C? 1

H _ H GG

Z

H Q__%

M¾ ¾e¾ «¹ !*,+*^&_/bq-z x/x ¤x ¤Bn*_/isr %?ba]a]misrr*nmo t mSz t r*iso/wUb K < ?isrem wfxSov¡:bUx

¤Qk/o/wUn*ixSo-m lQ¢6bFzb¤Qbz¬n x µS4¤xvzm¥q-z x/x ¤iso8n*_/b"wfx:ovn*bUxWnRx ¤¤Æk/lslvmr t uWo-mSaiswUr¢WwVxSn*iswUb

_cx &bU¡Sbzn*_-m n@n*_/b¤xvzDa"k/ljm¤xvz Qm o t n*_Wk/r¤xvz ¤xSlslx|¬r t ijzbUwUnDlu ¤z xWaÃn*_/bD(«k/lsbz²

%Bmvz*m o/SbbU±Wk-m n*ixSo¦m r*r xWwUijm n*b

t n xn*_/isreaiso/ijais´m n*ixSopq-z xS/lsba

9

W F 9

W 9 < ?,M

F 1

_/bzb < ;X1>C@?£m o t

< ; 1>C@?~m:zbn*_/b %Bm Wz*m o/Sbea@k/lsn*isq/lsisbzr¢

M¾ ¾ ¾ %V&¹W¾-¹ !*>(!*#6b a]mo/iq/k/ljm nDbFn*_/b¥wfxSoW¡SbUx8zbUljm n*ixSo/r*_/isq)y/¢:m wUwfxvz

t

iso/

n x¦nD_/b¤ xWz*a@k/lÍm6y/¢O |S®¤xvznD_/b0bUo/bzSu¶m o t ioWn*bUvzDm n*b"isn@io -ÁBn*_-mo-§Srn x¦nD_/b0wfx:ov¡SbUxWisnu

q-z x:q-bznu]r*nm n*b t iopn*_/bq-z xjx¤Fx ¤n*_/b"^&_/bfxvzba8y/¢OSÁS&bWxS/nmio

K < ! T

? 0K

!

F G-

2 P

!

QRB/

_/bzb"n*_/b -k/x_-m rn*_/b"bUxWq-zbUr*rDix:or*nmn*b t iso¶nD_/b"n*_/bfxvzba¦¢^&_/bUoBÁ~b@k/r*bn*_/b %?ba]a]m

y/¢Oen x t b t k/wUb

K ! T

M KD<

! T

? R KD< ! T

?

m o t nD_/ir¬q-z x ¡SbUr&n*_/b"zbUr*k/lsn¢

w-xSnDiwUbnD_-m n&n*_/bisovnz x t k/wUn*ixSox¤n*_/br xSk-zwUbn*bz*ar t xWbUrocxSnlsbm t n x]mwfxSo/r*bz¡ m n*is¡Sb

bUoWnz xSqWu#iso/bU±vk-mlisnu mr¦in¦rD_cxSk/lt b6¤z xvaQ ¢°SU¢^&_/bzb¤xvzb¦&b6lxjx:r*b8m o.iÍacq"xvznm ovn

q-z x:q-bznu]_/isw_BÁ/n xS:bUn*_/bzZisn*_¦ocxSo/bUvm nDi¡Wisnux ¤®mn*bz_/bUisS_WnÁ/Sis¡SbUr¬±vk-m t zDm n*iswYmcq-zixvzi

xSk/o t r-xSo]nD_/ber x:lk/nDix:oB¢ wbU¡:bzn*_/bUlsbUr*rÁWm oFbU±Wk/is¡|m lsbUoWn&io/bU±Wk-m lsisnu aisS_Wn xWwUwUk-zZ-bUwm k/rDbSÁ

bUx/mSaiso/iso/FnD_/b

Z

n*bzDar&io¦nD_/bbUovnz xSqvuio/bU±Wk-m lsisnuF&bWxS/nmio¦nD_/bYm q/q-z x|xWija]m n*ixSo

< E

E 2 ; Z C@?

!

Z

< ;@C?

! T

FIZ P

< ;C@?

!

< ;C@?

!

Q F ; ! T <

Z

G-P? 9 )

¹ (/!*>(* R j > ! L# j # ! Q ] f 4

H # *<!> # Q # ] I /"9 %,1 =;4 _ A _*> ! Q ] H$]

Q : _"!$# "#L&#j% *_ # I :C)!_46X,

j 4H, /"9 %,1 =;4 _"! ] %

(18)

&

% ,

6b¦wfxSo/r*it bz _/bzbFnx¦r*ijaq/lsb¥ovk-abziwml£n*bUr*nwmr*bUr@¤xvz¬_/iw_#m o m o-m lsuWn*iswm l£r xSlsk/n*ixSo

wm o¦bwfxvaq/k/n*b

t

¢©o-m lxS(xSk/r&q-z x:/lbacr~_-mM¡Sb-bUbUopq-z xSqxSr*b

t

io6m@$xWz*§SrD_cxSqFiso/isn*ijm n*b

t

Wu5(lsbUwUnziswUisn"

t

b0¯/z*m o/wUb¦Kr*bUbF O y4Æ~¤xvz®n*_/bq-z xWwUbUb

t

io/:r x ¤n*_/isrxWz*§SrD_cxSq-U¢!isn*_pn*_/b

r*bUwfx:o t n*bUrDnwm r*b~b¬r*_cx #n*_/belxSrDr x ¤Bb4FwUisbUo/wUux ¤Bn*_/b¬r*nm o t mSz t rDw_/bab]y/¢2S+*?yW¢QOM|?io

