Upload
combatps1
View
43
Download
11
Embed Size (px)
Citation preview
! "$#&%
'
)(+*,*-!/.
01234576
8 9 :=?A@B-C4DE/F!:G
H/IJH KMLNOPQRTSJLNVUXWYS[Z]\^_S`LN abdcfegih
jk
jlnmporqs
k4tu
sv
k
miw
xzy
s|{~}
j
j
j
j/ sv{
j
v
j
v
m
j
v
j
v
o
u4
w !& }
T__T_TXT>dT
x
f[nf&fffTnXXAf$/`!]XJ-$fffdX!!
Xff3[ff_X[z]T5[_!!"fX,/ff!!
$Xz!
o
3X
w
/ff/T!X`XT
k
f4_
!X-"fX
J[T"3]T/T_d[T$`_p][TYJ_
D-E)FHGJIKLMNPOQGRF/STMUVWM#SGRNXVF,KY,MY,Z-M(M[]\QGRY,KENPM^OQF,M_,_,KNQI`Y,ZQM#aQGUGNQSTM#ERbcMN-MFId
G_ e)f
e)g(h
h
h
h ijTk
5d`GJJJS0JfJe!fgD`SfS
S
v@bv#Df[
L!SfSh!nSMJS&!J!JS)SG!2UDMJ!RS
_U5aDUegVUhXJUhU
l
g=[Uj0JyS
S
\QMSJ_!SbjS&
!g!JQDbJJ]`SJM2SQbL
l-5 G&
=yUV7
-5
_byUV7
-5
5V7
v
7
LS#abJg=-hgNf#USlJ-SJ!aM#
!g!J_=2Ul0SjS
S
j-aN`!SRSJMal
Q_J=[JbfSd`QX[SSgN
gyS)5JLNU]bS!y
DS
SM`!_R=V[hD\!U!!_bS`
XM0#&XM0
Mh
b
n
S
n5hNfS2a
O
!JXh
h
S
X
"
#
$
%'&)(*,+-(/.102n`Q`SJM!Sd`
Z\[]_^
]_`)a-b8cd
[]^
]_`)aecfhgjik
Z\[lm^
lm`)ab@cd
[ln^
lm`)aecf"gYik
]^
gpo
d crq d
b
ots
Z
f
u
Z ]
`
k
ln^
gvo
dwcrq d
b
ots
Z
f
u
Z l
`
k
xzy{|)}4~y/{'{H5~:m|{;n{;
gY'
]
k
l
k
k
k
5
|Gm~{|~n{n:H5|\{'m4~5|/{HD!{~!~|/e
g
]
c
l
}\
g
]
o
l
8
o
gw
k
k
k
k
8
UW):
m:{
d
b
os
Z
f"gji
U{H:{n},8{DH5|m|!{~
n
~{H:
|/5~m~:y
}5~y/{O:{
]
m|
l
5{e|/5~
|4{{|4{|~
8{'!{~O
gYi
}~:y/{H|
]_^
]
`
g
o
d
u
Z
~8H5|E{Wm{{H-}n/{H~@{m5~:m|)}n~:y~z|e~y/8Hm:{
gji
}|EOy/yDHm:{U@{
m|
l
~:e{D/|_~:m|G:y~:y5~12
gwi
}/m|E~y/{'{H5~:m|{;n{;
gY
]
k
l
k
k
k
,)
@x
:
n!{
d
b
ots
Z
f$i
xzy/\m:{/{|~:m~~y/{y{mtn!{/4~G|/
]
5|
l
5{m/{
n|=/~:{;
d
b
os
Z
f i
_
x5n{
g
]
c
l
}
g_
]
o
l
5|G~:y/{1{H~n|
{Hn{H
z
c
gY
k
k
k
k
D,m4;, t
cr;
gw
b
c
H
,/:
4 "!#:%$\'&)(*#+-,/.10203/.
2455"$6$87:9$=?@02#03A "+-BCAED//03A
FHGIFHGI
p2 w4{-y/wy *YfV
/
! #"%$'&%&)(+*,-+&/.102(
&
-
3547684:9=?479A@B=?4DCE9FCE@GCH8IKJML#N)O29P/H)QSRUT#N89VP8CE@GCN89?W#JMN)Q
JXH8Q+4D9N)@YWSZN[N)@E=F\
^]_*F`ba2&/$c
1
&)dea2`7f`
a?`
Fg
`h&)d
jiBk>lnmVlnoqpr
-
ikslnmVl+Xtop
u
9@E=2CvWUT#N8O2Q#W74w;r4U;eCEIEIxH/PyP8Q+4:W#Wz@E=F4zW:N)I{O2@GCN89|Nn}~FH8Q+HyL#N8I{CT4#7OFH8@GCN89?WwO%WSCE9%rCE9FCE@4
C 4:Q4:9VT4H89VPrCE9FCE@4MI47Z'479F@XA4:@B=?NPW7\
p _hfw4{p}/8w *Yf
Cy1}:N8QW:N)I{6CE9b=%R~F47QS
L#N8I{CT4#:OFH)@CN)9?W7\
u
9~FH8QS@GCT7O2IH8Q#J_rCE9CE@4^N8I{O2Z4'A47@E=FN[P8W;eCEIEIML4U4+y@4:9VP/4#P@NP/4#H8I
;eCE@B=9N)9FI{CE9V4#H8Q=%R~F47Q7LN)I{CT4#:OFH)@CN)9?W7\
_B Y>rjenAFYns Es
UF/2G+%v)VF/#hB
'F1
+>^%%7/F2+1/_/
+87 F
"!#$&%')(*!,+-.+/0"1:23
,45)67) 85.+49
:@
A=[BDCE=[1)E=w/+=[x?GF?H=I
JK'EL71M$NO1)(*!,+GM: KK(G!P5:58K0
,QE7HR5SKTR7LF8EU7!8,+D| 227,+DA2"7#RS-E3
SGR74SG2.+49WV8287+DH8X)(*!,++*M7H:E?$YZ7EK0K77,Q
ERS[6Y\].+89
_^ _a`cbed2ne6fAznts |{}y8vb
BjrVS~s 8vivF
--UL-O7&7
_A d2nr
Esl
=7VrF8=b)
>
/EF?+#)5BDCb8=722vqE=)4q?S+1/
+8=72K
;
4U
7&
7
Tj5G#
47HYj5*8)lEAD P70c5l1D7"iG7T5Z
,"
8+9;: =@?BADCFEHG IKJ*LNMO1P%Q
0 1 2 3 4 5 60.5
0
0.5
1
1.5
2
2.5
x
g(x)
g1(x)g2(x)
IKJ*LNMO1PRP
1 2 3 4 5 6 7 8 9 100
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
0.45
0.5
k
|Ck|
g1(x)g2(x)
SUT%VXWYT%V+Z@[\]^[`_ abc_dfe/ghij_ kl_*mT%i,nWocWV,Wpi,q"\srRWtUuv`wlwx#y{z|}uu
~R^%FcX"^cKp;%R,*RpN`Ks%RKN
RpNcp3,;pp6NR%%^%;cXHRcp,;p
RRp%X3N,,p^p,NNR>#%NcK,HN3NKl^c
^%NcN;Np^R%UR^pNN3RU^cUNU^p_ abc_NN
YFpX/F_ kl_^,^%"s%;{^N^FNNKK^^js^}6^X^pKpj^N
ps}%^%c
8+9;8 HYGlRGRH?{RH
IKJ*LNMO1P
{K
bB
b
b
w
{
bB
b
b
bB
k
b
b
bB
b
b
d
k
H
kc
i,W#Tm
d
k
@^
kc
l
\qT%n\rTiX
e
N^ 6Xp3^pX^p+RX%6HU^pX^pFH@R%c^p;^
}^^^
"R%Fc+^cKp`/p;
#%NH c
NXY
^N*X^cNp
},pXc^+N@^%^j
R; j^cKp;%>R^cK^^+ c l j
"!#!$ %'&)(*,+.-/0 % 1
2$3465798
:
;=@? A)BDC/EGFIH6J
KMLONQP
RC$JS= RCTFIH6JUE
V;WC$JS= ;XCTFYH6JZE
[
R
^;,\^cK^N",^]pcNX^%/}G]p`NN_^
pp}^^^cRp%cN*R^c^pU`RN"XR%csa^`,%'R%N %K
pRBNK^6G]pNXbN*"dcfeNX+,X%%UNN,^]pcNX
cg^cK^%KN
hYiWj kml^n
o
>pqAr=(/E
kml^n
o
?dp9A= sut
vwyxZzr{ |~}$xZzY 9z 0yzY zY
$%&
$')(+* ,-".
%/&
0132546879
$%/&
$' (:* ,-".
% &= '
(@?AB(
%C
&
;:D % &"= '
AB(
%C
&FEG/H
&IKJ
% =MLN; '
AO( P
Q
&R
G
P
% C
& E GH
&IJ
ES
&T
D U VXW>YZ[\VX[3]^`_badc"VfeX_g
:
/
35k
/
| >3
"
F5
5
fl
/
M
M
kNik"fmFrr
k`b3b
j
m
35
0 2 4 6 8 10 121.5
1
0.5
0
0.5
1
1.5
x
u(x,
t)
t=0t=1
!
35"
#%$
#
$
'&ok)(+*
l
\
,(-*
\
.(-*
/
/
/
$
10
\
$
10
.(-*
M
M
352
>
$
K
$
354
687%9
68:=
7 9
=?;A@CBEDFEG HJILKNMOQPSR
T
= UVXW+Y[Z\)V
]
B^D _a`cbSdefSegXhCbJikjl;nm=ocBEDp;qm
]
@Crsutwvyx{zJV5\|x[v}s8rSF
~
mwD e8
mBED _a`cbSdefSegXhCbJikjl;>aD @.W+rhCXdL`chCbJiF
D `cbpg@Em
7%X
Z|CW8xEY)V5\tVXXWcuY[Wr
7J
F
Q1? .w+c^
VXrV58V5\%Y[Vx{zJV5Vas+z\)szW8W-Y)v}s8r VXzSVXrxsrY[VWc8V5V5rY[
~-
o+D@s8\WcV5rZ
V5\DFjVx)WcuY[SW-YwY)VWcVavyxw vyx{zJV5\|x[v}8V
s8rx{VLZV5r5Vs+! vyx{zJV5\|x[v}s8rvyxY)W-Y
Y[Vx{SW+zJVs+Y[Vx{s8}Z Y)v}s8rv}|WrVW8xY[VWc8Vz\[s8zW+LW-Y[Vx
Vx[V5Va\)s
Y[V
W
s-VY)W
}VY[W+Ys8\
T
;
]
@NY[VWcVVXLZW-Y)v}s8rSFjY)Vx{zJV5Vsz\[s8zW+LW-Y)v}s8rvx
s8rx{Y)W+rLYs8\W}
sVXxWrY)V5\)Vs8\[VWrvrv}Y[vyW+x{WzSVv}
Vz\[Vx{VX\[8VXWxvY
z\)szW8W+Y[VXx
Us8\
T
;
jl;mwD
W+rS?WcVXxv}Y[v DJv}z\[s8zW+LW-Y[VWs+Y
CWx{Y[VX\Y[WrY)VWc8VXxvY)x
W+ DjkWrY[VX\[V5s\)VWrLvrv}Y[vyW+5sr v}Y[vsrv}
|WrVvY|xx[W+zJVWxY[Vx{s8}Z Y)v}s8rVXs8}8VXx
VaWx[sprs+Y)VY[W+YV5r v}JVX\[VXr8YwV5\)v}-W-Y)v}8VwY)V5\
xW+\)Vz\)VXx[V5rLYjY[Vx[sZ Y[vsrv}
V v
v}YkW
V5WcvsZ\v|uvyxY)V\)VXx[ZYks+s
vrvraY[V v%V5\)V5rLY!Y)V5\
x!z\)VXx[V5rLY
Us8\v}rx{Y)WrVaW+rVXLZW+Y[vsrx[Z|Wx
7S
;qm
7%
7J
v}1V5 v
vY
s+Y[z\)szW8W+Y[vsr
W+rS V5Wc%Y[vxXW+rVXW8x{v}
Vx[V5VXr
rsY[vrpY[SW-YvrY)vxw5Wx[V=?;nmBEDmD
. E!{w>'!{
HJILKNMOQP
7J
7
;> ?@C
~+
F
v}Y[
7
@.8F;
7
@
~+
F
7
@CLF;
7
@
~-
F
V5VXY)s VY)V5\
vrVars8r
Y)\[vvW1zW+v\|x@
7
G1@CJF)SGSF
HJILKNMOQP
Yw5Wr
VVXW8x{vVX\[v}VXY[W+YY[VVXv}8V5r-W+ZVXxW\[V8
G
;
T
s\
T
;
]
~
55X
]
a5XX[8))[5+[}[?+)
8
"!#!"!
