This is a repository copy of
Efficient and accurate time adaptive multigrid simulations of
droplet spreading
.
White Rose Research Online URL for this paper:
http://eprints.whiterose.ac.uk/1756/
Article:
Gaskell, P.H., Jimack, P.K., Sellier, M. et al. (1 more author) (2004) Efficient and accurate
time adaptive multigrid simulations of droplet spreading. International Journal for
Numerical Methods in Fluids, 45 (11). pp. 1161-1186. ISSN 0271-2091
https://doi.org/10.1002/fld.632
[email protected]
Reuse
See Attached
Takedown
If you consider content in White Rose Research Online to be in breach of UK law, please notify us by
White Rose Consortium ePrints Repository
http://eprints.whiterose.ac.uk/
This is an author produced version of a paper published in International Journal
for Numerical Methods in Fluids.
White Rose Repository URL for this paper:
http://eprints.whiterose.ac.uk/1756/
Published paper
Gaskell, P.H., Jimack, P.K., Sellier, M. and Thompson, H.M. (2004) Efficient and
accurate time adaptive multigrid simulations of droplet spreading. International
Journal for Numerical Methods in Fluids, 45 (11). pp. 1161-1186.
$&%('*)+,).- /1032546)7825':9,);</>=@?BACBDBDBD1EGFGF.HJILK>M NO42QP>RS42L?T/1AL=U%>VXWYSZS[\:]1^_a`bc5dG254&AL=:eB%gfh>iijhakliBmakgnSoqpjn)UiSnQr
sutwv x5yz!{}|3zq~|vv |{y {x5y|~8|{x6yqa{x6x~x5q|{xzq3
~386y {qy|3~qxz
3 #B# G¡¡ I¢¤£
¥ ¦§¨©#ªS C.£ «
¬ G¡¡§* (
I
#®¯°
«
±8²O³¨´B³®
I
µ¶
%>VS=U%G2L264&=:%>V;</1=:?7·2¹¸a91RB%(=@¸6Aº25AL2LRB4Q¸&9¼»O4eB/P,½¾l¸&9,e6eB<e¹¿7·2L¸&91RS%(=:¸LRB<
¶
%>VB=:%G2¹254&=U%>Vj½
À
¾l¸a91e6eS<eQ¿ÁeB0Pg/>'Â=U%>Vj½
ä9128Ä%(=Upj264&AL=U'ÂÅeQ¿Æ¤2L2L?BAQ½Æ¤2L2L?BAÇÆ#¾ghÈmB+lÃ.½Ä.É3)
ÊGË̼ÌÍÎÏ
ÍÐqÑÒÓjÔÕÑjÐ(Ö ×¤ØlÙUÙ.ÍÚ¤ÚgÛaÜBÝlÔÕÞ8ßSÖaÔUÜ>зÊGÖaÜÛ6ß>à1ÑXá×GÍÊlâÌãØ¤ÙJÖaÔÕà>ÛaÔUä·ß>ÙÕà1ÜÛaÔJÖaå¤ÞæÔÕçØ¤ç&ÑBäÖaÜ!ç&Ü1ÙÕè,Ñ3ßÛ6ßФà>ÑÜéäGÛaÜ1Ú¤ÙÕÑÖ ç&ÚlÛaÑBß>älÔÕФà·ê.ÜBëìçÞqÜGälÑÙUÙÕÑBäíß>ç!ß·ÓjÜ1ØlÚ¤ÙÕÑBäîç&ÑÖ!Ü>éФÜ>ФÙÕÔÕФÑBßSÛ!ÙUØlïlÛaÔÕÓBßSÖaÔUÜ>ÐðÑBñ(ؤßÖaÔÕÜ1ÐlçjòóålÑßÙUà>Ü>ÛaÔJÖaå¤ÞôÔÕçé@ؤÙÕÙJõ ÔÕÞqÚ¤ÙÕÔÕÓjÔJÖß>Фäíå¤ß>çÑÞÈï#ÑBälälÑBäëìÔJÖaålÔUÐíÔJÖ!ß>Ðß1älß>ÚgÖaÔUè1ÑXÖaÔÕÞqÑöçLÖaÑjÚlÚ¤ÔÕФàqç&Ó6å¤ÑjÞqÑ!Öaå.ßSÖÑjФß>ï¤ÙÕÑjçÖaå¤ÑXçaß>ÞqÑ!ÖaÜï#Ñ Ü1ÚgÖaÔUÞqÔÕç&ÑBä¼ÔÕÐíß·ÓÜ1Ð(Ö&ÛaÜ1ÙÕÙÕÑBäîÞ8ß>ФÐlÑÛç&ؤïG÷LÑjÓÖ3ÖaÜãß8ç&Ú#ÑjÓjÔJø.ÓXÑÛ&ÛaÜÛÖaÜ1ÙÕÑÛ6ßФÓjÑ1òóålÑÈÞqÑÖaålÜGäÔUçø¤ÛaçLÖè>ß>ÙÕÔUä¤ßSÖaÑBä ß>à1ß>ÔÕФçLÖìßÛ6ßФà1ÑÜ>éßÐ.ß>ÙJõGÖaÔÕÓBß>ÙßÐ.äÑÝgÔÕçLÖaÔÕФà!ÐGؤÞqÑÛaÔÕÓBßÙ¤ÚgÛaÑBälÔÕÓÖaÔÕÜ1ÐlçÓjÜ>ÞqÞqÑjФç&ØgÛ6ßÖaÑëìÔJÖaå·ägÛaÜ>Ú¤ÙÕÑÖç&ÚgÛaÑBß1ägÔÕФà ß>ФäðÖaå¤ÑjÐùØlç&ÑBäÖaܼç&ÔÕÞÈØlÙUßÖaÑ·ßç&ÑÛaÔÕÑjçÜéìФÑëúÖaågÛaÑjÑöQälÔÕÞqÑjÐlç&ÔÕÜ1Ð.ßÙ ê¤ÜBëìçÈÓjÜ>Фç&ÔÕçLÖaÔÕФàîÜ>éägÛaÜ>Ú¤ÙÕÑÖTÞqÜ>ÖaÔÕÜ1ÐðÜ1Ð ç&ؤïlçLÖ&Û6ßÖaÑjçXÓjÜ1Ð(Ö6ßÔUÐlÔÕФà¼ÖaÜ1Ú#Ü>à>Û6ß>Úlå¤ÔÕÓß>Ð.äðëÑÖ&ÖaÔÕФàîå¤Ñ5ÖaÑÛaÜ1à>ÑjФÑjÔJÖaÔÕÑjçjò óålÑ·ÙUßSÖ&ÖaÑÛqßSÛaÑ·ÜéÚ.ßÛ&ÖaÔÕÓjØlÙUßÛÔÕÐ(ÖaÑÛaÑjçLÖ ß>ФäÛaÑè,ÑBß>Ù ålÜBëûägÛaÜ>Ú¤ÙÕÑÖaçÓBßÐ"ï#ÑãÞ8ß1ägÑqÖaÜç&ÚgÛaÑBß1ä"ÚlÛaÑé@ÑÛaÑjÐ(ÖaÔUß>ÙÕÙJõ"Ü1Щç&ؤïlçLÖ&Û6ßÖaÑjçÜBëìÔÕФàîÖaÜß>ЩÔUÐ(ÖaÑÛaÚlÙUßBõ ï#ÑÖ¹ëÑjÑjÐîägÔÕüÑ5ÛaÑjÐ(ÖÖaÜ>Ú#Ü1àÛ6ß>Ú¤ålÔÕÓÈßÐ.äç&ØlÛ&éÂß>ÓjÑ!ëÑ5Ö&ÖaÔUÐlà8Ó5å¤ßÛ6ß>Ó5ÖaÑÛaÔÕçLÖaÔUÓçjòýÜ>ÚGõGÛaÔÕà1å(Ö þÿÓ Ü>å¤Ð°ÔÕÙÕÑõ ÊgÜ>Фçjú Ö6äò
äGÛaÜ1Ú¤ÙÕÑÖaç1ç&ÚlÛaÑBß>älÔÕФà,ÞTؤÙJÖaÔÕà>ÛaÔUä>ß1älß>ÚgÖaÔUè1ÑÖaÔUÞqÑ5öQçLÖaÑÚ¤Ú¤ÔÕÐlàÙÕØ¤ïlÛaÔÕÓBßSÖaÔÕÜ1Ð8ß>ÚlÚlÛaÜBÝgÔÕÞ8ßÖaÔÕÜ1Ð>é@ؤÙÕÙJõ ÔÕÞqÚ¤ÙÕÔÕÓjÔJÖBò
"!#%$'&)()*)+,# -&)!