r xWabr*isn*k-m n*ixSo/rx ¤bU±Wk/ilsis-zisk-aisn*_¦¡|mSziÍm/lb@"xSnDn xva¦¢

6bbUSiso#isn*_#m6ocxSoWr*nmn*ixSo-mSzupn*bUr*n"wmr*bS¢ +n]isrcm r*ijaq/lsb t mSa@-zbmS§piso!m6w_-m o/o/bUl

isn*_¥ mn£"x:n*n xvaÃ

Z

MN/SU¢+Qo¦n*_/isrwm rDben*_/bbUoWnz xSqWun*_/bfxvzuKr*bUbgWbUwS¢y/¢dvisr&¡|m lsit ¢&^&_/b

iso/isn*ijm lwfxSo t isn*ixSo/remSzb

;=< /1>2@? M

; 1 ¤xvz,2GR0/1

;1 ¤ xvz,2 DB/1

Kd¢QO

C=< /1>2@? M /)

Kd¢°S

6b¦w_cxjxSrDb ;UD ;]n xp-b¥wfxS_/bzbUoWn"isn*_(n*_/b]q/_WuWr*iswm l£q/_/bUocxvacbUocxSo x¤¬m t mSa@-zbm:§

¤z xva n*_/belsb¤Qnn x@n*_/bYzi:_vn¢^&_/isr&n*bUr*nm lslx r&n x"isr x:lÍmn*benD_/b-bU_-mM¡vixSk-z x ¤n*_/br*w_/babRxSo

m@r*_cxWw §cq-zxS/lsba¦¢gWbUb0ÁbS¢Ü-¢jÁ- |Á$µ24ªÁ °B|Q¤xvzn*_/bwfxvaq/k/nm n*ixSox ¤Bn*_/bbUx/m wUn£r x:lk/nDix:oB¢

^&_/bwfxvaq/k/nm nDix:o-m l

t

xva]mio6isr"rDuÂa]acbUnzisw0mSz xSk/o

t

n*_/bq"xSisoWn2 M /¦m o

t

n*_/bn xSnm l

lsbUo/Sn*_ 6isr~/Á°:SabUn*bzr¢B6b_-mM¡SbZk/r*b

t

nx

t

iµ· bzbUoWn£¡ m lsk/bUr¤xvzBnD_/b~rDq-m wUb&r*n*bUqBÁ /j/

m o t /j/ /ioxWz t bzZn x"r*_cx .n*_/bb· bUwUn*rx ¤n*_/bYacbUr*_FrDi´UbS¢Z^&_/benDiÍacbUr*n*bUq¥ir¬wfxvaq/k/n*b t

n*_-mo-§Srn xpn*_/b¯%!wfx:o t inDix:o y/¢QOýdvU¢@¶b¥_-mM¡Sbn*bUr*n*b t n*_/b t mSa@-zbmS§AxSo m¥&bUn-b t

m o

t

x:om

t

zu]r x:ilQ¢

¯/xvzZn*_/b¨ zr*n£n*bUrDn£wm r*bSÁW&bq-zbUrDwzis-b; M mo t ; M /)6Á:n*_/be¯1i:k-zbd-¢QOQzbUr*q

d-¢µ]q-zbUrDbUovnnD_/b"m n*bz&lsbU¡SbUlQzbUrDqB¢n*_/b t irDw_-mSzSbZ¤xvz. 2 M / /Fm n¬n*ijab-3M

W

/j/.S¢

+ o ¯1isSk-zbUrd-¢y/Á/d-¢Üd¥&b"q-zbUrDbUovnn*_/b"rmSab0zbUr*k/lsn*r¤xvz 25M /j/ //¢6bwfxvaq-mSzbn*_/b

wfxvacq/k/n*b t r xSlsk/n*ixSo/rq/lx:n*n*b t isn*_ mºwfxSoWn*isovkcxSk/rlsiso/b¦isn*_(n*_/bFbUx/m wUnxSo/b8t xSn@liso/bUrý¢

^&_/b@q"xSrDinDix:o$x¤n*_/b@r*_cxWw §¥isrio6(x/x t m vzbUbabUovn&isn*_pn*_/b"bUx/m wUnr xSlsk/n*ixSoBÁ-/k/nRxSopn*_/b

wfxvm:zr*babUr*_n*_/b&r*_cxWw § isrrax/xSn*_/b t m rk/r*k-m lWisn*_]m¬¨ zr*nxWz t bzBrDw_/babS¢¯/xvznD_/b~rDbUwfxSo t

n*bUrDnwm r*bSÁ &b&wfxSo/r*it bzm t zu"b t xSo]nD_/b®zi:_vnq-mSznx ¤+n*_/b&w_-mo/o/bUlÆÁ:n*_/b&wfxvaq/k/n*b t m n*bz

lsbU¡SbUlm o t¦t isr*w_-mSzSbmSzb"m lsr xwfxvacq-mSzb t m nn*ijab*-,M j/.n x]nD_/bbUxÂm wUnxSo/bUr¬io6¯BisSk-zbUr

d-¢Md-¢|/¢16b¡Sbziµ¤Æu¥oWk-abziswm lslsu¦n*_-m n&n*_/b®mn*bzZ_/bUi:_vnqxSr*isn*is¡Winu¥isrq-zbUr*bz¡:b t ¢

(19)

0.5 0.55 0.6 0.65 0.7 0.75 0.8 0.85 0.9 0.95 1

-1000 -500 0 500 1000

X

level.calc level.exact

¯1isSk-zbd-¢QO i )m:a-zbmS§i18m n*bz&lsbU¡SbUlQÁ 2M /j//¢

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7

-1000 -500 0 500 1000

X

disch.calc disch.exact

¯BisSk-zbd¢°i)mSa-zbmS§i)isr*w_-mSzSbSÁ12 M /j//¢

(20)