$&%('*),+.-0/21.)436578/*98:;%)4:A@B;CD>ACEF;CGCE>?C(HBI>"JLK>"@NM;>OPCDB;QR?OSBQTQU>PR(VNB;@W;JX@,KYCDBZCE>?>"JLK4>P@*MF[]\N>S^OF@`_>
OB;ab_PJX@>WcJX@*CdB[}e
F;@W8
.f
=?>g@BCd>eF;[]RPB&CEFChJX@iFWW;JXCjJB;@kCdBcCXEN>l>PJLK4>"@NM;F;[]\N>PR
F;@Wk>"JLK>"@NM;>OPCDB;QRGRUENB;Hm@nF_B;M;>ToNp
rq
o
p
JsRtF;[uR"BAF;@n>"JLK4>P@*M>O"CdBQ
f
v
JX@O>CE>l>"JLK4>P@*aiBwW>PRgOB;JX@OPJW>eHmJXCXEtCE>yx8B;\ QTJ>"QabB#W>TRohH>lOF;@zQU>dKFQUW{B;\ QBQTJLKJX@F;[
x8B;\ QTJ>"Qz>S|TVNF;@NRTJB;@0FRzF@I>S|PVF@RTJB;@IJX@ICD>PQTa}R}B~z>PJLK4>P@*aiBwW>PR
f
@i~PFO"C(ogCE,JsRt[F;CjCd>"Q
JX@NCD>"QjVQU>"C(JB@bJsRl"\ JXCD>yK>"@>"QUF[ F@WFVV[]J>TRlJX@iRTJXCj\NF;CjJB;@RgJX@bH8E,JOEiCE>x8B;\ QTJ>"QgaiBwW>PR
>"JXCXEN>PQWB@B;CG>S|JsRTC(o{BQWBZ@BC{OB;JX@OPJW>HmJXCEYCE>>"JLK>"@NabB#W>TRB~bCE>W;J &>PQU>"@NCjJF;[
BSVN>"QUF;CDB;Q
f
P}ciG"."}
ly*m}Ag &, 4Xn
}[pX5gSUS )}8
&
U
y
hp*
U
{ )}8
"QU>AH>F;QU>F;RR\ azJX@4KYCXENF;CCE>>"JLK4>P@*MF[]\N>TRdh>PJLK4>P@*M>O"CdBQREF;M;>Z_>>"@rB;QUW>PQU>WIJX@
RT\NOSEFzHF}CEFCyJ ~GF;@Z>"JLK>"@NM;F;[L\N>&EF;R{Fiaz\ []CjJ V[]JO"JXCj6JXCHmJX[X[OSB@NCjQTJ_P\ CD>eCD>PQTa}R
CDBiCXEN>GRT\ abF;CjJB;@
f
nhTZ m# ;hUD ^ z jh*
nt*#U{UN&yz;S"i &Uz S?mStSN#g^8NUkU
iPS .U"NSUUsNw
.Uy`U t
UN"
c
t
8
l#U
s.UN#N;S;SeeNTSz Ul,U#zlS4UwGU
UeUN",;S"
,NbU&"",;lcNUPs#
UN#N;Sk;S
A!
"
#$%&
#('
#
#
)
)
!
"
#*$+%
,
'
#
,
#
,
'
#
,
&
#
'
# -
'
#
y
'
%
#/.102
Uy
"
#$%
'
%
#/.302
#
54
6y#,N# "U#UiYU#i*S}# S^bU S("U# Si87D #*#N "4
S"9
:#
&
;
:#=@?ABCDFE1>@GHI>JI?*B
;
K?SML6N"s;U*
OQPSR3T5UWVYX[Z\1][P^\1_`1XWa8P__\*]FP^`*b5Xcad\*b/edUfUNg8ZUhaW]iUhjQXkPbadUl\m_\1V[RUWVlUWPSRUWbn1\1_poUlqrP__(RUWbUWVN\*]sU
_\*VFRUWVt]FPuvUtn3\*VkP^\1][P^`1bXt]wT5\*]qrP__T5\1n1Ux]i`QedUtuy\*]sadT5Udj3z+]wT@Vh`1oR3T{]TUY`1VkPSRPb\1_Ud|Wo\1][P^`1bz
qrP]wT}T@PSR3T5UWVQX[Z\1][P^\1_IjU8VkPn1\1][Pn1UkXW~
`1V=]wT5U=TUh\*]Uh|Wo\1][P^`1b/XW`1_Sn3Udjy`1b\1VWeWP]FVh\1Vkvj`1uv\1PbXqUcqrP__UNg8ZUhaW]]TUQU8PSRUWbn1\1_poU8X
]i`ehUVNUd\1_\*bj9bU[R\1][Pn1UjUdad\1d~xb]wT5U}`*]TU8VMT\1bj1z8`1Vv]wT5Uvq\1n1UUd|Wo\1][P^`1b]wT5U
UWPSRUWbn1\1_poU8XyqrP__ehUZo(VNUW_pPuv\WRPb\*VkZVN`dZ5\R\*]FP^`*b*~vPb\1__p3zlqUyqrP__UNg8ZUda8]Y]wT5U
UWPSRUWbn1\1_poU8Xm]i`}edURU8bU8Vh\1_Iad`*u=Z_UNgbo(uyedUWVdXcqTU8bad`1bn1Uda8][Pn1U{\*bj}j*P Yo@XkPn3U{]iU8Vku{X
\1VhUxZVNUkXWUWb]Pb]wT5UUh|Wo\1][P^`1b~
3
rlrv
(c1k**F*cl+d8dF*mY((F1F3NwdFF*(WkhW
118M
"!$##%&'!"(*)+,$ &-/.0)#123254
67)#4&.89/2#":;?)5A@
B C$DFEGDHIJKDLMINOIE5PQIRC5STNU'VGWXY
ZO[\Z ]^`_a2b\c/d efchgjikmlnk\_9i oqpsrut/vw
xGyjz`{}|}yT~yyzn{}qy*Tuz`yqyy~~{}uyy~-u?y|1~ujuy*} yyqy
{qFu{n~~zny{uuGm{{y{qFu{yyu|{u|}y~u?my~*\
} 2yyqy~yy9yOuun~y9{q1{zny~&my|yqus~`z?{qFu
{}}n~
G`Q;qs-\\qF`sF\q1``n-sQ}
`&`}
2
11
q
hnOFn T
nnnFF n`q1s1F
sFs}sF
Mh Q !"#$%&h'
(
F/
)
*
,+.-&/0s
Qh21s\&```}nj43qFF`n!576
Fn
ZO[8 9;:=
NOPRQTSKUVWSYXZUZP\[]X7^_PVa`7bYc4SX7^E[edf^OP`bchgYXibjk[lSc!b'mRc4bYn,Pdo^qprPsUrVtPuP;^OS^G`7SdudHPodv^OVtbwrUxO
SYc]c^]O7Pq`7bY[]X^ld;yz{|'}x{x~|a'|HHK@ iZ]TB
7t2Yt
Z }
z
x$K
Z}
z
x$K
{
=
ZR
{
z
K$x
}x{
KtHrWr
{
ZR {
z
K$
ie$l $=l
iZ]T
H
G;
yz
{
$
|tz
{
|tz
{t
~
K7ru2Hru7r>ZtKtireZ
}yz~
riK@
riK_
_Z
i
_Z
i
}
{
G
H_Z'
riK_
_Z
i
}
{
riK_Z_
H@
HWH@Z@
HoiK@
}x{t
iZ]TB
>W'74rWte
yl
4u'>2HW4Ytr
~
{
$
{
{t
v
W
t
rWu
W
W
HZtKtx
W
W
uqrW
bYVPWZSj`c4PSdHPuubYXipbYVtpPVWSWn,SYVWpBp[ >PHVtPXuPSu`r`iVWbH[]j S^[lbXWSXuPn"Vo[]^^_PX
Sd}
{
y}x{}x{
G
}x{
~W2ykzi~H
iZ]TB
. y
xr2KtYYW4
~
{
$
{
{t
x
K
K
KWHWK
,Y_q]7Yio]iuq,WEu_oa]"]\]tu]7T]7_7Ya]7Yl
eRHWurqtt,;HWuoYYaY\e\uY7os7qYY7t o
Zt
!_ "t$#rY_%roY&r(' , ")t*2t+4uo'-,/.
7Y]7
0(1
24365798 :;3;7
p9; R;
ER
""!iX^$#i%
Z&C"% $'
i()i*
+E+ ,i-#.i0/
X.}$21
35476987:2;
@?
A
BDC
=FE
2k.$%2c2.%O2.2P%O(O7
x5
xm
O
m@
x
P
2
P
x
kO.M229
.
x
x
P
(
99k $c
2
K
99k $c
Om
m
x
x
P
52U.OkOk5N.9k7..9O,9.c2
cO$%
cM7Ox%$%.9z7Ok9O&.O97%
52ckOk.%
O97.
x
k2c9M2 Dz%PD
2
9 $O
D
9
99
D
99
D
.
9
D
.
9
9O...%9%O22F
M2
K
99k $c
Om
m
x
.2r729k5UO.OOx
52O.9@2c2M2
%.O.99
x
x
5N.$m(9U
Y5N.9k7$j%
YkOk.9((97Q..
"!#%$&(')+*,+-.#,/!, 0 1*,+23&45!6&-07
x
kOkt%9%O27OkO
.N%N.29k5kc
.kx25(97.08*2O2
cO&.%O@%t2k$(25..%9%
O2%N%&c*7O92
x
cN25(97%N.29k72%9%
O2)%t:92
M7O
;
Om
;
o44scRuBco/
5
!
!
/s
!
!
/sc
!
!
\ W
__ R! \
1_"
#4uo / osc
Zo4
$%'&( )+* ,
-/.10325476983:
-/.@?BADC
,
-/EF0G2H476983:
-/EJ?KA
L
8MONP8JQJRS:
-T.VUW-/EYX[Z
;\=
U
I+]^RQ
?
$
%
&_( )
*
=J`a`
C
I\`9>a`
Ccb
;
-/.
`
UB-TE
`
?
dDegfih
NP8JQJRS:
=kjmlonI+jmp
$
%
&( )
*mq
;!l
C
2
?
`
C
q
;rp
C
2
?
`
s
:
-/.+UW-/EYX[Z
tvurw xzy|{~}1Jy|{~ay @3+'
'
X
q
la`
C
q
p`
,>
`
s
:
-/.+UB-/EkXZ
pjl
'
X
q
la`
>
`
'_
;
e
`
?
s
:
T
R
! "#%$!&'&(#%)*",+-#.0/!132 45
6879;:=-@BADCFE>HGIE?C=JKHM-CN>HGI:N7O
!"#!$%!'&
(*),+ -.
!0/10 243567
.
!0/8&29/:
;
),+ -5=
?*@@!8
.
.
!035/2A!6$
.
!CBEDGFIH
.
24J@&
K
L1M
NO
O
O
O
O
O
O
P
K
QRSR
K
Q>RAT
K
Q>RAU VWVXV
K
QR4Y
K
Q T,R
K
Q T9T
K
Q T9U Z
Z
Z
Z
Z
Z
K
Q@U,R[Z
Z
Z
Z
Z
Z
Z
Z
Z
V
Z
Z
Z
Z
Z
Z
V V
K
Q\] R4Y
K
Q\ R VWVXV^V
K
Q\ Y ] R
K
Q@\ Y
_W`
`
`
`
`
`
`
acb
K
d1M
NO
O
O
O
O
O
O
P
K
e
RSR
K
e
R4T
K
e
RAU VXVXV
K
e
R4Y
K
e
T,R
K
e
TST
K
e
T9U Z
Z
Z
Z
Z
Z
K
e
U,RfZ
Z
Z
Z
Z
Z
Z
Z
Z
V
Z
Z
Z
Z
Z
Z
V V
K
e
\] RAY
K
e
\ R VWVXVgV
K
e
\ Y ] R
K
e
\ Y
_W`
`
`
`
`
`
`
a
hji'kClmnpoqrhji'ksotuhjvnwx4yuz{4|}lmn~5ink:x4yun@i0wx45i0h5{4uy8h%lvx9nw@0xAislmnh5i0k
k:x4vnlxAi0wvnwz*nlx45n{A|65wx4@nGEhji0kCICvnwz*nlx45n{A|
0
W
XWf
*I
**
XWf*
W
u
0
X
*XXf
**
*
^WX
W
u
n@lq8h5i0kt1'nlmny8hjlvx4@nwclm'h%l@5i6lh5xAiClm0ni5vy8hj{4xAnkGn@x45ni6nl5vwcjoq
hji'kGotpvnwz'nlxAn@{4|,wn@nw{Ax9k:nm#0w@:cn@h5iGvxln
o
q1
qjcq
q
o
t
t>t
t
mnvn
cq
h5i0k
t
hjvnlmnk:x9hj5i0h5{*y8hjlvx4@nwjn@x45ni#>h5{A0nw
|xAi'wnvlxAilm0nhj*%5n}n@:zvnwwx45i0wp5v8o
q
h5i0ko
t
x4i6lE5vuy8h%lvxAn60hjlx45i
zvn@y1{lxAz0{A|r#|
q
hji'kGz*wly1{Alx4z{A|G#|
t
:cn5:lhjx4i
cq
qE
t
q
t%ct
q
t6
n0ix4i
q
t
hji'k
1
q
t
nz0vz'6wnlmn{A{4%c5x4ih5{A5vxlmy
1
q
7t
>>
>@>
1>>
5v
@@
7
@
q
p
t
n 50wn@v5nlm0h%llmny6wln:z'ni0wxAnwln@z0w
hji0k
5i{4|1x4i#5{Anyp{Alx4z{4x4h%lx45i85
y8h%lvx9nwj7wx4@nE5vEm6'w@xAE#lmnp@wljlmnh5'%nh5{A5vxlmyw@hj{4nw
{4xAnpr>
nm0h>nhj{4vnh5k:|1xAi'k:x4h%lnkrh5'%nlm0hjllmnn@v|wz'nx9hj{*wlv0lvn5lmny8h%lvx9nw
q
h5i0ks
t
h5{A{4%5vh5iGn5n@iCyu5vnnrx4n@i6lxAyuz{4n@yunilh%lxAiG5lm0nyuhjlvxGzv:k#
0lw0wx4i0hwl105vx4n@vvh5i0wvy8w8,5vp'h5wl#x4in}vh5i0wvy8wx4ilmx9wph5wnl
5lhjx4ih1i0nhjv{4|r5z:lxAy8h5{w{A:lxAiC@wl8,{45
@
!"#
$&%(' )*,+(-.-/*,01)324*6587(9;: = ?A@/BDC;EGFIH
JLK"MNK&O,K8PRQTSIUWVYXZKM\[^]IK8KN_Z[`KS4URVYQaSQcb[^]IKed,S4VD[`KXTV f8KMNK"SAPRKhgiK"[^]IQjXk[lQSAQTSm`MNKRP[`nTS.oaprqsnTM
XtQaginaVDSIU"u
vGwxayzw8{s|/}Nw"yzw~(}Nwj{|~(Z.{s|I
h
{|
t|;
A
t|;
a;
\
wyNw
.