# .0/)1324657%281:9-.0;<7%=>=/?%;@7 9A.0/B/6C46;@/ED9'DF50G/E?A;<4E>8HI7-2H@59A;J2DK21ML2N'7 ;@DFO2H@OF;JDPRQ?A2=HJ/E9%7-=0?A/>Q;JDP 9A.0?-250P.S9A.0/T57-/T281>U135HJH@VWX;@G=0HJ;<46;J9YZ9-;@G/6W[>Q0>=9-;@O/T\]50H^9A;JP?-;<Q_>8H@P2?-;J9-.G`a54.bL2N'7]>?-/c21 /D02?AG257d7-;JPD0;Je4E>D46/f;@DSGg>DVih0?>8D4.0/7j281g7-4E;J/D46/c>DQb462GG/E?46/Y%;JDQ57k9-?A;@>Hl>8=0=HJ;<4E>89-;@2D7 ;@D46H@5Q;@D0PYm132?g/6n0>GR=HJ/Y:9A.0/]Q/=27k;J9-;@2Do281)4E2>89-;@D0P7>8DQp;@D0q7EYrQ0;J?A/469=>9k9A/E?AD0;@D0Ps211350D4t9-;@2D>8H H<>uV/?A7cQ5?-;@D0PvG;<46?A27-.0;J=w=0?A2Q5469-;@2DIYR7k=0?A/>Q;@D0P_281d=x/7k9-;<46;<Q/7o>8DQy9A.0/zL2N{281d2;JHK;JD|.0/>9 /En4.>8D0P/E?7l}J8YI~u# ./7A>8G/R>H@7-2>?-;<7k/;JDs289A.0/E?BQ;@O/E?7-/>8?A/>7'217-4E;J/D46/R?A>D0P;@D0Pg13?A2G h;J2HJ2PVY N%.0/?-/)9A.0/EVg132?-GGR/Glh0?>8D/7,2D`Gg>8GGg>8H@;@>DKH@50D0P7%>8DQ9-/>8?,eH@G7';JDK9-.0//V/YF9-2P/2H@2PVN%.0/?-/ 9A.0/EV13/>9A50?A/);@DjH@>uO>L2N'7B}
?A2h0H@/EGg7>8?A/'G2FQ0/EH@HJ/Qg57-;JD0P>H@2D0PN >uO/Y2?EtXY>=0=0?A2unF;@Gg>9A;J2DRN%;J9-.9-.0/B>QO>D9>8P/ 9A.>9j=0?A2F4E/E/Q;JDPo;@Dz9A.0;@7jN >uVU?-/Q54E/79-.0/cQ;@G/ED7-;J2D>8H@;J9"VhFV2D0/>DQzVF;@/EH<Q07`G2?A/]9-?>469A>h0HJ/
¦§,§8¨¹¤x¦0§¹,§(B¹,(R§(>.¤*¸(§,R§, B¦,
13250?k9A.W2?Q/E?=>8?-9-;<>8H Q;/E?A/ED9A;@>H/5>89-;@2D73=MQ/7B9-.>8Dp/D46250D9-/?-/Qf;JDp>]1350H@H !Z>uOF;J/?kW"a9-2q/7 132?-G50H@>89-;@2DI(/7k=0;J9-/r9-.0;<7M7-;JG=0H@;Je4E>89-;@2DIY7k;@PD0;Je4>8D9I2h7k9A>46H@/7?A/EGg>8;@DB9-2%hx/r2O/E?462G/Yt=0?-;@D4E;J=>HJH@V Q5/9-29-.0/R7"9A;D0/7-7Z;JD9A?-2Q54E/QdhFVj7k5?k1>46/9A/ED7-;J2D >8DQ`9-.0/RD0/E/Qj9-2?-/7k2HJO/7k.02?k9ZHJ/D0P89A. 7-4>8H@/7 4EHJ27k/Z9-2N,/69k9A;JDPRH@;@D0/7aF;@G;JH<>8?AHJVYF/6n=0H@;@4E;^9%9A;JG/7"9A/E=0=;JD0PgDF50G/?-;<4E>H7A4.0/GR/7 >8?A/Z21²H@;JG;J9-/QO>8H@50/ 7-;JD46/)9-.0/VK7-5/E?'>8Dj;@G=0?A>4t9A;@4>8H?A/7k9-?A;@469-;@2DK>=0?A2=x279-.0/B9A;JG/7k9-/=IY tY0N%.0;<4.KG57k9'h/>89'GR27"9
tYN%.0/E?A/ |;<7 9-.0/7-=>89-;<>8H?A/7-2H@59-;@2DIY9A2>uO2;<QDF50G/?-;<4E>H;JD7k9A>h0;@HJ;J9"V }8
aF/EO/E?>8H >59-.2?7.>uO/_7-2H@O/Q9A.0/v9A;JG/EWQ/=/DQ/D9SHJ5h0?-;<4E>89-;@2D /5>89-;@2D7 7k5446/7-7k1350H@HJV ;JD >yN%;<Q/bOu>?-;@/69"V281s=.V7-;@4>8Hd462D9-/6nF97E ?-2GR;@D0/D9>8G2D0P 9A.0/7-/_.>uO/ihx/E/D9A.0/v7"9A5Q;@/7U21 a4.FN >8?-9R>8DQs4628WN,2?Aq/E?7'N%.2K.>uO/l/>7-/Qd9A.0/g7k/O/?-/?-/7"9A?-;<4t9A;J2D 2D 9-;@G/7k9-/=sD289-/Q >h2O/hFV /GR=HJ2VF;@D0PR>H^9A/E?AD>9A;JDP8W[Q;J?A/469-;@2DK;JG=0H@;@4E;^9(Z%>8H@P2?-;J9-.G7,N%.0;<4.IYF132? 9-.0?A/E/EWQ;@G/ED7-;@2D>HL2N'7EY 57-/f>8HJ9-/E?AD>89-;@D0Po7-N//E=7K;@D />4. Q;@?-/4t9A;J2Di7-2T9-.>9j2D0H@Vz>Th>DQ/Q 7-VF7k9-/G 281l/5>9A;J2D7D/E/Q hx/K7-2H@O/Q>89l/>4.f9-;@G/K7k9-/E=z}um# .0/7-/7A4.0/EG/7YI21¶9-/EDp?-/E13/E?A?-/Q 9A2s>7g "!#¶X¶%$8Y }&8XY².>uO/ hx/E/D57k/Qo9-2T>8D>8H@VF7-/d>f?A>D0P/j21)=0?A2hHJ/G7;JD4EHJ5Q;JDPY¸132?/6n0>8G=0H@/Yr9-.027k/]Q5/j9-2f9-./]/'/469A7 21 7-50h7k9-?>9A/4E50?AOu>89-50?A/g>8DQf7-50?-1>46/R9A/ED7-;J2DfP?>Q;@/ED9A7)2Ds9A.0/9-.;JD0D;JD0P 281,4E2>89-;@D0P7g})(YMQ?A2=0H@/69 G289A;J2D7I2D.0/69A/E?A2P/ED0/2577-50h7k9-?>9A/7IN%;J9-.Gg>89-/E?A;<>8H<7I.>uOF;@D0P%N%;@Q0/EH@V)Q;*/E?A/ED9:/5;JH@;Jh?-;@50Gb462D9>4t9 >D0PH@/7)}+0Y%,>DQP?A>uOF;J9"VW:>8DQK7k5?k1>46/7-.0/>?,7k9-?A/7A7kWQ?A;@O/ED9A.0;@DK462>9A;JD0PlL2N'7}J-8X[DK>HJH219-.0/7k/ 7k9-5Q0;J/7EY07-=0?A/>Q0;JD0PRG289A;J2DK;<7,1>4E;JH@;^9>9A/Q`hFV7k=x/4E;^13VF;@D0Pg>9-.0;@Dj/ED0/?-P/69-;<4E>HJH@V7k9A>8hHJ/BN,/69k9A;JDPRH<>uV/?Y 2?.#xE!E/0Y281B9-.;@4qFD0/7-7132d2O/E?9-.0/ 7-50?-1>4E/]281B9-./ 7k5h7"9A?A>89-/Y>8Dz>=0=0?A2>4.p9A.>9`.>7 7-;JPD0;Je4E>D9'4E2G=059>9-;@2D>8HI>Q0Ou>D9A>8P/72O/? >H^9A/E?AD>9A;JO/Y07-HJ;@=`GR2Q/H@7l}^8YI~uX