0.5 0.55 0.6 0.65 0.7 0.75 0.8 0.85 0.9 0.95 1

-1000 -500 0 500 1000

X

level.calc level.exact

¯1isSk-zb@d-¢yi)mSa-zbmS§@i8m nDbz&lsbU¡SbUlQÁ2 M /j/j//¢

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7

-1000 -500 0 500 1000

X

disch.calc disch.exact

¯BisSk-zb"d-¢Üd@i )mSa-zbmS§@i)isr*w_-m:zSbSÁ 2 M /j/ //¢

(21)

0 0.2 0.4 0.6 0.8 1

-1000 -500 0 500 1000

X

level.calc level.exact

¯BisSk-zbd¢i )m:a-zbmS§x:o m t zuFb t i8m n*bzZlsbU¡SbUlQÁ)2 M /j//¢

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

-1000 -500 0 500 1000

X

disch.calc disch.exact

¯BisSk-zb@d-¢i)mSa-zbmS§x:om t zu]b t i )isr*w_-mSzSbSÁ 2 M /j//¢

(22)

0 0.2 0.4 0.6 0.8 1

-1000 -500 0 500 1000

X

level.calc level.exact

¯BisSk-zb"d-¢2 i )mSa-zbmS§!xSom t zu]b t i8m n*bz&lsbU¡SbUlQÁ 2M /j/ //¢

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

-1000 -500 0 500 1000

X

disch.calc disch.exact

¯1isSk-zb"d¢|i )m:a-zbmS§xSom t zu]-b t i )isr*w_-mSz:bSÁ 2M /j/j//¢

(23)

O " "

6bocx !r*nDk

t

ucm n*bUr*nZwm r*bwfxSo/wUbzo/iso/"n*_/br*n*islsl+m n*bzbU±Wk/ilsis-zisk-a¦¢&^&_/bwfxvacq/k/nm n*ixSo-m l

t

xvam iso0isr£mw_-m o/o/bUlWisn*_]mq-mSz*m xSlsiw&/k-aqxSon*_/bZ"xSnDn xva³ax

t

bUlsis´Ub

t

Wu"n*_/bZbU±vk-mn*ixSo

Z

< 2?,M

P

/)

W

/)/< 2 / ? 9 Q )

Kd¢yS

^&_/blsbUo/SnD_x ¤nD_/bw_-m o/o/bUl+isrSSa¦Á:~bk/rDb2M /j//¢^&_/biso/isn*ijm lBwfx:o t inDix:o/remSzb

C<->12? M /1 1

;=<-1>2@?

FIZ

< 2@? M W )

^&_/isrr*nDbm t uerDnm n*bZirocxSnq-zbUr*bz¡:b t Wun*_/b£r*nm o t mSz t §:iso/bUn*isw~rDw_/bab&m rr*_cx o"iso]¯BisSk-zbUr

Kd¢{SUÁ-Kd¢QOS1_/isw_¥q-zbUr*bUoWnn*_/br xSlsk/n*ixSoFmn£n*ijab3-,M j/.S¢g/xvab-x:r*wUislljm nDix:o/r®mq/q-bmSz xSo

n*_/b¤zbUbr*k-zª¤ m wUbm o

t

n*_/b¡:bUlxWwUisnu]¨-bUl

t

o/bm:z£nD_/b/k-aq¦isreocxSn3//¢

0 0.5 1 1.5 2

0 5 10 15 20

X

level.calc level.exact bottom

¯1isSk-zbd-¢{igWn*isll1®mn*bziB8m nDbz£lsbU¡SbUlQÁr*nm o t m:z t §:iso/bUn*iswr*w_/babS¢

'!D0 2 +)+0

6bocx q-zbUr*bUoWn]m6¡|m:zijm ovnx ¤n*_/b¦r*w_/babF¬_/iw_#isr m/lb¦n x¶q-zbUr*bz¡Sb¥n*_/b¦r*n*islslm n*bz

bU±Wk/islis-zisk-a m o t § bUbUq#n*_/b¥r*nm/ilsisnu8q-z xSqbzn*isbUrx ¤Y§:iso/bUn*iswr*w_/babUrK_/bUisS_Wn0qxSr*isn*is¡Winu

(24)

-0.1 -0.08 -0.06 -0.04 -0.02 0 0.02 0.04 0.06 0.08

0 5 10 15 20

X

disch.calc disch.exact

¯1i:k-zb"d-¢OigWnDilslBm nDbzi )isr*w_-mSzSbSÁ:r*nm o t m:z t §:iso/bUn*isw"r*w_/babS¢

m o t bUoWnz xSqWu¦iso/bU±vk-mlisnu/U¢©ocxSn*_/bz¬bUxvn*bUo/rDix:o$x¤~§Siso/bUn*isw]r*w_/babUr_/isw_6q-zbUrDbz¡SbUrn*_/b

r*nDilslm n*bzbU±Wk/islsi-zik-aÃ¬ilslBb"q-zbUrDbUovn*b

t

iso¶m ¤xvzn*_/wfxvacio/q-m qbz" °dÆU¢

0! 5

^&_/bc -k/x¶rDq/lisn*nDio/º¤ xWz*a x ¤~nD_/b'µw_-mSzDm wUn*bzisr*n*isw$r*w_/bab_/isw_(nmS§ bUrisovn xm wUwfxSk/oWnn*_/b

xSn*n xva

Z

isr@io/wfxWaq-m n*is/lsb]isn*_6n*_/bbU±Wk/ilsis-zisk-a wfxSo/r*bz¡ m n*ixSopq-z xSq-bznªu8¢^&_/b@ -k/xWbUr