xa|4
xayzwkt{stw|e
KnTURURprgK[D]4nT[k[^]IKRQTprSAXZnTMWQcb8\>PRnTSRKXaVDVYXZKRXGVDS4[`Q
nTSX
UWp4Pz]
[^]InT[
u&S3
VDMWVYPR].qK"[
I
RQTprSAXZnTMWPRQaSAXTVD[VYQTS4UnTMNKnzZAqVYKRX
O]IK"MNKRnUiLK"prginaS4SPRQaSAXTVD[VYQTS4U
j
1A
naMNKnzZAqVYKRXQaSGuV_.KRXLQaMhQZVDS4
[zIKRQTprSXtnaMWPRQTSXTVD[VYQTSIUVDS4TQTqjVDS/o[D]4KbRprSAP"[VYQaSnaSAX[^]IK8SAQaMWginaqXZKMWVDTnT[VDTKPRQTprqX
nTqUQRKKRnUWVDqVDSAPRQTM4QTMNna[lKRXZu
eYAY IY
?A@/BDC;EGFIF
KMNKR"prVDMNK3nSAQaS4mlUWVDS/otprqnaMgnzZAVDS.ozK"[O,KzK"SnMNKRP[`nTS.oaprqsnTMMNKloaVYQaS18nTSX[^]IK
XtQaginaVDSQ(bkVDS4[`KMzKWUW[u
^rN
Yr(
h
xa|
wrw"}NwyNi{s|Iwxa|w/I{sTxasw"|/}\IyNZIsw"}ziw~NtstwjZ|
WSQTMNXZKM3[lQU"QTqTKQTprMiMNQt"qK"gQaS18O,KSAKRKRXQaprMXTV f8KMNK"S4[VYnTqKRp4na[VYQaS[`QRK
Kz_WMNKUzU"KRXVDSGURp4PR]nO,nT[D]4nT[hnaqDq,XtK"MWVDna[VDTKWUnTMzK[lnjtK"SO>VD[D]MNKUIKRP"[\[lQ[^]IKiVDSXtK"m
4KSXtK"S4[,TnTMWVYnZqsKWUh3naSAXu
j
l4TD.TiD4\RThAzZDc8eZjWD/k^.iRtAD.TzTWT rY
WDAT/4RW"INz8rz(WY\Z"ANaYalRR.446IR" RTD
t"TT`z`iDrz4"R6iNt"laD
!" $# %'&(*),+.-/&021435146879(;:)9+ ?"@BADCFEHG4I
JiRDhAsRIaDKLrsa MhzYaYaKN,kIa^ITO.
PRQTS,UWV*XZY5P[QTS9QT\;U^];XUWV8QT\;U^]*XWX
_ `La b
cedgfh ij
k
Pml n \leP
fZo
]lP
i
Pmpqn \pP
f o
]eprP
i
sutvxw;tvyyz{|~}{y^y
\
l*
]
l*
\p
{w
]p
?"@BADCFEHG*
\n5\*QSFUWV*X Sn5S9QT\;UW]*X
]n5]8QTS,UWV*X Vn5V8QT\;U^]*X
\
]
n
\l \
p
]
l
]p
S
Vx
S
V
n
S
f
S
i
V
f
V
i
\
]
k
\
l
\p
]el \
p2
n
S
f
S
i
V
f
V
i
8
n
V
i
S
i
V
f
S
f
nS
f
V
i
S
i
V
f
?"@BADCFEHG*
P
l
n
QTV
i
P
f
V
f
P
i
X
Pmpn
Q
S
i
P
f
o
S
f
P
i
X
{>w
P
l.l
n
S
QTP
l
Xn ;\
l
\
o
]
l
]9
P
l
n
;V
i
\
V
f
]R
P
l
n
Pmppqn ?"@BADCFEHG>
9>{|~|
QTP
ll
o
P"pprXnx
;
ytey
Q=uP
fWfO
P
f^i
o2
P
ii
o
P
i
o
P
f
Xn
"r*>;">e^**
5>u2;
BW
L
L
2
>
B ^
L
5"
24
mT*'DKD;$[=D;LLDr=DLOD"D4De"D4
=>
"
'^DTL"8
"~
! "$#&%'"$()+*-,-./.~0."^21
3,41
)65 ~;879
:
;
#
(
#
%
(
) :
;
2
3%WrOA@"^*CB
ED[FDDmGBT*T*HD;mLL
I
J
LK&* MN
M
"
N
#
"
#
N
(
"
(
:
PO
T
"
#
"
(
)
N
Q
R"
#
20"$S)
N
T
U
^V "
#
%'"
(
)Z :
;
-%W>W)L
X4YCZ[Y\XGY\]_^Y
=
`acbed
a
gfh'e"i];LkjRDmGh BWD^lL
=Lm
honcp-qFqrr
`Rs
b
h
s
ZD"8XD>DVf
m
>=^.utL=
a
vu^CY
*0wChx^u";mhm^eyl
s
A@e'Dih'e"znDL|{q
=
"LDBB
p-qFq3}B
:
R$cz4-R-C4\+HH?|eg-&R683W26R
0&6kHRc-3FF
W
SMA-HPC 2002 NUS
16.920J/SMA 5212 Numerical Methods for PDEs
Thanks to Franklin Tan
Finite Differences: Parabolic Problems
B. C. Khoo
Lecture 5
SMA-HPC 2002 NUS
Outline
Governing Equation Stability Analysis 3 Examples Relationship between and h Implicit Time-Marching Scheme Summary
2
SMA-HPC 2002 NUS
[ 2 2 0,u x t =
GoverningEquation
3
Consider the Parabolic PDE in 1-D
0subject to at 0, atu u x u u x = = =
If viscosity Diffusion Equation If thermal conductivity Heat Conduction Equation
0x = x =
0u
( ), u x t =
] u x
= u
?
SMA-HPC 2002 NUS
Stability Analysis Discretization
4
Keeping time continuous, we carry out a spatial discretization of the RHS of
There is a total of 1 grid points such that , 0,1, 2,....,
jN j x j
+ = =
0x = x = 0x 1x 3x 1Nx Nx
2
2
u t
=
x N
u x
SMA-HPC 2002 NUS
Stability Analysis Discretization
5
2 1 2
2
2 ( j j
j
u uu O x x
+ =
2
2Use the Central Difference Scheme for u
x
which is second-order accurate.
Schemes of other orders of accuracy may be constructed.
1 2 )j u
x + +
SMA-HPC 2002 NUS 6
Stability Analysis Discretization
We obtain at
0
0
Note that we need not evaluate at and since and are given as boundary conditions.
N
N
u x x x u
= =
1 1 22: 2 )o
du x u udt x
= + 2
2 2 32: 2 )du x u udt x
= +
1 2: 2 )j
j j j
du x u u
dt x
+ = +
1 1 12: 2 )
N N N N
du x u udt x
= +
x u
1 ( u
1 ( u
1( j u
2 ( N u
SMA-HPC 2002 NUS 7
Stability Analysis Matrix Formulation
Assembling the system of equations, we obtain 1
1 2
2
2
2
1 1 2
2 1
01 1
0
1 1
1 0
1
o
jj
NN N
du u udt x du
udt
udu x dt
udu u xdt
=
0
0
A
2
2
2
+
SMA-HPC 2002 NUS 8
Stability Analysis PDE to Coupled ODEs
Or in compact form
du Au bdt
= G G G
We have reduced the 1-D PDE to a set of Coupled ODEs!
[ ]1 1where T Nu u u = G
2 0 0 T
o u b x
= G
+ 2 u
20 Nu
x
SMA-HPC 2002 NUS 9
Stability Analysis Eigenvalue and
Eigenvector of Matrix A
If A is a nonsingular matrix, as in this case, it is then possible to find a set of eigenvalues { 1 1, ,...., ,....,j = For each eigenvalue , we can evaluate the eigenvector consisting of a set of mesh point values , i.e.
j j
j i
V v
1 1 Tj j j
NV v v =
( from det 0.A }2 N
2 j v
)I =
SMA-HPC 2002 NUS 10
Stability Analysis Eigenvalue and
Eigenvector of Matrix A
The ( 1) ( 1) matrix formed by the ( 1) columns diagonalizes the matrix byj
N E N V
1E AE = 1
2
1
where
N
= 0
0
N A
SMA-HPC 2002 NUS 11
Stability Analysis Coupled ODEs toUncoupled ODEs
Starting from du Au bdt
= + G G G
1Premultiplication by yieldsE
1 1duE E Au E bdt
= G G G
( 1 1 1duE E A EE u E bdt = G G G
I
( 1 1 1duE E AE E u E bdt = G G G
1 +
)1 + )1 +
SMA-HPC 2002 NUS 12
Stability Analysis Coupled ODEs to Uncoupled ODEs
1 Let and , we haveU E u F E b = GG G d U Fdt
= + JG JGG
which is a set of Uncoupled ODEs!
1 1duE u E bdt
= + G G G
Continuing from
1= G
U
1 E
SMA-HPC 2002 NUS 13
Stability Analysis Coupled ODEs to Uncoupled ODEs
Expanding yields
Since the equations are independent of one another, they can be solved separately.
The idea then is to solve for and determineU EU= G GG
1 1 1 1
dU U dt
= + 2
2 2 dU U dt
= +
j j j
dU U
dt = +
1 1 1
N N N
dU U dt
=
u
F
2 F
j F
1 N F +
SMA-HPC 2002 NUS 14
Considering the case of independent of time, for the general equation,th
b j
G Stability Analysis
Coupled ODEs to Uncoupled ODEs
1jt j j
j
U e F = is the solution for j = 1,2,.,N1.
Evaluating, ( ) 1 tu EU E ce E E b = JJJJGG G Complementary
(transient) solution Particular (steady-state)
solution
( 11 1 1where j N Tt tt t j ce c e c e c e c e = JJJJG
j c
1= G
) 2 2 t N
SMA-HPC 2002 NUS
We can think of the solution to the semi-discretized problem
15
Stability Analysis Stability Criterion
( ) 1 tu E ce E E b = JJJJGG G
This is the criterion for stability of the space discretization (of a parabolic PDE) keeping time continuous.
Since the transient solution must decay with time,
for all j( )Real 0j
Each mode contributes a (transient) time behaviour of the form to the time-dependent part of the solution.j t
j e
as a superposition of eigenmodes of the matrix operator A.
1
SMA-HPC 2002 NUS 16
Stability Analysis Use of Modal (Scalar)
Equation
It may be noted that since the solution is expressed as a contribution from all the modes of the initial solution, which have propagated or (and) diffused with the eigenvalue
, and a contribution frj
u
G
om the source term , all the properties of the time integration (and their stability properties) can be analysed separately for each mode with the scalar equation
jb
j
dU U Fdt
= +
SMA-HPC 2002 NUS 17
Stability Analysis Use of Modal (Scalar)
Equation
The spatial operator A is replaced by an eigenvalue , and the above modal equation will serve as the basic equation for analysis of the stability of a time-integration scheme (yet to be introduced) as a function of the eigenvalues of the space-discretization operators.
This analysis provides a general technique for the determination of time integration methods which lead to stable algorithms for a given space discretization.