[Dg462D9A?A>7"9m9-2B9A;JG/EW7-=0H@;^9-9-;@D0PY\d50HJ9-;@P?A;@QRG/69A.02Q07r>8?A/ 1>7k97-2H@O/E?7Y2?-;@P;@D>HJH@VlQ/O/EH@2=x/Ql132?r9-.0/ 7-2H@59-;@2D210/EH@HJ;@=9A;@4=IQ/7u# .0/E;@?M/7-7-/ED4E/;<79A257-/>'7-;@GR=HJ/,;^9A/E?>9A;JO/¸9-/4.0D0;45/>7²>'7-GR2F29-.0/?YD0289 >7²>)7-2H@O/E?Y2D>)7-/50/ED46/,2814E2G=059>9-;@2D>8HFP?A;@Q7I9A2)?A/Q0546/,.0;@P.13?A/50/D46V/?-?A2?m462G=2D0/ED9A7²2D >=>8?-9-;<4650H<>8?²P?A;<Q}^uu# ./EVB.>uO/h//EDR>8=0=0H@;@/Q9A2;@D46?A/>7k;@D0PHJVBQ;JCg4E50HJ9¸7-V7"9A/EGg7:281=IQ/7m>8DQ>?-/ D02NT4E2GG2D0H@VB57-/Q9A2Z7-2H@O/9-.0/ Q;<7-4E?-/E9-/>D>8H@2P50/7:>8?A;@7-;@D0P%13?A2Gv>'N%;<Q/,O>8?A;J/E9"VZ210L2Nc=0?A2h0H@/EGg7 132?N%.0;<4.Gg>65"2?,;JG=0?A2O/GR/D9A7r;@D/EC46;@/ED4EV>8DQ?-2h057k9-D0/7-7>8?A/hx/E;@D0PR>4.0;@/EO/Q]}J 0YYx&Y7( # .0/98050H@H:=0=0?A2un;JGg>9A;J2Da9-2?A>P/;<8)a=MOu>?-;<>8D9I219A.0/\]50H^9A;JP?-;<QBG/69A.02QB132?:D02D0H@;@D0/>?I/5>89-;@2D7 }J+um;<7;@G=0HJ/G/ED9-/Qj9-2`7k2HJO/l>7-50;^9>8h0H@/eD0;^9A/RQ0;/E?A/ED4E/l>8==0?-2un;@Gg>9-;@2D`9A29-./l9A;JG/6W[Q/=/DQ/ED9 H@50h0?A;@4>9A;J2Ds/5>9-;@2D7132?B9-.0/L2N 281 Q?A2=0H@/697B2O/E?B7-50h7k9-?>9-/7B/E;J9-.0/?N%;^9A.f2?N%;J9-.0259N/E9k9-;@D0P >DQ>u2?g9-./ =0?A/7-/ED4E/]281B9A2=x2P?>8=.0;@4j./69-/?-2P/D0/E;J9-;@/79?Glhx/Q0Q/QUN%;^9A.0;@Dz9-./s>8H@P2?-;J9-.G ;<7K>D /ECg4E;J/D9Y%>Q0>=9-;@O/ 9-;@G/6W[7"9A/E=0=0;@D0PU7-4./EG/Y,h>7-/Q 2DA@/E50DCB7d7-/4E2DQFW2?Q/?=0?A/Q;<4t9A2?-W4E2?A?-/4t9-2? G/69A.02QT}J,uYN%.;@4. >59-2Gg>9-;<4E>HJH@VK>Q75"57k9A79-.0/l9-;@G/7k9-/E=s7k50h=5"/4t99-2K>7k=x/4E;^e/Qd/E?A?A2?%9A2H@/E?>8D4E/ /D>8h0H@;@D0PD 9-2hx/2=9A;JG;<7k/Q`;JDd>462D9A?-2HJH@/QGg>8D0D0/?
4E2G=>469,Q/?-;@O>9A;J2Dg2819-./P2O/E?AD0;@D0P/5>89-;@2D7r9-2lhx/Z7-2H@O/Q;@7=0?A/7-/ED9-/QR132? 462GR=HJ/E9-/ED/7A7EY 132HJH@2N/QdhFVdQ/69>8;@H@7)2819-.0/R50DQ/?-=;JD0D;JD0P`DF50G/E?A;@4>8H²G/69-.2FQC7>Q2=9A/Q]132?Z9-.0/;J?7-2H@59-;@2DI# .0/ \d5H^9A;JP?-;<Q 7-2H@O/?s;<7se?A7k9fOu>HJ;<Q0>89-/Q >8P>8;@D7"9f> ?>8DP/p21K>8D>8H@V9A;@4>8H>8DQ =0?A/EOF;@257-HJVW=050h0H@;<7k.0/Q DF50G/E?A;<4E>8Hl=0?-/Q;<4t9-;@2D7j132?Q?-2=0H@/69 7-=0?-/>Q;@D0P=0./ED02GR/D>>8DQS9-.0/D_/6nF9-/DQ/QS9A2 7-;JG50H@>89-/p> O>8?A;@/69"V217-=0?A/>Q0;JD0PZL2N'7m2DR7-50h7k9-?>9A/,N%;J9-.Q0;/E?A/ED9rN,/69k9>8h0;@H@;^9"Vl4.>8?>4t9A/E?A;@7k9-;<4E7m>DQ4E2D9A>;JD;JD0P 9A2=x2P?>8=.0;@413/>89-50?A/7
~%\E¸#F@G?\E¸# -+HGIc\s&)(G?9I
8²;@P50?A/s`;<7g>s7A4.0/EGg>89-;<4`7-D>=W7-.029281Z9-.0/`L2N281)>Q?A2=0H@/69YKJLNMPOQ9ORG6Y:2O/E?> 9A2=x2P?A>=0.0;<4 13/>9-5?-/YFS;NMEOTQtY%21R>8G=0H@;J9-5Q/LSVUf>8DQ H@/ED0P9-.7XW9Y >DQ[Z\YyN%;^9A.i?A/7-=x/4t9`9-2o9-./?A/613/?-/D46/ 4E22?AQ0;JD>89-/]NMEOTQ2D>7-50h7k9-?>9A/K;@D4EHJ;@D0/Q>9g>8D>8D0PHJ/X^S9-29A.0/j.02?A;*2D9A>H²9A7g>7k=x/469R?>9A;J2 _a`
î :¦íq>¨ °q.Ȳ¦."¦ ²¨
Z
X
Y
T
T
S
0
W
α
(0,0)
g
S(X,Y)
L
1
(X ,Y )
1
H(X,Y,T)
×ÔÕà1ØlÛaÑ>òÊGÓ5ålÑjÞ8ßÖaÔÕÓç&Фß>Úlöç&ålÜ>ÖÜ>éÖaålÑê.ÜBë Ü>éßqäGÛaÜ1ÚlÙUÑ5ÖBúùá 3â5ú¤ÜSè,Ñ5ÛßÈÖaÜ1Ú#Ü1àÛ6ß>ÚlåGõ(úá Èâ5ú¤Ü>Ð¼ß ç&ؤïlçLÖ&Û6ßÖaÑÔÕФÓÙUÔÕÐlÑBäßÖßÐßФà1ÙÕÑùÖaÜXÖaå¤ÑålÜ>ÛaÔjÜ1Ð(Ö6ßÙú¤ç&å¤ÜBëìÔÕФàqßXç&ÑjÓÖaÔÕÜ1ÐÖaålÛaÜ1Ølà1å·Öaå¤ÑäGÛaÜ1Ú¤ÙÕÑÖ é@Ü>ÛìÖaå¤Ñ3çaß,Ñ
ÜéÓjÙUßÛaÔJÖ¹õ(úlÔÕÐÖaålÑ ÚlÙUß>ФÑ!ßÐ.ä8Öaå¤ÑÚ¤ßÛ6ß>ÞqÑ5ÖaÑÛaçälÑø¤Ð¤ÔÕФàXÖaå¤Ñ3ÖaÜ>Ú#Ü1à>Û6ßÚ¤å(õ(ò
\]>?A>D0P2D0;I/ x/4t9A7B>?-/;@PD2?A/Q]7-2g9-.>9)9-.0/7k5?k1>46/9A/ED7-;J2DIYmY;@7)>8H<7k2K462D7"9>8D9# .0/lG29-;@2D]21 9A.0/Q?A2=0H@/69%;<7%P2O/E?AD0/QghFVg9A.0/B9-;@G/6W[Q/E=x/EDQ/ED9Z!Z>uO;@/E?-W"a9-2q/7/5>89-;@2D7YOF;*
e "!#
!
R%$&# '()# *
` +
(-,.$
f
(-/0# $
e 1 O k
( '#
`
-=O X~
N%.0/?-/
#
`
#
O32bOTZ f>DQ
,
>8?A/9-.0/UL50;@QwO/EH@246;J9"V >8DQ =0?A/7A7k5?-/U?A/7-=/4t9-;@O/HJV >8DQ1
`
1 7-;JD;^9O- O + 4627^ 6YN%.0/E?A/41 ` ,
'
+6587:9
/
;<7Z9-.0/>446/EH@/E?>9A;J2D]Q50/9-2`P?>uOF;^9"V#²>8qF;JDP J U >79-.0/ 4.>?A>4t9A/E?A;@7k9-;<4BQ?-2=0H@/69'9A.0;@4qFD0/7-7>8DQ W U >7'9-.0/H<>9A/E?>8HI/EnF9-/ED921¸;^97'132F289A=0?A;JD92DK9A.0/7-50h7k9-?>9A/Y ;J9%;@7 >7A7-50G/Q9A.>9<;
`>=@?