_-mM¡Sbn x¦nm:§ b"isoWn xm wUwfxSk/oWner xvabU_cx nD_/b0o/bUisS_W"xWziso/wUbUlslsr¢ ¬bUo/wUb~bisoWnz x t k/wUb]nx

±Wk-m oWn*isn*isbUr5;

m o t ;

n*_-m n¥islsl-b8q-zbUwUirDb t ljm nDbzUÁ/k/nm lsmMuvr¥_-mM¡SbnD_/b¤xSlslx iso/

t

bUqbUo

t

bUo/wUu

;

M < ; 1 Z 1; T1

Z

T>?1

;

M < ; 1 Z 1>; T1

Z

T_?)

n*_/b¤Qk/o/wUn*ixSoÃisrew_cxSr*bUo¥r xFm r&n xbUo/r*k-zbn*_/b¤x:llx iso/]wfxSo/rDnz*m isoWn*r

DB/r*k/w_¦n*_-m n ) 4 1 /RB;

R ; 1

& ¢QO

;

FIZ

M ; T

FBZ

T!M ;

MN;

T )

& ¢°S

(25)

¶bk/rDbn*_/bbUxvq-zbUrDr*ixSox ¤nD_/b& -k/xvbUr

( < &

?&m o

t ( <&

?ZioWnz x t k/wUb

t

iso y/¢OSm o

t

y ¢QOjOZm o t n*_/b t b¨-o/isn*ixSox¤ iso#Q/¢S«n xezisn*b

( < & ? M ;

0

< C

F

?

C F

!<?*1

( < & ? M ; 0

0

< C F

?

C F

!<?

m o

t

&bisoWnz x t

k/wUbn*_/bwfxvz*zbUwUnDb

t

-k/xvbUrK&bk/r*bn*_/brmSacbeocxSnm n*ixSo-

( < & ?,MN;

0 0

90

< C F

90

0

?

C F

90

0

!<?1

( < & ?,M ; 0 0

0 0

< C F

0

0 ?

C F

0

!<?*)

& ¢yS

6bFzbUwmlln*_/b@ocxSnm n*ixSo

M

0

9 ¢ ©nn*_/isrlsbU¡SbUln*_/b@r xSk-zwUb@n*bz*a isrnmS§bUobUxWq/liswUisnÁ

r x¦iso!y/¢2S&isnisr

t

bUocxSn*b

t

<

& !

?UÁ-isnisrocxSnr*qbUwUiµ¨-b

t

m nn*_/isrer*nm Sb"bUxWwUbUq/nn*_-m ninDr¨ zr*n

wfxvacq"xSo/bUoWn&irr*n*islslm rDr*k-ab t n x]¡|mo/irD_B¢

^&_-m o-§:rn xn*_/bUrDb]nz*m o/r¤ xvzDa]m n*ixSo/r&bx:/nm iso m¦o/bUr*w_/bab]n*_-m n_-mr"r xvacb0nD_/bfx

zbUnDiwml£q-z xSqbzn*isbUrn*_-m n@&b0q-zbUr*bUovn@iso n*_/bo/bUxvn@r*k//r*bUwUn*ixSoB¢ +Qo xvz

t

bzn xr*ijaq/lsiµ¤Æun*_/b

ocxSnm n*ixSo/r&&bisll t z xSq¥n*_/br*k/q-bzr*wzisq/n.#p_/bUo¦ocxFaisr*nmS§ birq"x:r*r*is/lsbS¢

O "

%'&¹W¾-M¹ !*,+ *L ; M ; ; M ;

T

Z M Z

T R G b

/"=$% .4 6/"= %J4V/ j%:=;4 $>> %

M¾ ¾~¾£$&¹ %'&¹W¾-M¹ *3+ *~6b_-mM¡SbZn xw_/bUwU§nD_-m n

( < & ? F ( < &

?,M ( <&

?_/isw_

¤xSlslx|¬r

t

ijzbUwUn*lsu]¤z xvan*_/b"mr*r*k-aq/n*ixSo/rx ¤n*_/bn*_/bfxvzbaº¢

%'&¹W¾-M¹ !*>(_>#$b / %31J4 , #$ 0 >N!># !!>A> Q

/"= %,1 =;4 5#$j "!"# ] 7 > !

G- ( *-,<

LC ! L F

! ? R 2 1

& ¢Üdv

] /= % .8476/"= %J4V/ %>=;49! H I ! &#j%

(26)

M¾ ¾0¾"$&¹ %'&¹W¾-M¹ !*>(!*6b]Sis¡Sb"nD_/b"q-z x/x ¤

t

ijzbUwUn*lsum nn*_/b0am wz xSr*wfx:q/iwlsbU¡SbUlQ¢

*Zu¥io t k/wUnDix:oBÁ-&b_-m¡:b

),4 ; !

.B/ )

¯/z xWa y/¢2:+*?y/¢{S&b_-mM¡Sb

; ! T

MN;

!

FP (

< & ! 8

T ? F (

< & !

? (

< & !

? (

< & !

T

?Q

m o t mr&bY§:ocx nD_-m n

& (

<&

?A.B/1

(

< &

?RB/1

isnzba]m iso/r&n xq-zx ¡Sbn*_-m n

MN;

! F P (

< & !

? (

<& !

?8Q=.B/)

x:o/r*it bziso/]n*_/b -k/xWbUr t b¨-o/b t iso /¢ySUÁ &br*bUbnD_-m n

MN;

!

F 00

0 0

< C F

;

?!<?