SMA-HPC 2002 NUS 18
1 11 1 12 2
2 21 1 22 2
du a u a udt du a u a udt
=
=
1 1 12
2 1 22
Let , u a
u u a =
G du Audt
= G G
Consider a set of coupled ODEs (2 equations only):
Example 1 Continuous Time
Operator
+
+
1
2
a A
a =
SMA-HPC 2002 NUS 19
Proceeding as before, or otherwise (solving the ODEs directly), we can obtain the solution
1
1
1 11 2 12
2 21 2 22
t
t
u e c e u e c e
= =
11 21 1
21 22
1
where and are eigenvalues of and and are
eigenvectors pertaining to and respectively.
A
( )j As the transient solution must decay with time, it is imperative that Real 0 for 1, 2.j
Example 1 Continuous Time
Operator
2
2
1
1
t
t
c c
+ +
2
2
=
SMA-HPC 2002 NUS 20
Suppose we have somehow discretized the time operator on the LHS to obtain
1 1 1 1 12 2
1 2 1 1 22 2
n n
n n
u u a u
u u a u
= =
where the superscript n stands for the nth time level, then
1 11 12 1
21 22
where and Tn n n a u u u u u A
a = =
G G
Since A is independent of time, 1 0
.... n n nu u AAu A u
= = = G G G
Example 1 Discrete Time Operator
1 1
1 2
n
n
a
a
+ +
2
n n a Aa
= G
2 n A
= G
SMA-HPC 2002 NUS 21
Example 1 Discrete Time Operator
1 0 1
2
0where =
0
n n n
n u E u
= JJGG
' 1 1 11 1 2 12 2
' 2 1 21 1 2 22 2
n n
n n
u c
u c
= =
1 1 0
2
' where are constants.
' c
E u c
= JJG
1
1 1 0
,
.... n
A E E
u E E E E E u
= =
JJGG As
A A A
n E
'
'
n
n
c
c
+ +
1 E
SMA-HPC 2002 NUS 22
Example 1 Comparison
Comparing the solution of the semi-discretized problem where time is kept continuous
[ 1 2
1 1 12 1
2 1 22
t
t
u e c
u e
= to the solution where time is discretized
[ 1 1 12 1 1 2 1 22 2
' n n
n
u c
u
= coTh nte di inuofference equation where time is hus exponential
solution as
. te
The difference equation where time is hasdiscretized power solution . n
] 12 2
c
] 12 2
' c
SMA-HPC 2002 NUS 23
Example 1 Comparison
In equivalence, the transient solution of the difference equation must decay with time, i.e.
for this particular form of time discretization.
1n <
SMA-HPC 2002 NUS 24
Consider a typical modal equation of the form
t
j
du u aedt
= where is the eigenvalue of the associated matrix .j A (For simplicity, we shall henceforth drop the subscript j). We shall apply the leapfrog time discretization scheme given as
Substituting into the modal equation yields
( 1 2 n
t t nh
u u aeh
+
= =
n nu e=
1
where 2
n du u u h dt h
+ =
Example 2 Leapfrog Time Discretization
+
)1 n u + ha+
1 n
t
=
SMA-HPC 2002 NUS
Solution of u consists of the complementary solution cn, and the particular solution pn, i.e.
un = cn + pn
There are several ways of solving for the complementary and particular solutions. shift operator S and characteristic polynomial.
The time shift operator S operates on cn such that
Scn = cn+1 S2cn = S(Scn) = Scn+1 = cn+2
25
( 1 1 2 2 2 n
n n n n n hnu u e u h u u ha eh
+
+ = =
Example 2 Leapfrog Time Discretization Time Shift Operator
One way is through use of the
)1 1n hu a +
SMA-HPC 2002 NUS 26
The complementary solution cn satisfies the homogenous equation
1
2
2
2
2
1( ) 0
( 1) 0
n n
n n
n n
n
c c c cSc h c S
S c h Sc c S
cS S S
+ = =
=
=
2( ) ( 2 1) 0p S S h S= = characteristic polynomial
Example 2 Leapfrog Time Discretization Time Shift Operator
1 0
0
2
2
n
n
n
h
h
SMA-HPC 2002 NUS 27
The complementary solution to the modal equation would then be
1 1 2 2 n nc =
The particular solution to the modal equation is 2 2
2
hn h n
h
ahe e p e e
=
Combining the two components of the solution together,
( ) ( )n nu p= ( ) ( )2 2 2 2 1 2 21 2 hn hn h ahe eh h h e e = + + + +
The solution to the characteristic polynomial is 2 2( ) 1h h h = = + 1 and 2 are the two roots
Example 2 Leapfrog Time Discretization Time Shift Operator
n+
1 h h
n c +
2 1 1
n
h h h
+
S
SMA-HPC 2002 NUS 28
For the solution to be stable, the transient (complementary) solution must not be allowed to grow indefinitely with time, thus implying that
is the stability criterion for the leapfrog time discretization scheme used above.
( ) ( )
2 2 1
2 2 2
1
1
h
h
= + <
= <
Example 2 Leapfrog Time Discretization Stability Criterion
1
1
h
h
+
+
SMA-HPC 2002 NUS 29
Im( )
Re( )-1 1
Region of Stability
The stability diagram for the leapfrog (or any general) time discretization scheme in the -plane is
Example 2 Leapfrog Time Discretization Stability Diagram
SMA-HPC 2002 NUS 30
In particular, by applying to the 1-D Parabolic PDE 2
2
u t
= the central difference scheme for spatial discretization, we obtain
which is the tridiagonal matrix.
Example 2 Leapfrog Time Discretization
2
2 1 1 1
1 1
A x
=
0
0
u x
2
2
SMA-HPC 2002 NUS 31
Example 2 Leapfrog Time Discretization
According to analysis of a general triadiagonal matrix B(a,b,c), the eigenvalues of the B are
2
2 cos , 1,..., 1
2 2 cos
j
j
jb c j NN
j N
= + = +
The most dangerous mode is that associated with the eigenvalue of largest magnitude
max 2
4 x =
( (
2 max1 ax max
2 max2 ax max
1
1
h h
h h
= + = +
i.e.
which can be plotted in the absolute stability diagram.
a
x
=
) )
2 m
2 m
h
h
+
SMA-HPC 2002 NUS 32
Example 2 Leapfrog Time Discretization Absolute Stability Diagram for
As applied to the 1-D Parabolic PDE, the absolute stability diagram for is
Region of stability
Unit circle
Region of instability
2 at h = t = 0
2 with h increasing
1with h increasing 1 at h = t = 0
Re()
Im()
SMA-HPC 2002 NUS 33
Stability Analysis Some Important
Characteristics Deduced
A few features worth considering:
1. Stability analysis of time discretization scheme can be carried out for all the different modes .
2. If the stability criterion for the time discretization scheme is
j valid for
all modes, then the overall solution is stable (since it is a linear combination of all the modes).
3. When there is more than one root , then one of them is the principal root which represents
( 0
an approximation to the physical behaviour. The principal root is recognized by the fact that it tends towards one as 0, i.e. lim 1. (The other roots are spurious, which affect the stability
h h
but not the accuracy of the scheme.) )h =
SMA-HPC 2002 NUS 34
Stability Analysis Some Important
Characteristics Deduced
1
4. By comparing the power series solution of the principal root to , one can determine the order of accuracy of the time discretization scheme. In this example of leapfrog time discretization,
1
he
h
= ( ( 1 2 2 2 2 4 42 2 2
1
2 2
1 .1 2 1 2 !
1 ...2
and compared to
1 ..2!
is identical up to the second order of . Hence, the above scheme is said to be second-order accurate.
h
h h h
hh
h e
h
+ + + +
= + + +
= + + +
+ ) )1 2 .
2
.
h
h
=
SMA-HPC 2002 NUS 35
Analyze the stability of the explicit Euler-forward time discretization 1n du u u
dt t
+ = as applied to the modal equation
du udt
= 1
1
Substituting where
into the modal equation, we obtain (1 ) 0
n
n
du u h h tdt
u u +
+
= = +
Euler-Forward Time Discretization Stability Analysis
Example 3
n
n
n
u
h
+ =
SMA-HPC 2002 NUS 36
Making use of the shift operator S 1 (1 ) (1 ) [ (1 )] 0n n n nc c Sc h c S h c + + = + = + =
Therefore ( ) 1 and n
h c
= +
=
characteristic polynomial
The Euler-forward time discretization scheme is stable if
Euler-Forward Time Discretization Stability Analysis
Example 3
1 h + or bounded by 1 s.t. 1 in the -plane.h = <
n h
n
h
1 < h
SMA-HPC 2002 NUS 37
Euler-Forward Time Discretization Stability Diagram
Example 3
Im(h)
0-1-2
Unit Circle
Region of Stability
Re(h)
The stability diagram for the Euler-forward time discretization in the h-plane is
SMA-HPC 2002 NUS 38
Euler-Forward Time Discretization Absolute Stability Diagram
Example 3
max 2
4As applied to the 1-D Parabolic PDE, x =
max
2The stability limit for largest h =
1-1
leaves the unit circle at h = 2
at h (=t) = 0
Re()
Im()
with h increasing
=
t
SMA-HPC 2002 NUS 39
Relationshipbetween and h
= (h)
Thus far, we have obtained the stability criterion of the time discretization scheme using a typical modal equation. generalize the relationship between and h as follows: Starting from the set of coupled ODEs
du Au bdt
= + G G G
Apply a specific time discretization scheme like the leapfrog time discretization as in Example 2
1
2
n du u u dt h
+ =
We can
1 n
SMA-HPC 2002 NUS 40
Relationshipbetween and h
= (h)
The above set of ODEs becomes 1
2
n nnu Au h
+ = + G GG
Introducing the time shift operator S
1
2
2
n nn
nn
uSu hAu hbS
S SA I u bh
= +
G GG
GG
1
1 Premultiplying on the LHS and RHS and introducing
operating on n E
I EE u
=
i G 1
1 1 1
2 nS SE AE E E E u E b
h
GG
1 n u bG
2 n + =
G
1 =
SMA-HPC 2002 NUS 41
Relationshipbetween and h
= (h)
Putting 1 , nn nU u F E b = GG G
1
2j j S S U
h
=
we obtain 1
1
2 n S SE U F
h
G G
1
2 S S
h
i.e. 1
2 n S S U
h
= G G
which is a set of uncoupled equations.
Hence, for each j, j = 1,2,.,N-1,
1 n E = G
j F
nE =
nF
SMA-HPC 2002 NUS 42
Relationshipbetween and h
= (h)
Note that the analysis performed above is identical to the analysis carried out using the modal equation
j
dU U Fdt
= All the analysis carried out earlier for a single modal equation is applicable to the matrix after the appropriate manipulation to obtain an uncoupled set of ODEs.
Each equation can be solved independently for and the 's can then be coupled through .
th
n n n j
j U u EU= GG
+
n j U
SMA-HPC 2002 NUS 43
Relationshipbetween and h
= (h)
1. Uncoupling the set, 2. Integrating each equation in the uncoupled set,
3. Re-coupling the results to form the final solution.
These 3 steps are commonly referred to as the
ISOLATION THEOREM
Hence, applying any consistent numerical technique to each equation in the set of coupled linear ODEs is mathematically equivalent to
SMA-HPC 2002 NUS 44
Implicit Time-Marching Scheme
Thus far, we have presented examples of explicit time-marching methods and these may be used to integrate weakly stiff equations.
Implicit methods are usually employed to integrate very stiff ODEs efficiently. solution of a set of simultaneous algebraic equations at each time-step (i.e. matrix inversion), whilst updating the variables at the same time.
Implicit schemes applied to ODEs that are inherently stable will be unconditionally stable or A-stable.
However, use of implicit schemes requires
SMA-HPC 2002 NUS 45
Implicit Time-Marching Scheme
Euler-Backward
Consider the Euler-backward scheme for time discretization 1 1n n du u u
dt h
+ + =
tdu u aedt
=
(
( ( 1
11
111
n n n
n n
u u eh h u u ahe
+ ++
++
= =
Applying the above to the modal equation for Parabolic PDE
yields
n
+
)
) ) n
h
hn
u a+
SMA-HPC 2002 NUS 46
Implicit Time-Marching Scheme
Euler-Backward
Applying the S operator, ( ) ( 11 n nh S u ahe + =
the characteristic polynomial becomes
( ) ( ) ( )1 0S S = = The principal root is therefore
2 21 which, upon comparison with 1 .... , is only2
first-order accurate.
he h = + + +
2 21 1 .... 1
h h
= = + + +
The solution is (
( 11
1 1
n u n
h
aheU h e
+ =
)1 h
1 h =
h h
)
) 1 h
h +
SMA-HPC 2002 NUS 47
Implicit Time-Marching Scheme
Euler-Backward
For the Parabolic PDE, is always real and < 0. Therefore, the transient component will always tend towards zero for large n irregardless of h ( t). The time-marching scheme is always numerically stable.
In this way, the implicit Euler/Euler-backward time discretization scheme will allow us to resolve different time-scaled events with the use of different time-step sizes. scaled events, and then a large time-step size used for the longer time-scaled events. hmax.
A small time-step size is used for the short time-
There is no constraint on
SMA-HPC 2002 NUS 48
Implicit Time-Marching Scheme
Euler-Backward
However, numerical solution of u requires the solution of a set of simultaneous algebraic equations or matrix inversion, which is computationally much more intensive/expensive compared to the multiplication/ addition operations of explicit schemes.