A
?
;<7 7kGg>HJH7-29-.>9 9-.0/!Z>uO;@/E?-W"a9-2q/7r/5>9A;J2D7 4E>Dh/7-;JG=0H@;Je/QK57-;JD0P 9A.0/'H@2D0PWXN >uO/Y2?'EtXYF>8==0?-2un;@Gg>9-;@2DM6'H@H>8D>8H@VF7-/7>8?A/%=/?k132?-G/Q>DQ?-/7k5H^97¸=0?A/7-/ED9-/Q ;@D`9-/E?AGg7 281¸462?-?A/7-=2DQ;@D0PlD2DW[Q;@GR/D7-;J2D>8HKHJ2N,/E?%4E>7k/7Ou>?-;<>8hHJ/7EYF9A.0/q/Vg2D0/7 h/;JD0PCB
1 ED OGFVO
`
JL<MEOQ9ORj
J U
OH7:ED OGF=
`
SFNMEOTQ
J U
O ED OGF=
`
NMEOTQ
W U OJI
` K J U
O
L
`
WbU
,
;
O EM OGN3OPO
;
`
#
O32bOTZ RVU
W U O
`
R
R U OARVU
`
f WKU
;RQ
O
N%.0/?-/9-./9A;JG/'7A4E>HJ/HR U ;<7²9-.>89rQ/E?A;JO/QlhV&Z?A4.>?AQ}~ -u132?²9-./,H@/EO/EH@HJ;@D0P)2817k50?-1>4E/,Q0;@7k9-50?Ah>D46/7E !289-/l9-.>89B7-;@D46/l9-.0/R7k50h7"9A?A>89-/7-50?-1>4E/;@7ZP;@O/D]hFV9A.0/9-2=x2P?A>=0.FV1350D4t9-;@2DS7:EDCOTF=7-2g9-.>9)9-.0/ Q?A2=HJ/E9,H@;J/7 h/E9"N//EDI
`
7B>DQUI
`
1
$
70*DQ/E?,9A.0/>8hx2O/)D02DW[Q;@G/ED7-;J2D>8H@;<7->89-;@2DIYD0/EPHJ/4t9A;JD0P 9A/E?AG7 21
V;
/
,2?,.;JP.0/E?YF9-.0/!Z>uOF;J/?kW"a9-2q/7?A/Q546/B9-2CB
!
/
M
!
I
/
`
!
L
!
D
+JWYX
;
7k;@D;^bO <:
! /
N
!
I
/
`
!
L
!
F
O
!
L
!
I
` + W4X
¦§,§8¨¹¤x¦0§¹,§(B¹,(R§(>.¤*¸(§,R§, B¦,
N%.0/?-/ W4X
` A
?
;@7Z9-./,2DQ]DF50Gh/?)G/>7-50?-;@D0P9-.0/?A/EH<>9-;@O/;JG=x2?-9A>D46/281P?A>uOF;J9A>89-;@2D>H9-2 7-50?-1>46/9A/ED7-;@2Df132?46/7E:# .0/7-/g/5>89-;@2D7>?-/7-2H@O/Qf7k50h=5"/4t9l9-2j9-./K462DQ;J9-;@2Df281%D02W7-H@;J=c>9l9-.0/ 7-50?-1>46/B21M9-./7k50h7"9A?A>89-/>8DQ/E?A2l9>8D0P/ED9-;<>8HI7k9-?A/7A7 >9%9-./B13?-//B7-50?-1>4E/)21:9A.0/Q?A2=0H@/69YD>8G/HJV
EM OGN
`
-=O-: >89I
`
7 >8DQ !
M
!
I
`
!
N
!
I
`
- >89I
`
1
$
7 "(6
?95>9-;@2D7H<:²>8DQ :;JD9-/P?>9A/Q9"N%;<46/ N%;J9-.R?-/7k=x/469m9-2IZ2O/E?²9-./ Q0?-2=0HJ/E9:9A.0;<4qD/7A7I }7 OT1
$
7E >DQY07-50h5"/469%9-29A.0/>8hx2O/)h250DQ0>?-VK462DQ;^9A;J2D7YV;@/EH<Q
M
`
!
L
!
D
+ W4X
;
7-;JDF^
*
I
+
77
~
I
+
77
+
1
*
O +
N
`
!
L
!
F * I
+
77
~
I
+
77
+
1
* '
,
# .0/'9-;@GR/EWQ0/E=x/EDQ/D9H@50h0?A;@4>9-;@2Dg/5>9A;J2D132?r9A.0/'/O2H@59A;J2Dg2819-.0/)Q?A2=0H@/69r9A.0;<4qD/7A7¸;@72h09A>8;@D0/Q hFV4E2D7-/E?AOF;JD0PRGg>7A7,9-.0?A25P.0259 9-.0/BL2NYOF;
!
1
!
` + ( '
O k-
N%.0/?-/
`
EM OGN Y
I[D9-/P?>9-;@D0P/5>9-;@2D7 <+:¸>8DQ <, ¸9-2132?-G
>8DQ7k5h7"9A;^9A59-;@D0P;JD9-2 /5>9-;@2D "- VF;J/H@Q7 9-.0/B132HJH@2N%;@D0P/5>9A;J2DK;@D`9-/E?AGg7 281²9-.0/=?-/7-7-50?A/e/H@Q
L
!
1
!
`
!
!
D
1
Q
!
L
!
D
+ W4X
;
7-;JDF^ * $
!
!
F
1
Q
"!
L
!
F * '
k
[D 4E2GG2D]N%;J9-. 9-.0/N2?-q`281r29-.0/?A7R}+Y ,ur>Q;<7N5"2;JD;JD0PK=0?A/7A7k50?A/B9-/?-G ;<7Z/EG=0H@2V/Q`9A2K>HJH@/EOF;<>9-/ 9A.0/ 7-;JD0P50H<>8?A;^9"Vp>89N/E9k9A;JD0PpHJ;@D0/7# .0;<7G2Q/HZ>7A7-50G/7g>9-.;JD=0?A/4E50?7k2?eHJG{281)9-.;@4qFD0/7-7 1V2 >.0/>Qp21Z9A.0/]7->G/j>DQo?-/H@>89-/7R9A.0/d2h7k/?-O/Qp4E2D9A>4t9>8DPH@/K132?g=>8?-9-;<>8H@H@VTN/E9k9A;JD0Pc7-V7"9A/EGg7R9-2 ;@D9-/E?AG2H@/4E50H@>?¸132?46/7¸9A.>9,h/462G/Z;JG=x2?-9A>8D9132?H@;45;@Q07281M7-50h0G;<46?A27A462=0;@4%Q0;JG/ED7k;@2D7)},uXF# .0/ Q;<7<5"2;@D0;JDPR=?-/7-7-50?A/Z9A/E?AGjY13tY;@7 P;JO/EDf}~0tMhFV
13
`
+
7'E5
+
7'k
+
4E27
1
2
+ 5 ;
/
1V2
1 *"!