0 0

90 90

< C F

;

?!<:? ,)

6b"k/rDbn*_/b"zbUljm n*ixSo/r*_/isq

0 0

0 0 C

A<:? 0 0

90

90 C

!<? . LC L1

m o t nD_/b¤xSlslx iso/]wfxSo/r*bU±Wk/bUo/wUbWx ¤n*_/br*k/q/q"xvznq-z xSqbznuºy/¢QO yS

0 0

0 0 ;

;

!<? .

W

;

; 1

0 0

90 90 ;

;

!<? R

W

;

; )

n x6xS/nm iso¨-o-m lslu

.B;

! P < LC L F

W <

;

; F

;

?_?8Qf)

(27)

¬r*iso/]n*_/bk/q/qbzxSk/o

t

xSo5;

mo t ;

Á-m o

t

n*_/b

t

b¨-o/isn*ixSox ¤

&bewfx:o/lk

t

b@n*_-m n

.B;

! P < LC L F

?Qf)

^&_-mo-§Srn xn*_/b,¯%¦wfxSo t isn*ixSoBÁ nD_/irq-z x|¡:bUrn*_/b&qxSr*isn*is¡visnuSx¤ mo t wfx:o/wUlk t bUrnD_/b~q-z xjx ¤*¢

" "/

6bocx !q-z x ¡Sb&n*_-mn£isn&ir&qxSr*r*is/lsbn x"q-zbUr*bz¡SbnD_/br*n*islsl+®mn*bzbU±vk/islsis-zisk-a¦¢Z¶bwfx:o/r*it bz

¨ zrDnn*_/b®mn*bzZ_/bUi:_vnbU±Wk-m n*ixSoBÁWn*_/bUo¦nD_/b"axvabUoWn*k-abU±Wk-m n*ixSo6¢

«¹ !*,+ _>#$ 35!_"!&# / j%:9;45QG Q>N!46: GL# I

> ! # >N!

C ! M /1 ; !

FIZ

M51 ) 4

/"= %/.476 /"= %J4V/ j%:=;4D$# _*

; ! T

MN;

! 1 ) 4 *)

t

ijzbUwUnwm lswUk/lÍmn*ixSoCx ¤n*_/b¨ zr*nwfxvaqxSo/bUovn,x ¤en*_/b -k/xWbUr

r*_cx r&bm rDilsu¦n*_-mn (

< & ? F ( < &

T>?,MN//ÁWn*_Wk/rq-z x ¡Wiso/0nD_/b %+ba]a]m/¢

¶bocx ³nzu¦n x¨-o t m o8m q/q-z xSq-zijm n*b t isr*wzbUnDirm n*ixSox¤£n*_/b"r x:k-zwUb"n*bzDa iso<xvz t bz¬n x

bUo/r*k-zbn*_-m n&n*_/br*bUwfxSo t bU±Wk-m n*ixSomlr x0q-zbUr*bz¡SbUr&n*_/br*nDilslBm nDbzbU±Wk/ilsis-zisk-a¦¢

%'&¹W¾-M¹ !*_>#$b $I 0/j%>9?4 # _ >

:< & !

?,M

/

U

; 09 0

1

& ¢S

H,

; M ;

F ;

W 1

H _:;

;

H, C3! 4Q H bG"#

Z

Q H,I5!_"! ]

;

FIZ

MN; T

F0Z

T!M ;

8

FIZ

M ;

FBZ 3

1

& ¢S

Z M Z

T!M

Z

M Z M Z

3 T)

& ¢2S

- * b !_ : > H "#$L#C#$%

(28)

M¾ ¾"¾ %'&¹W¾-M¹ * *e¯/z xva y/¢2S+*ByW¢{ UÁ-&/¢y:UÁxSo/bwm o¦wfxvacq/k/n*b

\ ! T

M \ !

FP ( < & !

8

T ? F ( < & !

? ( < & !

T

?Q %b- <& !

?

M \ !

F W P< ; T 9

T 9

;

T F ;

9

;

?' < ; 9

9

; F ;

T 9

;

T

?8Q 6G- < & !

?)

^&_/bUoBÁ ~bw_cxjxSrDb

;

M

U ; 9

9

; F ;

T 9

; T 1

m o

t

&bWxS/nm iso

\ ! T

M \ !

U

; < ;

;

4

? 6G- < & !

?)

¬r*iso/]n*_/b

t

b¨-o/isn*ixSo x ¤ <

& !

?&iso#&/¢SýÁ ~b¨-o

t

\ ! T

MN\

!

U ; P

< ;

FIZ

? B< ;

FIZ

?Qf1

m o

t

nD_/b"zbUr*k/lsnisrq-z x ¡Sb t ¢

; MN;

FIZ

(

J

<( *-,<

Z 1 Z

T ?1;

FIZ ?1

; MN;

FIZ

(

J

<( *-,<

Z 1 Z

T ?1;

FIZ

?1

Z

M <; FBZ

? 9 F

< ; T

FIZ T ? 9

W ;

)

^&_/bwfx:o t inDix:o &/¢QOisrrm n*isr¨-b t isn*_ M r x"n*_/b ¬¯ %6wfx:o t inDix:o KrDbUbcy ¢QOUdvUÁK&/¢dvisr

ocxSnax t iµ¨-b t ¢£}xWzbfx ¡Sbzisn*_¥n*_/bUr*b¬¤xvz*a@k/ljm b&bYzbUwfx|¡:bzn*_/br*nm o t m:z t r*w_/bacb¤xvz« m n

xSn*n xvacr¢

#

6b"_-mM¡Sbn*bUr*nDb t n*_/isro/bU.r*w_/bacbWxSo¦n*_/br*n*islslBm n*bz«q-z xS/lsba t b¨-o/b t iso¶gWbUwd¢°/¢jÁ:n*_/bUo

n*_/b¥bU±Wk/ilsis-zisk-a isrq-zbUr*bz¡:b t k/q n x¦n*_/b am w_/iso/bFmwUwUk-z*m wUu8m r"rD_cx|¬o8iso ¯1isSk-zb8&/¢QO