SMA-HPC 2002 NUS
Summary
49
Stability Analysis of Parabolic PDE Uncoupling the set. Integrating each equation in the uncoupled set
modal equation.
Re-coupling the results to form final solution. Use of modal equation to analyze the stability |(h)| < 1.
Explicit time discretization versus Implicit time discretization.
16.920J/SMA 5212
Numerical Methods for Partial Differential Equations
Lecture 5
Finite Differences: Parabolic Problems
B. C. Khoo
Thanks to Franklin Tan
19 February 2003
16.920J/SMA 5212 Numerical Methods for PDEs
2
OUTLINE
Governing Equation Stability Analysis 3 Examples Relationship between and h Implicit Time-Marching Scheme Summary
Slide 2
GOVERNING EQUATION
Consider the Parabolic PDE in 1-D
If viscosity Diffusion Equation If thermal conductivity Heat Conduction Equation
Slide 3
STABILITY ANALYSIS Discretization
Keeping time continuous, we carry out a spatial discretization of the RHS of
[ ]2
2 0,u u
xt x
pi
=
0subject to at 0, at u u x u u xpi pi= = = =
0x = x pi=
0u upi
( ), ?u x t =
2
2u u
t x
=
0x = x pi=
0x 1x 2x 1Nx Nx
16.920J/SMA 5212 Numerical Methods for PDEs
3
Slide 4
STABILITY ANALYSIS Discretization
which is second-order accurate.
Schemes of other orders of accuracy may be constructed.
Slide 5
Construction of Spatial Difference Scheme of Any Order p
The idea of constructing a spatial difference operator is to represent the spatial differential operator at a location by the neighboring nodal points, each with its own weightage.
The order of accuracy, p of a spatial difference scheme is represented as ( )pO x . Generally, to represent the spatial operator to a higher order of accuracy, more nodal points must be used.
Consider the following procedure of determining the spatial operator j
dudx
up to the
order of accuracy ( )2O x :
There is a total of 1 grid points such that ,0,1,2,....,
jN x j xj N
+ = =
2
2Use the Central Difference Scheme for u
x
21 1 2
2 2
2 ( )j j jj
u u uu O xx x
+ +
= +
j2
j1
j
j+1
j+2
j
dudx
16.920J/SMA 5212 Numerical Methods for PDEs
4
1. Let j
dudx
be represented by u at the nodes j1, j, and j+1 with 1 , 0 and
1 being the coefficients to be determined, i.e.
( )1 1 0 1 1 pj j jj
duu u u O x
dx
+
+ + + =
2. Seek Taylor Expansions for 1ju , ju and 1ju + about ju and present them in a table as shown below.
(Note that p is not known a priori but is determined at the end of the analysis when the s are made known.)
uj uj uj uj
ju
0 1 0 0
1 1ju 1 1x 2
112
x
3 116
x
0 ju 0 0 0 0
1 1ju + 1 1x 2
112
x 3 116
x
1
1
k
j k j kk
u u=
+=
+ 1S 2S 3S 4S
( 1 )
This column consists of all the terms on the LHS of (1).
Each cell in this row comprises the sum of its corresponding column.
16.920J/SMA 5212 Numerical Methods for PDEs
5
where
1
1 2 3 41
....
k
j k j kk
u u S S S S=
+=
+ = + + + +
3. Make as many iS s as possible vanish by choosing appropriate k s.
In this instance, since we have three unknowns 1 , 0 and 1 , we can therefore set:
1
2
3
000
SSS
=
=
=
(Note that in the Taylor Series expansion, one starts off with the lower-order terms and progressively obtain the higher-order terms. We have deliberately set the iS pertaining to the lower-order terms to zero, thereafter followed by increasingly higher-order terms.)
Hence,
1
0
1
01 1 111 0 1
1 0 1 0x
=
Solving the system of equations, we obtain
1
0
1
12
01
2
x
x
=
=
=
( )( )
1 1 0 1
2 1 1
2 23 1 1
3 34 1 1
1
1 12 2
1 16 6
j
j
j
j
S u
S x x u
S x x u
S x x u
= + +
= +
= +
= +
16.920J/SMA 5212 Numerical Methods for PDEs
6
4. Substituting the k s into 1
1 2 3 41
....
k
j k j kk
u u S S S S=
+=
+ = + + + +
yields
( ) 21 11 12 6j j j ju u u x ux + = + higher-order terms
In other words,
( )1 1 2 ....2j jj ju udu
u O xdx x
+
= = + +
i.e. the above representation is accurate up to ( )2O x .
Some useful points to note:
1. These 4 steps are the general procedure used to obtain the representation of the spatial operator up to the order of accuracy ( )pO x .
2. For other spatial operators, say 2
2j
d udx
, we simply replace j
dudx
in (1) with
the said spatial operator.
3. For one-sided representations, one can choose nodal points , 0j ku k+ . This may be important especially for representations on a boundary. For example
( )0 1 1 2 2 .... pj j jj
duu u u O x
dx + +
+ + + + =
One possibility is
( )1 2 23 42j j jju u udu O x
dx x+ + +
+ =
which is also second-order accurate.
(We can also use a similar procedure to construct the finite difference scheme of Hermitian type for a spatial operator. This is not covered here).
( ) , 2pO x p =
16.920J/SMA 5212 Numerical Methods for PDEs
7
STABILITY ANALYSIS Discretization
We obtain at 11 1 22: ( 2 )odu
x u u udt x
= +
22 1 2 32: ( 2 )
dux u u u
dt x
= +
1 12: ( 2 )jj j j jdu
x u u udt x
+= +
11 2 12: ( 2 )NN N N N
dux u u u
dt x
= +
0
0
Note that we need not evaluate at and since and are given as boundary conditions.
N
N
u x x x x
u u
= =
Slide 6
STABILITY ANALYSIS Matrix Formulation
Assembling the system of equations, we obtain
Slide 7
11 2
2
2
2
1 1 2
2 1
01 2 1
0
1 2 1
1 0
1 2
o
jj
NN N
du uudt x
duudt
udu xdt
udu uxdt
= +
0
0
A
16.920J/SMA 5212 Numerical Methods for PDEs
8
STABILITY ANALYSIS PDE to Coupled ODEs
Or in compact form
We have reduced the 1-D PDE to a set of Coupled ODEs!
Slide 8
STABILITY ANALYSIS Eigenvalue and Eigenvector of Matrix A
If A is a nonsingular matrix, as in this case, it is then possible to find a set of eigenvalues
{ }1 2 1, ,...., ,....,j N =
( )from det 0.A I =
For each eigenvalue , we can evaluate the eigenvector consisting of a set of mesh point values , i.e.
jj
ji
Vv
Slide 9
STABILITY ANALYSIS Eigenvalue and Eigenvector of Matrix A
The ( 1) ( 1) matrix formed by the ( 1) columns diagonalizes the matrix byj
N N E NV A
1E AE =
[ ]1 2 1where TNu u u u =
2 20 0 0T
o Nu ubx x
=
du Au bdt
= +
1 2 1 Tj j j j
NV v v v
=
16.920J/SMA 5212 Numerical Methods for PDEs
9
Slide 10
STABILITY ANALYSIS Coupled ODEs to Uncoupled ODEs
Starting from du Au bdt
= +
1Premultiplication by yieldsE
1 1 1duE E Au E bdt
= +
Slide 11
STABILITY ANALYSIS Coupled ODEs to Uncoupled ODEs
Continuing from
1 1 1duE E u E bdt
= +
1 1Let and , we haveU E u F E b = =
1
2
1
where
N
=
0 0
( )1 1 1 1duE E A EE u E bdt = +
I
( )1 1 1 1duE E AE E u E bdt = +
16.920J/SMA 5212 Numerical Methods for PDEs
10
d U U Fdt
= +
which is a set of Uncoupled ODEs!
Slide 12
STABILITY ANALYSIS Coupled ODEs to Uncoupled ODEs
Expanding yields
11 1 1
dU U Fdt
= +
22 2 2
dU U Fdt
= +
jj j j
dUU F
dt= +
11 1 1
NN N N
dU U Fdt
= +
Since the equations are independent of one another, they can be solved separately.
The idea then is to solve for and determine U u EU=
Slide 13
STABILITY ANALYSIS Coupled ODEs to Uncoupled ODEs
Considering the case of independent of time, for thegeneral equation,th
bj
1jtj j j
jU c e F =
is the solution for j = 1,2,.,N1.
16.920J/SMA 5212 Numerical Methods for PDEs
11
Evaluating, ( ) 1 1tu EU E ce E E b = =
( ) 11 21 2 1where j N Tt tt tt j Nce c e c e c e c e =
The stability analysis of the space discretization, keeping time continuous, is based on the eigenvalue structure of A. The exact solution of the system of equations is determined by the eigenvalues and eigenvectors of A.
Slide 14
STABILITY ANALYSIS Coupled ODEs to Uncoupled ODEs
We can think of the solution to the semi-discretized problem
as a superposition of eigenmodes of the matrix operator A.
Each mode contributes a (transient) time behaviour of the form to the time-dependent part of the solution.jt
je
Since the transient solution must decay with time,
( )Real 0j for all j
This is the criterion for stability of the space discretization (of a parabolic PDE) keeping time continuous.
Slide 15
Complementary (transient) solution
Particular (steady-state) solution
( ) 1 1tu E ce E E b =
16.920J/SMA 5212 Numerical Methods for PDEs
12
STABILITY ANALYSIS Use of Modal (Scalar) Equation
It may be noted that since the solution is expressed as acontribution from all the modes of the initial solution,which have propagated or (and) diffused with the eigenvalue
, and a contribution frj
u
om the source term , all theproperties of the time integration (and their stabilityproperties) can be analysed separately for each mode withthe scalar equation
jb
Slide 16
STABILITY ANALYSIS Use of Modal (Scalar) Equation
The spatial operator A is replaced by an eigenvalue , and the above modal equation will serve as the basic equation for analysis of the stability of a time-integration scheme (yet to be introduced) as a function of the eigenvalues of the space-discretization operators.
This analysis provides a general technique for the determination of time integration methods which lead to stable algorithms for a given space discretization.
Slide 17
EXAMPLE 1 Continuous Time Operator
Consider a set of coupled ODEs (2 equations only):
111 1 12 2
221 1 22 2
dua u a u
dtdu
a u a udt
= +
= +
1 11 12
2 21 22
Let , u a a du
u A Auu a a dt
= = =
Slide 18
j
dU U Fdt
= +
16.920J/SMA 5212 Numerical Methods for PDEs
13
EXAMPLE 1 Continuous Time Operator
Proceeding as before, or otherwise (solving the ODEs directly), we can obtain the solution
1 2
1 2
1 1 11 2 12
2 1 21 2 22
t t
t t
u c e c e
u c e c e
= +
= +
11 211 2
21 22
1 2
where and are eigenvalues of and and are
eigenvectors pertaining to and respectively.
A
( )jAs the transient solution must decay with time, it is imperative thatReal 0 for 1, 2.j =
Slide 19
EXAMPLE 1 Discrete Time Operator
Suppose we have somehow discretized the time operator on the LHS to obtain
1 11 11 1 12 2
1 12 21 1 22 2
n n n
n n n
u a u a u
u a u a u
= +
= +
where the subscript n stands for the nth time level, then
1 11 121 2
21 22
where and Tn n n n n a a
u Au u u u Aa a
= = =
Since A is independent of time,
1 2 0....
n n nn
u Au AAu A u
= = = =
In later examples, we shall apply specific time discretization schemes such as the leapfrog and Euler-forward time discretization schemes.
Slide 20
16.920J/SMA 5212 Numerical Methods for PDEs
14
EXAMPLE 1 Discrete Time Operator
As
1 0 1
2
0 where =
0
nn
n n
nu E E u
=
' '
1 1 11 1 2 12 2' '
2 1 21 1 2 22 2
n n n
n n n
u c c
u c c
= +
= +
1 1 0
2
'
where are constants.'
cE u
c
=
Slide 21
Alternative View
Alternatively, one can view the solution as:
01 1
1 2 02 2
n
n n
n
U UU U
=
0 1where n nU U U E u= =
EXAMPLE 1 Comparison
Comparing the solution of the semi-discretized problem where time is kept continuous
[ ] 12
1 11 121 2
2 21 22
t
t
u ec c
u e
=
to the solution where time is discretized
[ ]1 11 12 11 22 21 22 2
' '
n n
n
uc c
u
!
! !
= " #" # " #
$% $ %
" #
$ %
1
1 1 1 0
,
....
n
A E E
u E E E E E E u
=
= &'& ((
A A A
16.920J/SMA 5212 Numerical Methods for PDEs
15
difference equation where time is continuous has exponentialsolution The
.
te
The difference equation where time is discretized has powersolution .n
Slide 22
EXAMPLE 1 Comparison
In equivalence, the transient solution of the difference equation must decay with time, i.e.
1n <
for this particular form of time discretization.
Slide 23
EXAMPLE 2 Leapfrog Time Discretization
Consider a typical modal equation of the form
t
j
duu ae
dt
= +
where is the eigenvalue of the associated matrix .j A
(For simplicity, we shall henceforth drop the subscript j).