+
1V2
1 *"#
O k~
N%.0/?-/$f>8DQ 5 >8?A/Z9A.0//EnF=x2D/ED9A7 21²9A.0/;@D9A/E?>469-;@2D`=29-/ED9A;@>HM>8DQ% DCOTF%;@7 9A.0//50;@H@;Jh0?A;@50G 4E2D9A>4t9Z>D0PH@/!'29-/9A.>9)9-./Q;<7N5"2;JD;JD0PK=0?A/7A7k50?A/9A/E?AG ;<7GE/E?A2g132?)>1350HJH@VjN/E9k9A;JD0Pj7-VF7k9-/G 7k;@D46/
`
î :¦íq>¨ °q.Ȳ¦."¦ ²¨
# .0/r=0?A/7A7-50?-/²e/H@Q)9A.0?A250P.02599A.0/Q0?-2=0HJ/E9I;@7M2h9>8;@D0/Q)hFV);JD9-/P?>9A;JD0P /5>9-;@2D& N%;J9-.?A/7-=x/4t9 9A2YIN%./E?A/%9-.0/)462D7"9>8D921I;JD9-/P?>9A;J2DX7k/E9k9A;JD0P9-./=0?A/7A7k5?-/'Q0>89-50Gy9A2>9AG27-=0.0/E?A;<4m;<7P;JO/EDghFV
L
` + ( /
1
$
77
+
1 2D8I
`
1
$
7 O k
N%.0/?-/ 9A.0/
(
/
1
$
77²9-/?-Gy;<7¸9-./7kGg>HJHx7kH@2=x/'>8=0=?-2un;@G>89-;@2D9-29A.0/7-50?-1>46/'4E50?AOu>89-50?A/%2819A.0/'9A.0;JD eH@GjY0H@/>Q0;JD0PR9A2
L
` + ( /
1
$
7
+
13 $ WYX
4627 ^91
$
7
+
I:
'
k :
9;<7g;@GR=x2?-9A>D9R9A2cD0289A/`9-.>897k;@D46/j9A.0/dH@50h0?A;@4>9-;@2DU>8==0?-2un;@Gg>9-;@2Do;@7g132?AG>HJH@VTO>8H@;<Qo2D0H@Vc132? L2N'7132?gN%.0;@4.9A.0/`13?-//]7k50?-1>4E/`7-H@2=x/`;<77-Gg>8H@HY¸;J9A7g>4E465?A>46VT46250H<Qphx/]>/469-/QohFVT9A.0/d7"9A/E/E= 7-HJ2=/7B2819-.0/9A2=x2P?>8=.0;J/7B462D7k;<Q/E?A/Qs.0/?-/ @'2N,/EO/?Y?A/4E/ED97"9A5Q;@/7g}~0Y:~ >DQs4E2G=>?-;<7k2D N%;J9-./6n=x/E?A;JG/D9}~FY 7-.02NpH@50h0?A;<4E>9A;J2Dl9A.0/E2?-VB9A2h/%?A/EGg>?-q>8hHJV?A2h057"9¸;@D7-54.46;@?A4E50Gg7"9>8D4E/7 =0?A2OF;@Q0;JD0P)9A.0/>7A7-2F4E;@>89-/Q$%/VFD02H<Q07r>DQ+,>8=0;@HJH<>8?AVDF50Gh/?A7r>?-/%D289¸9-2F2H<>8?AP/8!'289A/>8H<7-2Z9A.>99-.0/ 132?-G 21²/5>9A;J2D]k/D7k5?-/7,9-.>89%9-.0/ IgQ/=/DQ/D46/;@D`/5>9A;J2D " %.>7 D02g;@DL5/ED4E/2DK9-.0/ /O2H@59A;J2DK21M9-./BeHJG9-.0;<4qFD0/7A7%>8DQK;J9';<7,9-.0/?-/E132?A/)2G;^9-9-/QK13?A2G7-50h7-/50/D9>8D>HJV7-;@7
?A/EOF;@257 N,2?Aq/?A7 .>uO/7k;@G=0H@V`7-50h7k9-;J9-59A/Qd/5>9A;J2DL" %,;JD9A2/5>89-;@2DL",9A2gVF;J/H@Qj13250?-9-.W 2?AQ/? 9-;@G/6W[Q/E=x/EDQ0/ED9ZH@50h0?A;@4>9A;J2D`/5>9A;J2D7%=050?A/EH@V`;JDj9A/E?AG7281²9-.0/leH@G9A.0;@4qFD0/7-7Y1M@2N/O/E? ;@Dd9-./l=0?A/7-/ED9Z7k9-5Q0V`;J9B;@7'13250DQ >QO>8D9>8P/257%9-2K7-2H@O/B9A.0/4E250=HJ/QdD02DHJ;@D0/>?/5>9-;@2D]7-/69Z132? 1U>DQ
L
Y¸/5>89-;@2D7 k7>DQ[" %?-/7k=x/469-;@O/EH@VYr7-;@D46/`9-./]7k/462DQFWX2?AQ0/E?Q;/E?A/ED9A;@>H%2=x/E?>9-2?A7 >?-/g7-;JG=0H@/E?9A2]Q;<7-4E?-/E9-;<7k/g>8DQ>j7-50h7k9-?>9A/gN/E9k9A>h0;@HJ;J9"V]1350D4t9-;@2D 7DCOGF=6YI132?lOu>?-;<>8hHJV]N,/69-9A>h0HJ/ 7-50h7k9-?>9A/7Y04E>8D`hx/;JG=x27-/Q`N%;^9A.02509';JG=x27-;JD0Pg4E2D7k9-?>8;@D9A7,2D`;^97'Q;*x/?-/D9-;<>8h0;@HJ;J9"V }+
?b5>89-;@2D7k7%>8DQ " %%>8?A/7-2H@O/Qd7-50h5"/4t9Z9-27-VFGRG/E9-?AVKhx250DQ0>8?AV`4E2DQ0;^9A;J2D7>89'9-.0//QP/7 21:9A.0/462G=0509A>9A;J2D>8HIQ02Gg>8;@DIY0D>G/EH@V
!
L
!
`
!
1
!
` -=O k
N%.0/?-/;@7,9A.0/D02?-Gg>8H9A2R9-.0/7-50h7k9-?>9A/)hx25DQ0>8?AV
0%!Z*\E?$%-+H Ic\E?#F@)&)(
6#X 4!EE6X4!EX
HJHr=?-2h0HJ/Gg7>8?A/7-2H@O/Q 2D>d7T5>8?A/462G=0509A>9A;J2D>8HrQ02Gg>8;@DsN%;J9-. DCOTF
`
-=OE7 <- OE 57-;@D0P>g\d50HJ9-;@P?A;<Qd>8=0=0?A2>4.I# .0/eD0/7k9P?-;<QjHJ/O/Hb3;X/l9-.0/>4t9A5>8H:P?A;@Q`2DdN%.0;@4.j9A.0/7-2H@59A;J2D`;<7 Q/7k;@?A/Q m.>7j~
$
mD2FQ0/7r;@D/>4.Q;@?-/4t9A;J2D7k29-.>899-./Z7-=>9A;@>H462WX2?AQ0;JD>89-/7¸281;J9A7P?A;@Q=x2;@D9A7
>?-/P;JO/EDThFVSD `
9
/
>8DQ F `
9
/
;JDc9A/E?AGg721%9-.0/KeD0/KP?A;@QpQ/ED7-;J9"V=>8?>8G/69A/E?:[D
9A.0/=?-/7k/D9B7"9A5QV
}&=OE-m462?A?A/7-=2DQ;@D0Pg9-2`P?A;<Q07>89Z9A.0/leD0/7k9)H@/EO/EHmN%;J9-. hx/69"N,/E/D\&:g>8DQ -~ D02Q/7¸;JD/>4.Q;J?A/469-;@2DI ?b5>9A;J2D7F"²>8DQk %m>8?A/ >=0=0?A2un;JGg>9A/Q7k=>89-;<>8H@HJV57-;JD0P4E/ED9-?>8H Q;*/E?A/ED4E/7}Y~ &uIHJ/>Q;@D0PR9-29A.0/B132H@H@2N%;JD0P7-/462DQFW2?Q/E? >44650?>9A/7k=>89-;<>8HIQ;<7A46?A/69-;<7A>9-;@2D7
!
1!#"
!
`
/
1
Q
%$
'&
"
L
"
+
L
#"
+
1
Q
($
9
&
"
L
#"
+
L
9) "
$
1
Q
$ *"
&
L
*"
+
L
#"
+
1
Q
$ *"
9
&
L
#" +
L
#" 9)
+
WYX
;
7-;@D;^ 1
/
*"
1
"
+
1 9)
"
~
*
$
/
¦§,§8¨¹¤x¦0§¹,§(B¹,(R§(>.¤*¸(§,R§, B¦,
L
#"
$
/
1
"
$
7
"
$
1 9
"
$
7 9)
"
$
1 *"
$
7 #"
$
1 *" 9
$
7 #" 9)
+
1 #"
$
7 *" $
+
' 5
+
"
+
4627
1
2
+
5 ;
/
1V2
1#" *"!
+
1V2
1#" * #&
+
WYX 1 *"
$
7 #" 4E27=^
$
/
` -=O k7(6
132?/>4. 3;XY5;@D9-./4E2G=059>9-;@2D>8HZQ02Gg>8;@DIY,N%.0/E?A/\ {;<79A.0/s7-=>9A;@>H;@D46?A/EG/D9 ` ~9
132?Z9A.0/eD0/7k9P?A;@Q HJ/O/H4Z>8DQ]9-.0/
Q
$
&
"
Y
Q
$
#"
&
9A/E?AG7B>?-/9-.0/#xT'66[!B>7A7k246;<>9A/QdN%;J9-.
9A.0/Q;<7A46?A/69-;<7A>9-;@2DM @'/?-/?A/7-50HJ9A7'>?-/2h9A>;JD/Qj57-;JDPx.02?-D0;J9A7-q>uV>>8DQ&/?k9A2 E;B7}~8 %!t¶¶ ,T ! E¶%$ ',132?%N%.0;@4.K9-./=0?-/EW1>4t9A2?7;@Dj/5>89-;@2D "&:,4E>8D`hx/7k.02N%DK9-2ghx/P;@O/DKhFV
1
Q
$
&
"
`
~
1
/
*"
1
/
"
1!