_/bzben*_/bbz*z xWzxSoFnD_/bem n*bzlsbU¡SbUl+isr&q/lxSn*n*b

t

m o

t

iso¯1i:k-zbc /¢°S_/bzben*_/b

t

isr*w_-mSzSb

isrq/lx:n*n*b t ¢

(29)

0 5e-08 1e-07 1.5e-07 2e-07 2.5e-07 3e-07 3.5e-07 4e-07 4.5e-07 5e-07

0 5 10 15 20

X

level.error

¯1isSk-zb ¢QOjigWnDilslm n*bzi+8m n*bz®lbU¡:bUlBbz*z xvzýÁSbU±Wk/ilsis-zisk-aÃr*w_/babS¢

-1.5e-06 -1e-06 -5e-07 0 5e-07 1e-06

0 5 10 15 20

X

disch.comp disch.exact

¯BisSk-zb /¢°igWn*islslm n*bz i )isr*w_-mSzSbSÁ:bU±vk/islsis-zisk-aÃr*w_/bacbS¢

(30)

^&_/b¤xSlslx iso/6n*bUr*nwmr*bUr"wfxSo/wUbzo n*_/brmSab]:bfxvabUnzu¦¬inD_ mºwfxSo/r*nm oWn

t

isr*w_-mSzSb

q-zbUrDwzis-b t xSo¥n*_/bioW "x ."x:k/o t mSzuSÁWn*_/bUo¥n*_/bewfx:o t inDix:o/r0Q/¢ÜdWmSzb¡Sbziµ¨-b t mo t &bwm o

t b t k/wUbn*_/b&bUxÂmwUn®mn*bzB_/bUisS_Wn¢^&_-zbUb&wm r*bUr£mSzb~wfx:o/r*it bzb t Á t bm lsiso/"¬inD_Fmr*k/-wzinDiwml

mxWz¬ -k/¡vijm lj "x "Ám¥nzDm o/r*wzisn*iswm lxSo/bisn*_cxSk/nr*_cxWw §#KnD_/b" "x -bUwfxvabUrn xvz*zbUoWn*ijm lxSo

n*_/b/k-acqm o t n*_/bRxSk/n "x .ir¬n xvz*zbUoWn*ijm lj£mo t m"nz*m o/r*wzisn*iswm lxSo/bisn*_¦r*_cxWwU§¶KnD_/b "x

bUwfxvabUrn xvz*zbUoWn*ijm lxSo@n*_/b&/k-aq0m o t n*_/b x:k/n "x 8isr -k/¡vijm ljU¢^&_/bUr*bZn*bUr*nBwmr*bUr_-mM¡Sb£mlr x

bUbUo¦nzbm n*b

t

io¥n*_/b<xvz*§:r*_cxSq]iso/isn*ijm n*b

t

Wu(lsbUwUnziswUisn"

t

b¯/z*m o/wUb] QO y ÆU¢^&_/b

t

i°·+bzbUoWn

wm rDbUrYmSzb t b¨-o/b t Wu¦n*_/b@"xSk/o t mSzu¦wfxSo t isn*ixSo/rÁ~b@q-zbUr*wzisb0nD_/b"isoW "x t isr*w_-mSzSb\

!

m o t nD_/bWxSk/n "x .®mn*bzlsbU¡SbUl

5

bUxWwUbUq/n&¤xvzZn*_/bn xvzDzbUovnDiÍmlcxSk/n "x i

¯1lsk/¡vijm l1 "x i,\

! M ) W .j1

5 M W )

^z*m o/rDwzisn*iswm l "x|.isn*_cx:k/n~rD_cxvwU§5i,\ ! M j) . 1 5 MN/ )

^z*m o/rDwzisn*iswm l "x|.isn*_¦rD_cxvwU§5i,\ !

MN/)

. 1

5

MN/)

¶b@k/r*b 2 M /j//¢^&_/b"iso/isn*ijm lwfxSo

t

isn*ixSo/remSzbSis¡SbUoWuFwfx:o/r*nm oWn

t

m nm@bU±vk-mlBn x

n*_/b"xSk/o t mSzuwfxSo t isn*ixSo/r¢]¯/xWzbm w_6wm r*b&b0_-mM¡Sbq/lxSn*nDb t nD_/b0wfxWaq/k/n*b t m n*bz¬lbU¡:bUl

m o t n*_/b t isr*w_-mSzSb@wfxvaq-mSzb t n xFnD_/b0bUx/m wUnRxSo/bUriso8¯BisSk-zbUrF/¢QO² /¢S¢^&_/bwfxSo/r*nm ovn

t isr*w_-mSzSbisreocx:n~q-zbUr*bz¡Sb t Á:n*_/bbz*z xvz«isr-x ¤n*_/b xWz t bzx ¤ bUxWwUbUq/nison*_/br*_cxWw §¬_/bzb

isnir¬lÍm:zSbzU¢

2,@ 2 0+-+!¬

- !&$%0®

6bocx|#m oWnn x"m q/q/lsuxSk-z«o/bU!r*w_/bab&n x"nx t ijabUo/r*ixSo-mlvq-z x:/lbacr¢B6b-bUSiso¥isn*_

n*_/b@q-zbUr*bUoWnm n*ixSozx ¤n*_/b¨-o/isn*b"¡BxSlsk-ab0abUnD_cx t xSomcSbUo/bz*m l+nzijm o/:k/lÍm:z&vzit m o t r xvab