We shall apply the leapfrog time discretization scheme given as
1 1
where 2
n ndu u u h tdt h
+
= =
Substituting into the modal equation yields
1 1
2
n nu u
h
+
( )tt nh
u ae=
= +
n hnu ae= +
Slide 24
16.920J/SMA 5212 Numerical Methods for PDEs
16
Reminder
Recall that we are considering a typical modal equation which had been obtained from the original equation
du Au bdt
= +
EXAMPLE 2 Leapfrog Time Discretization: Time Shift Operator
( )1 1 1 1 2 22n n
n hn n n n hnu u u ae u h u u ha eh
+
+ = + =
Solution of u consists of the complementary solution nc , and theparticular solution np , i.e.
n n nu c p= +
There are several ways of solving for the complementary andparticular solutions. One way is through use of the shift operator S and characteristic polynomial.
The time shift operator S operates on nc such that
1n nSc c +=
( )2 1 2n n n nS c S Sc Sc c+ += = =
Slide 25
EXAMPLE 2 Leapfrog Time Discretization: Time Shift Operator
The complementary solution nc satisfies the homogenous equation
1 12 0
2 0
n n n
nn n
c h c ccSc h cS
+ =
=
16.920J/SMA 5212 Numerical Methods for PDEs
17
2
2
1( 2 ) 0
( 2 1) 0
n n n
n
S c h Sc cS
cS h SS
=
=
Slide 26
EXAMPLE 2 Leapfrog Time Discretization: Time Shift Operator
The solution to the characteristic polynomial is
2 2( ) 1h S h h = = +
The complementary solution to the modal equation would then be
1 1 2 2n nn
c = +
The particular solution to the modal equation is 22
2 1
hn hn
h hahe ep
e h e
= .
Combining the two components of the solution together,
nu ( ) ( )n nc p= + ( ) ( )2 2 2 21 2 2 21 1 2 1
hn hn n
h hahe eh h h h
e h e
= + + + + +
Slide 27
EXAMPLE 2 Leapfrog Time Discretization: Stability Criterion
For the solution to be stable, the transient (complementary) solution must not be allowed to grow indefinitely with time, thus implying that
( )( )
2 21
2 22
1 1
1 1
h h
h h
= + + E&~
U
Y\^
QNciE&i
NSN
R
T}U
f
_Ni
R
M
T
U
sQ`
M
R
T
U
R
R
R
Q
R
R
R
a
bdcefhgSikjlb#clmonZpqfsr
itjugwvxzyn{pqf}|k~H%7-}7-J@7llH?>HH}d}dEHzJE?sEdH-H}N##'H
c'ef
g
v
c
ef
m
c
pqf
m
~Jg
77!
?(
}oN ?
known valuesunknown values
%
E7
c'ef
g
f
v
c
ef
m
c
p%!hq}>}
}'>@uHlHz7uJEHd'7-
known valuesunknown values
ElJE>?Sv
%'79h'
h'
'~
c'ef
g
f
v
c
ef
m
pqf
m
~
"!$#&% ' (*),+.-&/1032
465798;:=?a
m
:[WYZPT:[89n3WofgWp:[qR6r
is@CBtDXk6`vuZwyx>?Fz`Xuv{1a
|~}g3T OpT p"}TvgT[l3Ts.6"}6Z 3}T6T* o.
* .6T[3"]vvp[3T.G[ Z3=.;O.}; O6T*T
s.}l.pZ v ;ZTgOO&
>i@CBFk x D
t ==y$3T
Wh[WW:ZqP :\S7;893Wl89SL:ZqW,P3eR3c8fgW:[qRr*:[qWR3YrWY_8;SLVq89eqL:ZqW\S3R VSh=PTYZW
\5~rP :ZWr89S]P3\hZh46W8rW7eqP3SQ3Wh:[qWvYZWh[\7:RT:ZqW8;:[WYZPT:[89n3W5Y[RepWr\YZW3OSWepR\7r
h[VWW5:[qYZR3\Qq]:[qW_\S6SR VSh89SP3hZepWSr8;SQ3rWh[eWSr8;SQ3RYvPT7;:[WY[SPT:[WR3YrWYZhyqW
7P ::ZWY5YZRepWr\YZWo8heP37;79Wrh[,fgfgWp:ZY[8elPT\h[h4W8rW7
SRT:[qWY_W*We:[89n3Wlh:[YP :ZWQ36SR VSP3hYZWr6c7P3eP3\h[h46W8rW78;:[WYZPT:[89R3S89h:ZR\5
rPT:[Wo:[qWWn3WS\S,SR VShXYh:iY[Wrk
uZwyx
Z
>
j
u
Z
wyx
{
u
Z
x
{1
Zz
PTSr:ZqWS:[qWR6rreR3fg5*R3SWS:hi"c7P3ek
uZwyx
Z
wyx
>
uZwyx
Z
wyx
{
u[wyx
[
{1
v
Z
wyx
qWgYZWr6c7P3ePT\hZh46W89rW78:ZWYP :Z8;RS8h5R5\7PTYo"R3Y5P3YZP37;79W7epRfg5\:ZPT:[89R3Sth89SepW
:[qWYZWri"c79Pek5R8;S,:ZhR3S7;Y[W,\8;YZWL:ZqWc7P3ei"YZWrk5*R389S:hP3Sr$:[qWh[WeP3ScW
\5~rP :ZWrL89SPTS6R3YrWYqWfgWp:ZqRrLYZWP3r8;79W,:ZWSrh:[Rf\7;:[89579Wr89fgWSh[8;RSh
m
S
M6"R3Y89Sh:ZP3SepW:[qWYZWrPTSrc7P3e5R8;S,:Zh=PTYZWoh[qR VScW7;R V
Red
Black
(*),+.-&/103
Zy sCtXzvCE
[y O& jtX
sE
"!!#$!% '&)(+*-,./#0 1
235476%8:9
;=
?
fmjIkL|>fl5p}DP
W~
jIn:{n:fMuglovGhgun
~
P
z?
P
P
b
P
WW)tI P
r=y
t
'h]pnOfh]fkOf
S
YhjIhk:ufh]l5fih>jIk:fmlon:ph>jgjIn:jlu]pn:f+kOpkL|huBuglo{>x_hMwYkOpYfml5f
ugl5p|h]l5f
jIh>wl5jIhgu
W
fhjIhuglov:wYk:xluhBk:ugl5x|n:+wfikL{>x_hk:pvMn:pxjIh>wl5jIhgu
g' 'OIi-QBcOO[G$GOI
J7o GYQ 57%L
)I- ") '
""
BS
! "$# %
&('
+*),+
J7 -M.0/1/235460487 57%9
:
;=?@A=>BDC
BEGFH
! "# %
&5IKJ
+L),MONHY
P$QSR?TVUXW>YZ[\Q^]_ZQ`Qa]YVZDbcdYeYfg_Z@QhRc@iT\TXZQjR?T^kHYlmbnoQ^]_YeYX[$[c[Z[AYpiqc@Q`g8TXYrgWrc[sq[AZUXt
QjRUAZ@WDsqg[js_cuTXY$TvTR^iwUY
xtyFM`xz0{
R^W^W(iwcQbY}|iwc
{
i~bYrc@[YpQa]Ysw[cbXW>YXR?T}TXcWduYl
Y
{
R^W^W(TXYYv]c
{
YdY[
z
Q^]_ZQbXnoTQjg_lnR^i8 -M.0/1/235460487 57%
Fwc K$u*_
x
*
0.3
0.2
0.1
0
# Iterations
log(
||e|| L
2)
n=10n=20n=40
0 10 20 30 40 50 60 70 80 90
5> m01K1q0 qK^O
_v8eqXH@5>A@_AK1oeDA>V@>_@08DXAX
?vDpoXy>OX8Dv>
>
pv
y
>
De
w
>_Dyx>
k
z
{|2}J~V|QvyyJ"%@0};Q"%^{|2"0%D^88|2Q}F|yy|~FAQG}CD%ye{|2"
'H
0%"
y"%8|2Q}F||"DL9jAQ}62l{?"QDkJIF|g2DF|@]
|}F[%}F{Bdk"lQFAD DF|@
yA
0%8{O80lQFA@|y}_f"bWdk2
fv ChF V_5C
N"%9|@D"AHyO?-?W)SY@WQV@5L9Q@5V@MeD%}X_|2{BQ~%f|}F|FD=
86A"|@DAQ}F{2}62"Qj=|""y}%0D2|}e|2"~FLDD|2"y}%A"8|{BQ}FA"y2} ]
g_Bdy =D"8{?DAY%y|222}M|yA^QA}F|2}_L
6
"
B
2|2A
N"Fd|"DLY80"_BD"8{]M_vA"y2dF}%A"D%}7|FDv
y%A"|@DAQ}
{2}62"Qc2|2}_YA}%A"8|avQADAQ}
C?' e6j%ahQ'%fV2M0"?@v=6v9b)6a6W5w6H-%j
"gG2
Q"%OQ|Q{B2~FA"D|"y2}
O
AI{|}C~M^l|2"Ay;"}JDF|@eF|2
"8|Q2}F|2M}QD"yN{|}[~V"%@0} -]D"%2Q"y}f%2Dc"c"F|0|
|@D"AY0A"9"_Py}["%d|2A}78|Q2}F|2
bW
B
}M{B
V-
cB
B
|
!
"$#
!&%
"$#
'
'
'
'
6B
6l
'
'
'
'
w)(
"Fdy|Q=0y}%_F|2AA=2yy@k2D"%%y|222}M|2y}F|}M{Bj
C?+* ejlc2J-,/.
00132
Gkv
254-6
vg=2
7f%2D FAQ 8eh
9
:= ?@BADCFEG HJI$KML5NPOQO
RTS0UWVYXZ\[]SJ^_Za`bdceSJV_SJ`PVfUgZihQ`P^kjlVYmY`en3V_SomY`WX\pq`SJnl`+ZMho`3V_X&`rtsuhQ`b vJsw[xsyZzVPs&hQV_{ZMho^YZ
|]^knlVkj}}MZz`X&^YZ\}VYS~{}MbMbn3V_SomY`WX\pq`UWV_XyVY[]XrV0k`b/vDX&Vkjb`reVY{F`m`Xl-s}MSJnl`UWVYXyV_[]X
r)Vq`WbFvJX&VqjWb`Wr{F`enl^YSnl^Ybn[]b^YZz`PZihQ`e`W}p$`SomY^Ybd[o`WsPVfUu`3WvJbd}n}MZ\bdc\s`l`vJX&`m}V_[xs
b`lnZ>[]X&`WsanlV_SomY`WX>p$`SJnl`nl^_SPjl`-s&hoVY{SP_}MX&`lnWZ\bdck
g/YF
xq3kl
B
xkW&q3
HJI$KML5NPOQ
0_
-
-
-
-
-
-
Pu Q
kW
$qW&kl
_
]
q
F
x
$YkQ0
>F
)
Wq
x
a
HJI$KML5NPOx
>_
W$
x
q
x
qW
|]^qn3VqjW}n3V_SomY`WX\pq`WsUWVYXtVY[]X-r)V0k`b]vJX3Vkjb`Wr)
0
Yz
HJI$KML5NPOQ
0 2 4 6 8 10 12 14 16 18 20-1
-0.8
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
0.8
1n=20
(R
J)
mode k
v2 (mode k=2) v15 (mode k=15)
0.00.20.40.60.81.0
J$M5PQ
MW! "#%$'&)(+*-,/.1001023&
4658798:=
* ?
@
$BADCFE
$
#%$ #%$'.%(G
98HI:7KJ/:1L'M/:1N98OP5QO/R3SUT
;
C^
*
SF^_;W=
* ?
@
$BA)CFE
$
[
Y
$[
SUT\8\^
#%$
`%acbdafe gUhdi>jlk3mfn
J$M5P!o
p
l_>fq/"rstsvu>f_Mxw)xfysPMos\f]"B"8]"lY!{z|_}YM~q/]
]>_v"B"8BYqYM}(*-,Yq(*!
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 11
0.5
0
0.5
1
1.5
2
x
sin(pi x) sin(5 pi x)
e 0n=20
0 2 4 6 8 10 12 14 16 18 20-1
-0.8
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
0.8
1
(R
J)
mode k
0.00.20.40.60.81.0
J$M5P>
>ZDuW"w8>w> ]]{uW\fY>"B"8""P s}"]\fY
>Z]}"
q/]zIF>(+*1qP_6"PqYs{zI/"B ]qP
,/,
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10.5
0
0.5
1
1.5
2
%cdc !]!'D/]W s%6>
!Wvv'%B xP1
%/8P1 fU P +-/11
6/1/8F>
1B
>
1Df
P
f%!8
Z
d1!
s%6W
)]
rB]sK
v}
%
P
%]6
P
81P
+
Z
)PW//K/1KP_P'//1PPP !
Z
P
Z
W|
+
>
/
B]s
P
!