"
$
1*"
O k+
N%;J9-.d7-;JG;@H@>?%/6n=0?A/7A7k;@2D7132? 9-.0/
Q
$
9
&
"
Y
Q
$
*" &
>8DQ
Q
$
*"
9
&
9-/?-Gg7
#08- 4!EE6X4!EX
"Q/>8H@HJVY²9-;@GR/EW7k9-/=0=0;@D0Pf7-4.0/G/7l132?R9-.0/`DF50G/E?A;@4>8H 7k2HJ509-;@2Dp21'9-?>8D7k;@/ED9lL2N'7R7-.0250H@QTh/jhx289-. E<E>8DQ]AE-8"E# .0/'132?AGR/?/D7k5?-/7m9-.>897-G>HJH9-;@G/Z7k9-/E=7>8?A/'>uO2;@Q0/QRN%.0/ED9A.0/Z7-2H@59A;J2D O>8?A;@/7 7kH@2N%H@VRN%.;JH@/)9-.0/BH<>9k9A/E?%7-.0250H<QK/ED>h0HJ/Z9-./)/?-?A2?9-2Rhx/462D9A?-2HJH@/Q9A.0?A250P.0259,9-./B7-2H@59A;J2D =0?A246/7-7\d27"9,=0?A/EOF;J257rDF50G/?-;<4E>H7k9-5Q;@/7281IH@50h0?A;<4E>9A;J2DWX9"V=x/'L2N'7r.>uO/%1324657A7k/Qg2D;JD4E?-/>7-;JDP 9A.0//ECg4E;J/D46V)281F9A.0/7-2H@59-;@2D>DQYE;@D=>?k9A;@4E50H<>8?Y62O/?A4E2G;JDP9A.0/7k9A>h0;JH@;J9"VZ?A/50;@?-/G/ED9132?:/6n=0H@;@4E;^9 7A4.0/EG/79-.>89I9-./¸9-;@G/7k9-/E= 7-.0250H<Q)h/
6xW>8D;@G=0?>4t9A;@4>8H8?A/7k9-?A;<4t9-;@2DBN%.0/D/EO/EDBG2FQ0/E?>9-/ 7-=>9A;@>H¸?-/7k2HJ509-;@2D;@7B/GR=HJ2V/QMaF/O/?A>Hm7k9-5Q;J/7.>uO/R>4.0;J/O/Q]9A.0;@757k;@D0Pd>8HJ9-/?-D>9-;@D0PWQ;@?A/4t9A;J2D ;@G=0HJ;<46;J9<Z(,>8H@P2?-;J9-.G7/EG=0H@2VF;JDP>8HJ9-/?-D>9-/)7kN,/E/=7;JD`/>4.Q;@?-/4t9-;@2DK7k29-.>89,2D0H@V>h>DQ/Q 7-V7"9A/EG_281/5>9A;J2D7:D0/E/Qlhx/,7-2H@O/Ql>89²/>4.9A;JG/%7"9A/E=`}X# .0/7k/) "!#¶X¶%$ >8H@P2?A;J9-.0Gg7:/ED>h0HJ/ 9A;JG/7k9-/E=7'7k/O/?A>H2?AQ/?A7,21²Gg>8PD0;J9-5Q/BH<>8?AP/?9A.>8D 9-2ghx/57-/Q`7k54E46/7-7k1350H@HJV
80/Nz7"9A5Q;@/7r.>uO/'?A/E=x2?-9-/QR>89k9A/EG=9A79-2R462D9A?-2H9-./Z>44650?>4EVl21x9A;JG/EW7k9-/=0=0;@D0P7A4.0/EG/7>8DQ;^9 ;<7r2D0HJVO/E?AVl?A/46/D9-H@Vl9A.>9 >8Dg/7k9-;@Gg>9A/2819A.0/H@24E>8H9-?A50D4>9-;@2Dg/?-?A2?.>7h//ED57-/QN%.0/ED7-2H@OF;JD0P /5>9-;@2D7 281²9-.0/132?-G "7 >8DQ k :9A29-/7"9'N%.0/E9-.0/?%9-./EVK>?-/Bhx/E;@D0P7-2H@O/Q9-2>g7-=x/46;Je/QK/?-?A2? 9A2H@/E?>8D4E/}~6&# .0/BH@>89k9A/E?B7 7A4.0/EG/Y.2N/O/E?YQ02/7 D0289 2=09-;@GR;<7-/Z9A.0/)9-;@G/B7k9-/=132?%>lP;JO/ED/?-?A2? 9A2H@/E?>8D4E/# .0/=?-/7k/D9BN,2?Aq`>Q0Q?A/7A7k/7'9A.0;@7);<7-7-50/RhFVd57k;@D0PK9-;@G/6W[7"9A/E=0=0;@D0P`h>7-/Q 2DsHJ24E>H:/?-?A2? /7"9A;JGg>9A/7 13?A2G >8Dj;@G=0HJ;<46;J9Y7-/462DQFW2?Q/E? G/69A.02QjN%.0;<4.d?-/Q546/7 9-2 @'/E5D B7G/69-.2FQc}J,:;@Dj9-.0/ 4>7-/21,>e0n/Q 9-;@GR/7k9-/=II# .0;<7B57-/7B>8D/7k9-;@Gg>9-/219-./RH@24E>Hm9-?A50D4E>89-;@2Ds/E?A?-2?Yx2h9>8;@D0/Q 13?A2G 9A.0/gQ;/E?A/ED46/h/E9"N//EDs9-.0;<77-2H@59-;@2Df>8DQ>8D/6n=0H@;<46;J9B=?-/Q;@469-2?Yx9-2j;@D46?A/>7k/l9-.0/9A;JG/g7k9-/E=;@Df> 4E2D9-?A2H@HJ/Q`G>D0D0/?N%.0;@H@7k9)>899A.0/7A>8G/9A;JG/lG;@D0;@GR;<7-;JD0P9-.0/R462G=0509A>9A;J2D>8H:/6n=x/ED7-/>7-7-246;<>9-/Q N%;J9-.j?A/E=x/>9A/Q9-;@G/7"9A/E=`1>8;@HJ5?-/
# .0/j>8=0=0?A2>4.f?-/5;J?A/7l/5>9A;J2DA"& 9-2sh/`?-/N%?-;J9k9A/EDp>7>s7kV7k9-/G 21ZQ;*x/?-/D9-;<>8HJW>HJP/Eh0?>8;<4 /5>9-;@2D7 ;JD`9-./B132?AG
!
1 *"
!
` 1!#" O
L
*"6O1!