r*bUoWn*bUo/wUbUrm "xSk/n@n*_/b]SbUo/bzDm l§:iso/bUn*iswFnD_/bfxvzuiso#$)¢B^&_/bUo8iso(r*k//r*bUwUn*ixSo/r /¢y¦m o

t

/¢Üd

&b¦bUxWq/lÍmio.n*_/b ) §:iso/bUn*isw¶rDw_/babmo

t

Sis¡Sb¦r xvab

t

bUnm islrzxSo#n*_/b¦oWk-abziswm l¬iÍacq/lb

abUoWnm nDix:oBÁW¨-o-m lslsu¶iso6r*k//r*bUwUnDix:o /¢c~b@q-zbUr*bUoWnenD_/b0$) bUxvn*bUo/rDix:ox ¤£n*_/b@bU±vk/islsis-zisk-a

r*w_/babS¢

VO "

^&_/bSbUo/bz*ml-abUn*_cx t isr~wUlxSrDbn x"nD_/bO )¨-o/isn*b¡(xSlsk-abeacbUn*_cx tt bU¡SbUlxSq-b t isoFr*k//rDbUwUn*ixSo

y/¢OS¢

%+bUn

-b"m@nziÍmo/Sk/ljm n*ixSox ¤,68:9¬_/isw_¦¡Sbzn*iswUbUr~m:zb t bUocx:n*b t' ¢^®_/b t k-m l+wUbUlslsr ,

mSzbWxS/nm iso/b t Wue{ x:io/iso/¥n*_/bwUbUoWn*bzrx ¤a]mr*rx ¤nD_/bnzijm o/SlsbUr&r*k-z*z x:k/o t io/bm w_¦¡SbznDbUx

'

¢¶b@k/r*bn*_/b¤xSlslx|¬io/ocxSnm n*ixSo/r"KrDbUbY¯1is &/¢2:*i

ÁWr*bUn-x ¤ocx t bUr

rDk-z*z xSk/o t iso/ Á

L, L

Á/mSzbmSx¤ , Á

ÁW"x:k/o

t

mSzu¥b

t

Sb-bUlxSo/Siso/]n x]wUbUlslr5,

mo t , Á

(31)

0 0.5 1 1.5 2

0 5 10 15 20

X

level.comp level.exact bottom

¯BisSk-zb /¢QOjixSo/r*nm ovn t isr*w_-m:zSbSÁ/¯Blsk/¡Wijm lB "x "ÁW8m n*bzZlsbU¡SbUl

4.39 4.4 4.41 4.42 4.43 4.44 4.45 4.46

0 5 10 15 20

X

disch.comp disch.exact

¯1i:k-zb/¢µi x:o/r*nm oWn t isr*w_-mSzSbSÁ/¯1lsk/¡vijm l1 "x "Á )isr*w_-mSzSb

(32)

0 0.2 0.4 0.6 0.8 1 1.2

0 5 10 15 20

X

level.comp level.exact bottom

¯1isSk-zb ¢yi QxSo/r*nm oWn t isr*w_-mSz:bSÁÂ^ xvz*zbUovn*ijm l "x ¬inD_cxSk/n&r*_cxvwU§Áv8m nDbz&lsbU¡SbUlQ¢

1.51 1.515 1.52 1.525 1.53 1.535 1.54

0 5 10 15 20

X

disch.comp disch.exact

¯BisSk-zb ¢ÜdixSo/r*nm ovn t isr*w_-m:zSbSÁ/^xWz*zbUoWn*ijm l/ "x .isn*_cxSk/n&r*_cxWw §Á)isr*w_-m:zSbS¢

(33)

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5

0 5 10 15 20

X

level.comp level.exact bottom

¯1isSk-zb /¢ixSo/r*nmovn t isr*w_-mSzSbSÁ/^ xvz*zbUoWn*ijm l "x| isn*_FrD_cxvwU§-Á mn*bzZlbU¡:bUlÆ¢

0.175 0.18 0.185 0.19 0.195 0.2 0.205 0.21 0.215

0 5 10 15 20

X

disch.comp disch.exact

¯BisSk-zb /¢ixSo/r*nmovn t irDw_-mSzSbSÁ/^xvzDzbUovnDiÍml/ "x .isn*_¦r*_cxWw §Á)irDw_-mSzSbS¢

(34)

Á/lsbUo/Sn*_x¤ Á

#

ÁWk/o/inocxvz*am l-n x

Á/xSk/n®m:z t n x , # #

U¢

¯BisSk-zb /¢2i)k-m l+wUbUlsl , ¢

¶bm lsmuWr£wfx:o/r*it bzmr*uWr*n*ba x ¤¨ zr*nxvzt bzZwfxSo/rDbz¡|m nDix:oFljmMr

E'&

E - F HJLK

( <& ? MN/)

& ¢QO

¯/xvz¬g/m isovnbUo-movnrDuvr*nDba POS¢OUÁOPO:¢° isn*_

TZ

MN//ÁW&b_-mM¡Sbji

& M ;

\

1 ( < &

?,M

\

\ASI\

; F

UWX;

9 )

& ¢°S

^&_/bUo¦nD_/b¨-o/inDb"¡(xSlsk-abr*w_/babezisn*bUr

& ! T

M & !

0 <& ! 1 & !

1 # ?