! 1
#"+
$>
%
]
f
8'&(Ws8
6
/
6
%
)
P
*,+.-/+0* 13245 6879 6:47?@6BAC-)DFE GH8IKJ,LNMPO
QR:S.TUVUWYXZT[S.T3\'UV^]'_3_`bac_dVeb_efacghajikghUVehaKRlC_'_3eSKmn_dVoT[SpgV)TqSKVNUnT[arlCUStsZR:a u(gl[ikUlCe
mfUV)V_(ldv
wyx{z%|}n~0ndx)C}bx))x)~}=ox)t}bCoN)=0~P
| Z=
p )
p
}8 r:
,
}b K(
c
pZ
}8 p
}8
)
oCd}== n
[anS.TNSKm)gl[ayUV afacgVgac_NaRoUafSKVd3UVeUWTdgSKVq
n
vQRS.T
\'gmf](SKV_'eiSKaKR^aKR)_u(U=\daaRoUaaRo_fV mn]'_dlnguV:VgiV)T3SKVBUVeFS.T
UVen/mfUb_[TnSKay_(lCUajSK_fmf_daKR)geT)gay_(V ajSpUWKWYXUararlCU=\dajSK_u(glnUVe{Tro_'\dSpUWKWYXZ
lg=]dW_(m3Tdv
j :b,C,
GH8IKJ,LNMo
QRo_F\'gV Z_dljs8_(V\'_{UVUWYXZT[S.Tu(glaKR)_U:T'T'cP_(Speb_(Wmf_(aRogeS.TUVUWgsbg:TacgaRoUagu
aRo_Ub\gb](S,mn_daKR)gebvQR)__`(lC_(T'T'SpgVoT/u(glaKR)__dStsb_dV)_'\(aygl'TUVe_dSts8_(V ZUWY)_[T':u(glaRo_
mfgeb_(W8lCg=]dW_dmT'\'UVUWTdgh]'_u(gVenSKV\dWgZTd_'ehUWYaKR)g=sZRaRo_lCg\'_elC_S.TTdgmf_(iR)Ua
mfglC_SKV)gWY_'e=v
/jN'
k
[a\'UV]'_TR)giVcajl[XnSKa NaRoUa
p
/
}8
j:
k
op
c
=
?o
!" #%$ &(')+*!,.-0/++132045,6-879$):&;-8,1?=,@#6464)A -8/=BC2D*+-8BFE;4G!,H"IJ28=#KL28&;MD*+28&
I+N7!+MD,H-O/HE;/J=!!&,P,QR!/=/-8/S*+-8BFE-8&;!&,@!T U;VW6XZY\[[
n=20
mode k
v2 (mode k=2) v15 (mode k=15)
0 2 4 6 8 10 12 14 16 18 20-1
-0.8
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
0.8
1
(R
GS)
]Z^_]Z^a` bdc;ef(gh8i(gekj9glh8mong
U;VW6XZY\[p
qsrRrutovJw9xGyzwR{9x_|9}y~_}!|u}!~r03 ruy|9}!J{9}y }uu}u {(Go
!)9
!
;
)
;N
~_r9{;0
N
oQ
"
Q
20BDS/20,.d2)No29*+-9I!#K1I!7o,P/+!7o#6/ "-0&45d284 AS,6N&(7oBDI+!/"-A#6,.!/20,@#a-0&!T
dz"33=Rd
U;VW6XZY\[
w0J
.v}Jw0vx_r9y J
++D+.!>J
&45d#6&lN1$9#6'8#6,.!/20,@#6'8BO ,Q-M)L -0BDS*J-8=,29M0')20&,H2!$S-8'8!/d287+=#a28&!45#6BR#6&;20
,#a-8&;T
!
RzL 3 d
!#"$%&('*))+"
,.-0/214365879-.:3 ?A@B1(; CEDGFIHKJML
;=>3 N?PORQ#SI@1; TUDGVWXWZY\[^]IL`_^]Ia
b
'*cedgfehifjkilI'*$m^)'nk0f(opkiq fr&Iqs'*c tMf(uIv
, -i/1 3 wxNy5z, - :
! !!"$#&%('*)
,+.-$/01"-$/)2435768:9A@B5C3578D=.E(F)HGI"JI/KLM"ONP$QR/K
ST
!#"$%&(')
*+%),.-
/ 0214365879;:=)? @
ACBEDGFIH
J$KML(NL(OQPSRTLVUWPXNSLVY[Z\]LOQ^_`ZbaScedSLf^gYh_jiklYh^gmnUoZprqVsutXvwR]x]y]OQz{_r\]LfUW|nY_rZ4Rn^gRTL(Y
}
z~Z\)R]Yh^S!Zz{(^SO YLf_rx]O~Z_V
J[L(Yh_lZ\nL4RPC__z~ynz~OQzZ`dePg_PSOQNuz~mn^R]YPSynO~LV
}
zZ\
}
PY8^SmnU_rZPY^SSLR]YPSRTPSY
Zz~Pmn^gO,ZPZ\]L4mux]yLVYjPgx]m]um]P
}
mn_V
JL(OQO6UoLVNSLVO~PRLfU>PSYOQz~mnLV^gYLVO~OQz~R]Zz{R]YPy]OQL(_[i^gR]R]OQz{(^XZz~PmZPPSZ\]LVYLVCxn^X
Zz~Pmn_jz{_j_`Zz~OQOG^Sm^ZzQNSL4^SYLf^PSYLV_LV^SYh\,
lXgg`gSC`(Q`hhf8S EVr8CgVEf ,uG`Vfh
;ChCVCVC] u!V`4roogVCVVrVoVfVS
0214!39(=)3V>!
r lr[,;
@
ACBEDGF
>>;EEu(!Xh(CXEn]WXWgE(]XXS(~(EEuhgEnh
TE! ~!`4oge!]f]XT!(!hhXhhSgE(T8
o ~gE(]g]6Ehh4nX6(h6;EEjV( (X)E~b!!ShXo
E!gXe(EnVX!h( E ]e~Xh~h(oXSX!~h$!geShX8h(~(X
]E(XXVh!h
qx]OZz~Yz{U z{_^gm z~ZLVY^gZzQNSLL(Z\]PoU ^
ff8EEgSnh
}
zQOQOlYLVU]xn!LZ\]L
mux]yLVYPgzZL(Yh^XZz~Pmn_
ZP_PSOQNSL
yud8zZL(Yh^XZz~Pm
}
Lb(PSx]O{UeZ^gL
!#"
}
\]L(YL$%!& !'
((
(!
)
gT
!XEn)X*g+Xh!X>`o-,gTg;]hX!Xh
Xh!Xhnh!(]gb!ohX g{(!(/.!Eu goeV!S(~( X
gW`/.f102
)
3
4
hg]EnX[]eEnEXgn!h!gEnE!gX
f!hgg0${65T7Egh98hX En(Xog `EnS(~( X
S>`/.f
a
02>u!]6X>hSTEghhgu!
X
X
Xh!Xh8]g[]:X>X ;M!(n~(CSEu >8nXE!he~X(]g u{T7Xo=$>j{(~hXEnX]MVhhXoXoETh{h(ohhX?
]g{h;EnT!Q! X>]S
a
0+hg>g (8h(XXh8E!XX];;E>X
EEgCSnhT:fSh4(X*VEu
%@
)@
A
@
!B7C78);nXEn;XoEnX
E(XXXhVgh(!4n{nh](hhgu(D{MXE~(hlX](!g
q
EGFIHJKH1LNM+O/HM!M+P Q:JKR$SIHJ+LE=HJT:EULVQ:WIX&T:WZY\[!H!]!T:^-MH4_HP HJ+LE=HJKT`EaLVQ`WIMbT`JKHJKH!c^dLJKH!Y:e
LVfHCf4ghLNMiMkjlT`OO/HJf
m#ndo+pp7p oKqIr$s-nIt1m9r7uwv`x2y*oru!z:oKy{v|s9}sIrrC~dr7~\oKv
}v/riB41
}oy/{l
g
`
LWZ]!HiMQ:jHM!jQCQ:EGF]!Q:j%SIQ`WHW#EaM$Q_$EGFIHbHJkJKQ`J$PLOOMkEULOOJKHjlT`LWf
pz:xoru%zbxGr7y/oKr7uz`oKy{v|s#}7-oKq#r
HJkJQ:J$LNMiMkjlQCQ:EGF
#)r+v|nIN~3}v/rxGvuoKqIrruuKvu
vsz
]!QCT:J!MHJ$jlHMKF
drp
&
p
QCQCYLVY|H!T[!H]!T`^-MHC
A
tlv-v`oq\xGn#s#koy/vs#}wz|s\mr$urIur7}KrsoKrC~vsvz`u!}Kru|uy{~#}
vz|u}KrB|uy{~}v/nIoKy{v|s#}z`ur$!qIr7z|9r7u7p
F-LNMLVYH!TLNMLW3_T|]EiEF#H]!HW#EUJKT:O)LVY|H!TQ_jb^dOEUL
JkLVYE=H]!F-W9LVc^#HMfWQ:JY|HJE=QEU^dJkW
EGF-LNM$LVY|H!TLW#E=QTBSZJKT|]EULV]T`O2T`O
Q`JkLEFdje)MH:HJT:OLW
JKH!Y`LVHW9EVM$PLOO2[!HJKH!c^dLJHYf
a Iw
xdoKq#r
FdL
F_!JKH!c^#HW]
vtl9vsIrso}v`xdoqIr r7uKuv|u~dr77z%xVz}oruoKq#z|s$oKqIr
O/Q:P_!JKH!c^#HW]
vtl9vsIrso}7d4ri}zoq#z:ooKq#r$y*oru!z:oKy{|ritlroqIvd~y{}wz
MkjQ7Q`EF#HJ
p
NV I#
-1-0.8-0.6-0.4-0.2
00.20.40.60.8
1
n=19
(RJ)
mode kv2 (mode k=2) v15 (mode k=15)
0.00.20.40.60.81.0
LOW MODES HIGH MODES
0 2 4 6 8 10 12 14 16 18 20
}&z|vmIyzl}tlv-v`oqIruk pp7p!
+!bGI:ZV`4 9V:7d Z|V7::)kZN+#+-*I!!K!#
k )N/!`+#:d|
|
N+
4l
7
#
! "$#%'&)(+*
-,-B/. 1032546.)7
-1
-0.5
0
0.5
1
mode k
(R
J
)
=1.1(UNSTABLE)
=1/2
=2/3
=1
$8dV9,- :.) 8IV:0A;2-46.)=
? @2=BACABAC=
EDGI`
F.HGI2|Bd|
|
B!`6|
3
C
7
#KJVML
GI
`a
N{PO3QSRTa
,-
9V
,-
7 >L
=:U. WVYXPZ9A3:
GI:Z
[.:\]2GI
G
!
:
id#k
/^
k+
&
D:a;_k!)
+
`
?
8#V9,-7)\W2J
:
lIk#>`kV!
>`#:V1
d|
|
&
C
lC
GI$Nlk
L
L
D:a!
$GI
L!K!#Z
|
"$#%'&)(ba
cedfCgihjdlknmo pqglfrYsktgfBudlmvgfBuswCfFhPoxfCgglmYgzyvmufF{|h~}ShwdlmYg5mP}q2B
0 2 4 6 8 10 12 14 16 18 200
100
200
300
400
500
600n=19
Num
ber
of it
erat
ions
mode k
= 1=2/3
ZI1`Ki`IK
Ll#l#
b K`a
I1dK!
K!:#!lGIlGI
`ZU#|
$!|!k
|
&|
Y|\i!$GI`#ik=::K-
|
'. @2C
:
k
dK
:9=:U
I
` 3GI\-*I!4!K!#
l
|
#
GI%I`GI+dE=K/B!d
2!I
)*
Z:=k$GI%d*C
l
|
D:k3dVi:*
d
#
GIi#::
L !K!#
k1 |b
!GI:
LGk `:K -
|
Nb
:Ni``#` 3KI:w!#:1dNk
L
LAGI
DaZ
4GI
L!K!#
l
|4N
K!:*z`
{
:$#:a
i-k
`!
kK#k!L L
b
/1
`#!`GIb
D:a!
$GI
|
n !S
"$#%'&)(+
5fwChPn
Z
n=19mode k
v2 (mode k=2) v15 (mode k=15)
(R
GS)
0 2 4 6 8 10 12 14 16 18 20-1
-0.8-0.6-0.4
00.20.40.60.8
1
;[P l3CC3vPCE CC
'ivCYCjiji3->UP$~Y$j5ij'SYEq''t~MijKCP
iP1EMBP''$ESEv;[PEeSYtBj'KjqvCYCjj3
'eCljSjUvPE
$')+
eCijtY1lYtBlvBCPK
CEljFMBY5
~SlY5PqB
0 2 4 6 8 10 12 14 16 18 200
20
40
60
80
100
120
140
160
180
200n=19
Num
ber
of it
erat
ions
mode k
BCiF5
BvEMBP'
tBP'KP1~ viC;'CljSjli'liP~vj-$'YM;~iY
MBYEM;x
vPECMCj6;KYjUPCE'$6'xEMBP''
$ESEU35PiieSY-CMBEEKjjMiCiP+ijj
l9iliC$Cvvv'~CqYiPMP1viElPe'jK
tPKiliC$CjEC
q :)Y3$;z
$')P
Klj'>li'lM$BiijljUjCEE''BPEvPSPCCESC
$Bi-35jCEE'jijij $5iBPiC[MEl-P'tiEEESSj
>C1
iPn
!