" O
L
" O1!#"
O
L
*"
HO k,
î :¦íq>¨ °q.Ȳ¦."¦ ²¨ >4E465?A>89-/);JD`9-;@G/>8DQ`=0?A246//Q07,hFVK7-2H@OF;JD0P 1 ! *" ` / 1 ! 9) *" $ + / 1 ! #" $ ! " $ 1 ! #" O L ! *" O1 ! " O L ! " O1 ! #" O L ! *" O X~6- N%.0/?-/ ` &
>DQd9A.0/7k50=x/E?7A46?A;J=9 TQ/ED289-/7ZQ>9A>K>9)9-.0/ 9-.]9-;@GR/7k9-/E=Mx94E>D]hx/R7-.02N%DIY OF;<>K>`#:>uVFH@2?)7k/?-;@/7)/6n=>D7-;J2D]281/5>9A;J2D ~ - 6Y9A.>9B9A.0/HJ24E>HM9A?-5D4E>89-;@2D /?-?A2?.<I#F?H%132?)9-.0/ =0?A/Q0;@469-2?'7k9A>P/);@7'P;@O/EDKhFV
WKR ! "
#" ` ! / ! k $ & ! Q 1*" ! Q $ #$&% O X~
N%.0/?-/)9A.0/B9-.0;@?Q`Q0/E?A;JO>9A;JO/)9-/E?AG ;<7%/EO>HJ5>89-/QK>89%>R9A;JG/ ' N
! O ! t # .0/7-2H@59-;@2Dj7"9>8P/Z;<7%;@GR=HJ;<46;J9>8DQ`;<7%>8H<7k2g7-/4E2DQFW2?Q/?,>4E465?A>89-/);JD`9-;@G/Y0P;JO/ED`hFV 1 ! #" + ! ~ 1 ! #" O L ! *" OT1 ! " O L ! " O1 ! #" O L ! *" ` 1 ! *" $ ! ~ 1 ! #" O L ! *" O1 ! " O L ! " OT1 ! #" O L ! #" X~~ 132?%N%.0;@4.`9A.0/I#F?;@7 <WbR( #" ` + ~ ! Q ! Q 1 #" ! Q $ #
O\)" N
! O ! ' X~8 Z7ZQ/7-4E?-;@hx/Qd;@Do}J,uY9-.0/>7A7k5GR=09-;@2Dd9A.>9Z9-.0/9-.0;@?QFWX2?AQ0/E?Q/?-;@O>9A;JO/9-/?-G O>8?A;@/7'2D0H@Vj>K7kGg>HJH >GR250D9%2O/E?,9A.0/B9-;@G/7"9A/E=j/ED>h0H@/7,9-./WKR(v9-2h//7"9A;JGg>89-/Q`hFV WKR #" ` $ ~+* -, ,/. 1 ! #" + 1 ! *" ' X~:
82H@HJ2N%;@D0Pj}~:(XYF9A.0;<7%/6n=0?A/7A7k;@2D;@7 57-/Q9-22h9A>;JDj>8D`/7k9-;@Gg>9-/B21M9-./)2O/E?>8H@H9A?-5D4E>89-;@2D`/E?A?-2?hFV eDQ;JDP;J9A7C?546H@;<Q/>DD02?AGjY10'WbR(208YtN%.0;@4.;<7M9-.0/D57-/Q)9-2Z7k=x/4E;^13V)9-./D0/En9:9-;@G/7k9-/E=B9A.0?A250P.0259 9A.0/Q2Gg>;JDIY ! / YOF;<> ! / ` -' ,: ! R W 0 WKR(20* & O X~
;J130'WbR(2054 R WYN%./E?A/>7;^160'WKR2087[R WU9A.0/;J9-/?A>89-;@2D];<7Z?-/7"9>8?-9-/Q]N%;^9A. .>H^1¸9A.0/R4E50?A?-/D9 9A;JG/7k9-/E=M
# .0/B;JG=0H@;<46;J97kV7k9-/EG281²>HJP/Eh0?>8;<4Z/5>9A;J2D7 "(6,>8DQ]~~ ,>8?A/B7k2HJO/Q>89%/>4.K9-;@G/7"9A/E=j57-;JD0P 9A.0/8ZaKGl5H^9A;JP?-;<QG/E9-.02QjQ/7A46?A;@h/QKhx/EH@2N
=+9d$t;: 7E
¦§,§8¨¹¤x¦0§¹,§(B¹,(R§(>.¤*¸(§,R§, B¦,
P?-;<QK9-29-./D0/6nF9>DQ`N%.0/E?A/5<
U
Q/D0289A/7 9A.0/l462>8?7k/7"9%P?-;<QK>DQ <
A
9-.0/eD0/7k9x!'289A/B9-.>89Y132?>HJH 7-2H@59-;@2D7 =0?-/7k/D9-/QKH<>9-/?%9-./462>?A7-/7k9 P?A;@QKHJ/O/H <
U
;@7>K(n7(2D/7k2R9-.>9%9-./BeD0/7k9P?-;<QH@/EO/EH
<
A
Y8;X/89A.0/'2D0/%2DN%.0;<4.R9-.0/7-2H@59A;J2DR;<7r>4t9-5>[email protected].>7~
A
$
%D2FQ0/7¸;@D/E;J9-.0/?Q;@?-/4t9A;J2D 2?Y0;@Dj9-/?-Gg7281²9-.0/leD0/P?-;<Q`=>8?>8G/69A/E?%;@D9-?A2FQ0546/Qd>8hx2O/Y!
`
W
$
\d5H^9A;JP?-;<Qd7k2HJ509-;@2D7'>?-/ ?A/E=x2?-9-/QdN%.0/E?A/9A.0/eD0/7k9BP?A;@QdH@/EO/EH²?A>D0P/7%13?A2G <
/
N%;^9A.]&:un=& RD02Q/7'9A2 <
N%;J9-.p-~ un-~ D02Q/7E7802?:/En0>8G=0H@/YE;J1>Z\d50HJ9-;@P?A;@Q7-2H@59A;J2D;@7²Q/7-;@?-/Q2Dl>l~ ,n~ ,P?-;<Q)9-./ED9A.0;<7:4E2?A?-/7k=x2DQ79-2 P?-;<Q <
Q
N%;J9-.;@D9-/E?AG/Q;<>9A/7-2H@59-;@2D72h9>8;@D0/Q2DgP?A;@Q7 <
U
k7(n(6tY < n0:m>8DQ <
/
<&:un=& hx/6132?-/ 9A.0/%eD>H7-2H@59-;@2D2D <
Q
;@7>4t9A5>8H@H@Vl2h09A>8;@D0/Q!'29-/%9A.>99-.0/'P?-;<Q7k=>46;@D0P>7-7-246;<>9-/QlN%;J9-. />4.`P?A;@QKHJ/O/H < ;<7 50D0;J132?AG >DQK/5>8H;@Djh29-.dQ;@?-/4t9A;J2D7%>8DQK;<7'7k;@G=0H@V
`
/
# .0/'D02D0H@;JD0/>8?,Q;@7A46?A/69A;@7-/QR/5>9-;@2D7 "(6¸>DQ ~~ ¸>?-/'7-2H@O/QR57k;@D0Pl>l462Gh0;@D>9A;J2Dg2819-.0/j8050H@H =0=0?A2unF;@Gg>9A;J2Dfa9A2?>8P/8Za=BG/E9-.02Qc>8DQ]850H@H\d50HJ9-;@P?A;@Qs9A/4.0D0;450/]}JYr0Y7FYr&Yr7(FYr+u 8²;@P50?A/)~;JH@HJ57"9A?A>89-/79-./Z7k9-?A54t9A50?-/Z281I9A.0/Z7-2H@59A;J2D=0?A2F4E/7A7r2O/E?r9A.0/Z13250?,P?A;@QHJ/O/EH<7¸9A.>9%>8?A/'57-/Q N%.0/DR>)7-2H@59-;@2D2DIY7A>uVYP?A;@QlHJ/O/EH <
Q
Y;/u~6,nu~6,0YE;@7²2h09A>8;@D0/Q6'DR;JG=x2?-9A>D9¸>Q0Ou>D9A>8P/2109-.0;<7
2
3
G
G
G
1
: Full Multigrid interpolation
3, 3
3, 2
3, 1
3, 0
0
G
Ncyc
G
: smoothing
: exact solution
G
G
G
×ÔÕà1ØlÛaÑ
ÿ
òÑjç&ÓÛaÔÕÚlÖaÔÕÜ>зÜéÖaå¤ÑפؤÙÕÙ.ÌãØ¤ÙJÖaÔÕà>ÛaÔUäÖaÑjÓ5ålФÔUñ(Ø¤Ñ ëìå¤ÑÐqç&Ü1ÙÕØlÖaÔÕÜ1ÐlçßSÛaÑÜ>ïlÖ6ß>ÔÕÐlÑBäqÜ1Ðqà>ÛaÔUä ëìÔJÖaå ÿ ФÜGägÑjçÔÕÐ8ÑjÔJÖaå¤ÑÛälÔJÛaÑjÓÖaÔÕÜ>Ðò ÛaÑjÚlÛaÑç&ÑjÐ(ÖaçÖaå¤ÑÔÕФÔJÖaÔUß>Ù#ç&Ü>ÙUØgÖaÔÕÜ1зÜ1Ð ßSÖÖaålÑÚgÛaÑjèGÔUÜ>ؤçÖaÔÕÞqÑçLÖaÑjÚú
µ
Öaå¤Ñ ç&Ü>ÙÕØlÖaÔÕÜ1мßSé@ÖaÑÛÖaå¤Ñø¤ÛaçLÖìø.ÐlÑà>ÛaÔUäöQÓ5õlÓÙUÑ>ú
À
ÖaålÑç&Ü1ÙÕØgÖaÔUÜ>Ðîßé:ÖaÑÛìÖ¹ëÜöÓõgÓjÙÕÑjçjúlÑÖaÓ1ò
9A/4.0D0;450/l;<79-.>89)>8Dd;@D0;J9-;<>8H²P50/7-7'9-29-.0/R7k2HJ59A;J2D >9)>P;@O/DdHJ/O/HM;<7=0?A2OF;@Q0/QjhFVK;@D9-/?-=x2H<>9A;J2D 21,9A.0/K7-2H@59A;J2Dc2Df9-./D0/6nF9462>?A7-/7k9HJ/O/H ,9R>8H@Hh59l9A.0/462>8?7k/7"9P?-;<QfH@/EO/H>je0n/QcDF50Glhx/E? 21'W[46V4EHJ/7B;@7>=0=0H@;J/QY9-./7A>8G/DF50Glhx/E?21W4EVF4EHJ/7EY " " YIh/;JDPj>=0=0H@;J/Qs>89l>8H@H¸;JD9A/E?AGR/Q;<>9-/ P?-;<Q H@/EO/EH<78< 132?N%.0;<4. - W ,99-.0/K462>?A7-/7k9BHJ/O/EHr>Df/6n0>469BD2D0H@;JD/>8?7-2H@O/?BG>uV h/ >=0=0H@;J/Q2?¸>BH@>?-P/DF50Gh/?¸2817kG2F289A.0;JDP);J9-/?A>89-;@2D7m=x/E?-132?AG/QB;@DR9-.0;<7¸N,2?AqB9-.0/ 132?-G/?r>8=0=0?A2>4. ;<7>Q02=9A/QYIh>7-/Qs50=2Dc!'/EN 9A2D B7G/69A.02Q ,99A.0/O/E?AV]h/P;@D0D0;@D0Pd281,9-./1350HJH,\d50HJ9-;@P?A;@Qf4EVF4EHJ/ 9A.0/Z7-2H@59-;@2D13?A2G 9-./=0?A/EOF;@257m9A;JG/)7"9A/E=g;<757-/QR9A2=0?A2OF;@Q/Z>8Dg;@D0;^9A;@>H?A/7-;<Q5>8H>8DQg>7>7k9A>?k9A;JD0P =x2;@D9%132? 9-./\d50HJ9-;@P?A;@QK=0?A2F4E/Q5?-/
[Dj2?Q/E? 9A2g/6n=0H<>8;@D`4EHJ/>8?AHJVg9A.0/l7k9-/=7 9A>q/D`Q5?-;@D0Pg>g7k;@D0PHJ/W4EVF4EHJ/YF9-.0/lP2O/?-D;JD0P/5>89-;@2D7 k7(6,>DQ ~~ tY0>89,9A.0/=>?k9A;@4E50H@>?%P?A;@QKH@/EO/H YGg>uVhx/?-/N%?-;J9k9A/ED`;JD`9-./B132H@HJ2N%;@D0PN,>uV B
'
* 1
!