& ¢yS

isn*_

M

b-

L, L 1

& ¢Üdv

_/bzb+ < & 1 & 1 # ? t

bUocxSn*bUr"mo8isovn*bzq"xSljm n*ixSo<x ¤&n*_/b]ocxvzDa]m lwfxvaq"x:o/bUovnWx ¤&n*_/b -k/x

( < & ? #

mlx:o/¦n*_/b]b t Sb

¢0^&_/isr@ioWn*bzqxSljm n*ixSo6isr"k/r*k-mllsu6q-bzª¤ xvzDab t k/r*iso/6mzxSo/b

t ijabUo/r*ixSo-mlBr xSls¡Sbz&rDio/wUblxWwm lslsu¦n*_/bq-z xS/lsba lx/xv§Sr¬lij§ bm@q/lÍmo-mSz t isr*wfxSoWn*isovk/isnuS¢

(35)

O "5

k/wUb A<:?ýÁÂm@qxSr*isn*is¡SbSÁ/bU¡SbUo¤Qk/o/wUn*ixSo

t

b¨-o/b

t

xSo 68*9emo

t

rmn*isr

¤Quviso/

68

!<? 0M

)

& ¢S

+ o¶m t/t isn*ixSo¦&b"m r*rDk-abenD_-m n !<?£isrwfxvaq-m wUnDlur*k/q/q"xvzn*b t ÁWiÆ¢bS¢

4568 1r*k/w_¦n*_-m n !<?,MN/¬¤xvz

L L . )

& ¢S

!<? M

W'Q6

0

1M j1

W )

& ¢2S

+ o¶$)ÁWn*_/b t bUo/r*isnux ¤q-mSznDiwUlsbUr <-1>2X1" ?isrezisn*n*bUo

<->12X1" ?M

;<->12?

9

!<

C<->12?

?

& ¢|S

isn*_

t b¨-o/b t vu8Q/¢2SU¢

¶b@_-mM¡SbYm$))nD_/bfxvzbabU±vk/is¡ m lsbUovnn xF^&_/bfxvzba/¢°/¢

"

6bZ¤ x:llx rB_/bzb£gWbUwS¢y/¢°/¢+6bm lsmMuvrBm r*rDk-abiso"n*_/isrr*bUwUn*ixSo

T Z

MN/)M^&_/b

t

isr*wzbUnDi´mn*ixSo

x ¤n*_/b"aiswz x:r*wfxSq/iswbU±vk-mn*ixSoFWu¦mr*ijaq/lsbk/qv¬io

t

r*w_/babSis¡SbUr

! T

< ? !

< ? F G-

L, L 0 # !

< ?,M /1

& ¢{S

isn*_¦nD_/b -k/xvbUr !

< ?Zwfxvacq/k/n*b

t

Wu¥n*_/bk/qv¬io

t ¤xWz*a@k/lÍm

!

< ?,M !

< ? ¤xvz' # .B/1

!

< ? ¤xvz' # RB/)

^&_/bUo¦&bwfxvacq/k/n*b & ! T

Wu

& ! T

M 68

! T

< ? )

& ¢QOS

(36)

^&_/bbU±Wk-m n*ixSo /¢QOSwmo¦-b"zisn*n*bUo¥isn*_¦n*_/b¤xvz*a&/¢y:isn*_

< & 1 &

1:# ? M ( < & 1 # ? F ( < & 1 #

?1

& ¢QOjO

m o

t ( < & 1:# ? M

! 0 #

< ? 1

& ¢QOS

( < & 1 #

?,M

! 0 #

< ? )

& ¢QO yS

+Qn£isr&wUlsbmSznD_-m n£n*_/b¬r*w_/bacbisr~wfx:o/r*isr*n*bUoWn£m o t wfxSo/r*bz¡|m n*is¡SbS¢6bwm oFm lsr x@q-z x ¡Sb&n*_-m n

n*_/bm nDbz£_/bUisS_Wn&q"xSrDinDi¡Wisnu¦ir¬q-zbUr*bz¡Sb

t

k/o

t

bz&n*_/b ¬¯ %wfxSo

t

isn*ixSo

G-)(+*-, < LC ! L F

! ? R L, L

0 )

& ¢QOUdv

6b¥Sis¡Sb_/bzbr xvab t bUnm islsrWxSo8nD_/b0ijaq/lsbabUoWnm nDix:o$x¤~n*_/b]§:io/bUnDiwr*w_/bacb t b¨-o/b t vu

& ¢ySUÁO /¢QOO &/¢LO y U¢¯/xvzbmw_6b

t

Sb

Á1~b

t

b¨-o/b]m¥lxWwm l-m rDir < 2!T1>2

9

?m r*r xvwUijm n*b t n x

n*_/bocxvz*a]m l

t

ijzbUwUn*ixSo8m o

t n x¦n*_/bnm o/SbUoWn*ijm lxSo/bS¢6b

t

bUocxSnDb

&

ÁBn*_/b¡SbUwUn xvz t b t

k/wUb

t

¤z xva

&

Wu¥n*_/b"z xSnmn*ixSo]ison*_/isro/bU -m r*isr¢,+K¤#

M^< # 1 # ?ýÁv&b_-mM¡Sb

& t b¨-o/b t vu

&

M

&

isn*_ NM

/ /

/ #

#

/ # #

m o

t ( < &

1:# ?=M T

( <

&

?1 T M / /

/ #

#

/ # # )

6b":i¡:bn*_/b

t

bUnm islsb

t

bUxWq-zbUr*r*ixSozx ¤

( < &

?zbUljm nDb

t n xn*_/bb

t

:b

(

<

& ?,M

;

9 68

T

C

)

& ¢QOS

xvz

( <

&

?,MN;

0

68 <

C

T F T

?

C F

A<:? 1

& ¢QOS

&bWxS/nmio (

WuFm omo-m lxS(xSk/r&wfxvaq/k/nm n*ixSoB¢