"
"$#
B3lnln&%$CijlY('SMjl*)$+
,
-/.01$2*3547681:9=?35@>A$93B.C4EDGFHIJKLDNMF
OP"Q&R SUTVOQ&R PS W
;X9ZY
T\[X]^^^?]_/`ba
P
c
Iedgf:hiGihddkj>lmIknbo?MpedkDGl8qrisDNJDGKtun8Kvf$hKxwy
[
DCdmhFzI{uIF|F:j>lm}"I?p~EBIkdkKLpkDNJ?KLDNMFz}t
DGFHI"JKDNMFpFJKLDNMFMFKGf:IExFrIlI?dFJKDNMF$nhi*MK8M7oDGFo?MpklmhKLDNMFDCdi*MdkKj>pkDGFKGf:Iep3548
T\[X]^^^?]
w
[
9
UN NN: GU&
0 0.2 0.4 0.6 0.8 1-1
-0.5
0
0.5
12nd Eigenvector (n=19)
0 0.2 0.4 0.6 0.8 1-1
-0.5
0
0.5
118th Eigenvector (n=19)
0 0.2 0.4 0.6 0.8 1-1
-0.5
0
0.5
12nd Eigenvector (n=9)
0 0.2 0.4 0.6 0.8 1-1
-0.5
0
0.5
12nd Eigenvector (n=9)
>>N5e&&C&ENz$>>E&&CX
! #"%$'&()+*,-./"0' 1/ 23"%/4-' 056&/78"%
")90:9%;>CZ*]E^$>X5?&Bb$1b:Z>>&R
cd/"%9C! 5"e"%fgA)/9% F7g"%fO
hjilk mon/p@qp@rsutwvxp@r
GU1y
zI7C{4"%/l- ["G9%f%gM*@"z2@7g")7E|"[")gD>}~)7%*
"03I%$J)! !"%%O
&
!/)JF(4/G/f/%J)) wF!-
e1
1
e1
g
1
1
J
7!RH1
o >(%Ca(Cf(
=wBAA1)/A11!%AJF|>%A}zA%Al
N
l11/A1!w%C/GA({}14A/0F/(%(ww%/@!0uj}R},
A17d/)wFJJA)CF:3)3AF})(1-1F/uA!1,l1gJ1A}
GA(>%1C)A{JG{/1}JFA>1/4F/}(Co)7%1F1
)}31A>JRJ51}1
1}(C7l1BJ/%AJFoA7FA/%4A:/G1A/14.
"!$#&%
' (*),+ .-0/ ' 1)2+ -
'3(*) 4657+ .- / 8
9;:
'()2+ -
" ~Q;;3("kh c
~I`{"> *1
d
X
Z`
~!
#"
6%$&
'
)(
*,+
.-
0/
6%$&
1
3204
6587
489;:
7
-3=
=
GYX
Z7[S
GTJ489
U
:GL484
:
7
9;GLH
5
:\
6]!^_
\
9`HLacb
7d5T
4
V
H
9`H39
7
9
JIb
655
fe
H
Ug
X6GL4:69`HhJgR
:
7
GL4
^Z_
\
GIb
7
aib
7
9
5T7
\
H
Sj
aai4G
9
]L7
9;GLH
7
G
7
\
V5
G
b
7
9;GLH$
*1
^lk
4
e
HWmGYX
7
\
4
"hI7
9;GIH
e
4
8587
489;:
7
9;GLH
e
H
U
ac4G
GIHKJ
L7
9;GLH
5
:\
6]65VUX85
:489 P
U
e
:
H P
b
5"U^
n
4
:
B7
\
o
bp9`R
>
Hc:
P
67[Sk
H
5
G
R!9`HhJqX6GI4
r
GL4
A
s
-
^ut87
7
bp48H
5
GIb
7v7
\
L7wS
489
7
9`HKJ
7
\
:G
4
5"
JI489
U
:GL484
:
7
9;GIHD9`H
7
4
]5
GYX
7
\
484GI4
> Ux5Q7
G
5
9
]
a
>
4
H
U]
GL4
V587
4
9Jy\
7
X6GI4
Sw
4
U
X6GI4
]
b
I7
9;GIH
^
H
U
VZVXWS#USB[\]^_[`![aSQbLUQY>]1Y>a\cd\>eUOVf\VgWSihkjBUlVZ[`![)a SlmZY]^nVgWYVomiYpBVfSjQ\>]a5`
Vrqi\M[`![)aS)UBmiVgWSQU)\ase*Vrbo\>]8bLUtYasu\>UlVv[B\>]xw6SjycdSB^!z
{|d}X~x
T* !L!!*k ! #!
0, 5, 10, 15, 20, 25,14,
12,
10,
8,
6,
4,
2,
0,
2,
4,
# Iterations,
log, 1
0,||r|
|, L, 2,
Weighted Jacobi, ,=2/3,2level multigrid: ,
1,=2, ,
2,=2,
1WbLUihce*jSQbXaXaseUlVyjY>VSlUQVgWStue[BWOp)Y>UlVfSj[B\>]xw6SjycdS)]1[S\!VY>bX]SB^MeUlbX]dcMVXWSQue*asVrb5cjlbo^
j\[BSB^>e*jS.z1WS[B\>]xw6SjycdS)]1[SMaSw6SaibLUnUW\>q']w6SjBUleUVgWSM]e*unBS)j_\pnbXVfSjYVybo\]U\>]
VgWSJhi]SunSlUWz]1S_[BY>]VgWbX]*D\pVgWSn]e*unSjn\pbXVfSjYVybo\]UbX]VXWSJhk]1SunSlUWdm#Y6U
BS)bX]cY
!
j\bXunY>VS)a5`
j\
\>jlVrbo\>]1Y>a1Vf\tVgWS#Y>un\>e*]xV'\p[B\>u
e*VY>Vrbo\>]1Y>aqi\>jbX]w>\>a5w6SB^!z
TB'T)k)T
D!8
{|d}X~
1WSMVyqi\Mcjlbo^TU)[BWSunS
jSlUS]VS^8Y!B\>w>S_a SYw6S)Une*]jS)U\>asw>SB^VgWSMeS)UBVrbo\>]\ptW\qVf\
U\>asw>Sp)\>j_BbX]VgWS[B\YjBU)S8unSlUWz1WSY>]UlqiSj8[B\>unSlUOpBj\>ujSBY>a5b)bX]cVgWYVtVgWS
j\!aS)uUOZ
Y]^nBxBa\.\DSY![Vras`VXWSUY>unS_Y>]1^MWS][BSVgWS
UY>unS
j\.[SB^e*jS[BY]BSOeUSB^nV\U\>a5w6StVXWS)unz
O)* Z
Q
L>%
.J!t
T 5 !!.v
t
fkM
)!v! *D
}X
i
#
Q
Bxr
B
)J
B
B
}X
%
v
5% L
.
1 !"$#%g"&(')*++,.-/*0213.465702#1$8:9g;$-/*
?A@B-/C#Z8&'M#kg*+-/4602#HPlB8&46-/0O8$#8:B!#
SUTWVYX
\^]`_ab2]dceaFfGgihjcj[klafm,bamknf[goegpf;fFq9r[kla6st@_#agr9u[a;j]dq`v,agWhxwzyij[cj[klaFf{q|f{]g}j;g~j;a~y
]db3m,]a2j_HgWho]D_#ag,b3g,~]`_aj;g=m,b;f[aGaf3_#afs
[ F?[B}-
^i ^>Yz#i@iW 0`Y#
B#i3#l-U3#[o6[
@g}hm,b'a_m,v,a5g,~kc:f[ag,~@agih']D_#aHmq`~q9am,f'q`~v,g,kv,a;q`~*k]dqbFq9st@_#a
q9a;mgWhofF]m,bF]dq`~0}q`]`_m0g=g6}q`~q`]dq9m,k@af;f-_mpf{~@g,]ra;a[~Hq`-ekla[a~]a;}ca]st_*q|fq|f
mj;j;g,-ekq|fi_a;}q`]D_]D_#ah;*k`kY*k]dqbFq9f[j_aa6s
2{>
0`Y#
33#3|#Fp6036
6#;|xG#l3U
[[x3#6
,I3*=Ul3#[636
t@_#amr;gvpam,baf[ga}]g;eq9jfJ_q9j;_m,b3a]D_#af;r9u[a;j]2gWhmj[]Nq`v,ab3aFf[a;m,bj_m,~@J_q9j;_
m,bara[cg,~@]D_#aGf[j;geagWhG]`_*q|fGkla;j[]N*ba6s{~e#mbF]Nq9j[*klm,b;-mk a=r[b3m,q9jG*k]dqbFq9a[]`_g6,f
m,ba-0m,q`~q`~q`~@jbam,f[aoe#g;e*klmbFq`]Ncra;j;m,f[aGgWho]D_#a[q`beg,]a[~]dq9m,kJm,f{ma[~a[b3m,ka[m,]dq9g,~
f[g,kv,a[b[s{a[b3aa_m,v,ampf;fF*a]D_#m]-g,*bm,]dbFqm,fG]D_#ab3afF*k]>gWh,q|f[jba]dql[q`~0Hm
o
g,~m}q`v,a~5bFq9gWhfFql6asF~]D_q|ff[a[]N]dq`~0-q`]oq|ffF]Nb3mq=_*] hgbFm,b3]gHa9~a
]D_#a5j;g6mb;f[abFq9He@b3gr[kla[f[s9hg,*ba;m]Nq9g~fFcpfF]aG>g=aFf~g,]}j;ga
h;b3g]D_#aq|fj[b3a[]Nql=m]Nq9g~5gihGm
o
^gb{]`_a,q|f[jba]dql=m,]dq9g,~eb3g=j;aFf;f{q|f{~@g,]>m,v,m,q`klmrkaF
]D_#a-ebg6j;a;,*b3afq`vpa[~5q`~5]D_q|fkla;j[]N*bajm~~g,]-r;af[a;,q`b3a;j[]Nkcs
a;mkq`~0q`]D_HfFj_
0a~@ab3mkUfFcpfF]a[f;q`]D_#g*]ba[*q`bFq`~0}q`~6hgbFm]Nq9g~:mr;g,*]]`_aq|f[jb3a[]dql6m]Nq9g~e@b3g=jaf;f;
q|f2]`_a2fFr9u[a;j]^gWhmk a=r[b3m,q9j*k]dqbFq9a[]`_g6,f[s
=
@;x#
3 "!;
#$%'&(*)+,-/.0.1324*5,6 7 8 9:*;#-?@!AB1DCEF(*G/.@IH7JK
@L1BMNE$O
"!$#%&'()+*,-/.01234 5'
6
87:9;=
9?@A B3CC
D EF&GHIGJLKMN-OPJ/QGSR@TMUNWVXG&Y+O
Z@[\Z ]_^`)acbedfb\g=ach
i8jki8jki lUm8no(prq(sutuvxwys8p{z5m|q~} 8r5;
SIIy\x)
8
8
8 _
\ fuuu~&-|\xxrre
kIfxu-uu(~x5I
u)f
i8jki8j )q~wk}wyvxwks|pzmq~} 8r5
-8&IIx
8
8
8
8
|
1
8
8
_
8
8
;
8
8
_
&ux
~fA~f4|S
%uIxu&fyu|
UuI8IIfu-yuu
5
-IxxuSyAu(r~8xIx
8
|
X
|
|
_
&ux
1
(
|
x
f)/\uISI-
uuIxf(%uxIr(_ux&)|xuI_uuxucI( &
u|x~I|u'IIBy-cc~4xxxy'Ie uxIf
&(e'u\xx%uUx&uI
I|fIu(|
r~&B
8k8 (~u&
!#"%$&'(*),+.- /0213/4657*8 9
:=@?A
?BC
?ED2FGAEH
?
/JILKM.NPO
K
K
B
: ;
A
KM.N.Q
-
D2FGA
4
H
Q
D2FGA
0
H
5
RTSVUXWYZ[U[\]3^_`ba]dceWfgZGSVUhcafdZi\UYU]3^_ZSyUXfa_ZzWoaY A ^'YudTa]ZGSVU
D ^']3U&uWfklaYZnWYya
in 2v
(o o6
{
"!#$&%(')!*+,$&'.-0/1!,*!2435')!,*6')$&*-8793:-8/;!*[email protected]
E
whNC/;!*$&w63:-0/1!,*/qN3n6:8,0P$&w63:-0/1!,*(}t/;%1$
-8t6$\%1/;*$i3P73,#+$i'.-0/1!,*l$iw63:-0/1!,*l/qN"06)r9:)t/;N{5$&3P*hN5-8t63P--0t$3#+$&')-8/;!*
#/1>whNC/;!*$&w63P-8/;!*3P%;3Nt63,NNC{5!!P-0tN8!%;w-8/;!*hNr$r+$&*/12-8t$/;*/1-8/q3P%#3P-03v/qN#/qNO
')!,*-8/;*w6!w6N&}t/1%;$z-8t6$%1/;*$&373#+,$&')-8/;!*Y$&w63:-0/1!,*Y3,#{5/-9N