O
L
!
.
`
-FO X~6&
*71
!
O
L
!
.
`!
*71
!
O
L
!
.
î :¦íq>¨ °q.Ȳ¦."¦ ²¨ N%.0/?-/9-./leH@G 9A.0;@4qFD0/7-7-/7B>DQ]=0?A/7A7-50?-/R>9)9-.0/ 9-;@G/g7"9A/E=P ! >8?A/lqFD02N%D>8DQs>8?A/lP;@O/D hFV 1 ! ` 1 ! >8DQ L ! ` L ! Y>DQ 4E2?A?-/7k=x2DQ079-29-./)?A;@P.9kW.>8DQK7-;@Q0/)21²/5>9A;J2D]X~~ 6 ;JO/ED 1 U ! >DQ L U !
>7>8D;JD;^9A;@>H¸P50/7-7)9A2K9A.0/7-2H@59-;@2Ds281 /5>9A;J2D7.~ & )>8DQ ~:(66Y9-.0/
4E2G=059A/Q]O>HJ50/7B>1¶9-/?462G=0H@/69A;J2D 21r9-.0/ W4EVF4EHJ/>8?A/PB 1
! >DQ L ! ?>4. 'W[46V46H@/RN%;^9A.0;JD 9A.0/g7k2HJ59A;J2Ds=?-246/7-7)Gg>uV]hx/gQ/6eD/Q;JDf>`?-/4650?7-;JO/lGg>8D0D/E?57-;JDP`=?-/46;<7k/HJVj9A.0/g7->GR/=7-/E5Q28W 4E2FQ0/'132?AGg>8H@;@7-G|>7;JDK#M?A289-9-/EDFhx/E?AP}~6+B[D1350D469-;@2D>HD289A>89-;@2Dg9A.0;<7W4EV46H@/'13?A2GwHJ/O/HqGg>uVh/ N%?A;^9-9-/Dj>7 *71 ! O L ! . ` _
S Q * VO1 U
! O L U ! O O '
O ' O '
.
'
# .0/B135D4t9A;J2DX8)a++zGg>uVhx/h0?-2q/DKQ2N%D`;JD9A29-.0?A/E/Q;<7"9A;JD4t97k9A>8P/7B
!"$#% B
&
=0=0H@V '
9-;@G/7>f$%/QFWH<>4q >857A7"W"aF/;@Q/H'7-4./EG/c}~ ,u'9-2c>H@;JD0/>8?A;@7-/Qp132?AG 21 /5>9-;@2D7 ~6&:,>8DQ ~:(6,7-29A.>9Y0;@DK135D4t9A;J2D>8HD029A>89-;@2D B
*"'1 U ! O( L U ! . `*) W _ M ! ' *1 U ! O L U ! O O ' . '
~0,+-#-./ 1032%.%2 1453 6#% B
& +2GR=59-/B9A.0/Q/613/4t976B ` + *"'1 U ! O7( L U ! . O ' ` ' + ' *'1 U ! O8( L U ! . '
!'289A/)9-.>89 ;@Dj250?%4>7-/Y13?-2G|/5>89-;@2D ~6&:tY
'
A
`
-Y.02N/O/?9A.0;@7 ;<7 D029,9A.0/4E>7k/);JD P/ED0/?A>H132? WY0>7%7-.02N%DKhx/EH@2N)
&
$%/7"9A?-;<4t9)9-./RQ/E13/469Z9A29-./RD/6nF9)4E2>8?7-/P?A;@Q]HJ/O/EH²57-;JDP`>.>8HJ1¶WXN,/E;@P.9A;JD0P?-/7"9A?-;<4t9A;J2D 2=/?A>89-2?Y
) 9)
}JuB 9)
`9) 9)
O ' 9
`9) 9)
' ' &
$%/7"9A?-;<4t9 '1 U
! >8DQ9( L U ! ;@D`9-.0/7A>8G/BN,>uV B '1 U ! 9)
`*) 9
* '1 U
! . O:( L U ! 9
`*) 9
*;( L U ! . ' & +2GR=59-/B9A.0/?-;@P.9-WX.>DQ7-;<Q/>9 9-./D0/6nF9462>8?7k/?H@/EO/EHB 9 ` 9) $ 9) *"'1=U ! 9) O( L U ! 9) . O ' 9) ` ' 9 $ ' 9 *'1 U ! 9 O8( L U ! 9 . ' & +2GR=59-/%>8D>8==0?-2un;@Gg>9-/,7k2HJ509-;@2D=< O 9) O>< O ' 9)
219-./ 4E2>?A7-/P?-;<Q/5>9-;@2DR2D < 9) B
9) O 9) O O ' 9) ` 9) O X~6+ '
9) O
9) O O ' 9) ` ' 9) ' X~6, 1mq ` PBFeDQK9A.0//6n0>469'7k2HJ59A;J2DK57k;@D0PR9-.0/4E2>?A7-/ZP?-;<Q`7k2HJO/E?
1xq 7 B7k2HJO//F7=~ + :>8DQ ~6,::hV=x/E?-132?AG;JDP9A.0/F8Za46V46H@/57k;@D0P '1=U
¦§,§8¨¹¤x¦0§¹,§(B¹,(R§(>.¤*¸(§,R§, B¦, * < O 9) O/< O ' 9 .
` _ S Q * + O'1 U
! 9) O8( L U ! 9) O 9) O ' 9)
O ' O '
. ' & +2GR=59-/B9A.0/462?-?A/469-;@2D7B < N 9 ` < O 9) + '1 U ! 9 O< N ' 9) ` < O ' 9 + ( L U ! 9) ' & [D9-/?-=x2H<>9A/)9-.0/4E2?A?-/4t9-;@2D757k;@D0PR9-.0/h;JH@;JD/>8?%;@D9-/?-=x2H<>9A;J2DK2=/?A>89-2?Y 9) }JuB < N ` 9) * <N 9 . O><N ' ` 9) <N '
9 '
&
+2GR=59-/B9A.0/462?-?A/469-/QK>8=0=?-2un;@G>89-;@2DK2D < B
1 U ! ` '1 U ! $ < N O L U ! ` ( L U ! $ < N '
5 1 !"$#% B
&
=0=0H@V '
9-;@G/7 9A.0/?A/EH<>n0>9A;J2DK7A4.0/EG/PB * 1 ! O L ! . `9) W _ M ! '
* 1 U
! O L U ! O O ' .
# .0/>7A7k246;<>9A/Qj?A/EH<>n0>9A;J2D]7-4./EG/4E2D7-;@7k9A7Z281>`7-;JD0PHJ/R!'/N 9-2D ;^9A/E?>9-;@2Ds7k2HJO/QdN%;J9-.>ge0n/Q DF50Gh/?,21²7-N//E=7 ' %2? '"
281²$%/QFW,H@>4q
>857A7kW[aF/;@Q0/EHXF# ./ZH@;JD/>8?A;<7k/Qg132?AG 21M/5>89-;@2D7 X~6& >8DQ\~:(6¸;<7,2h9>8;@D0/QhV4E2G=059A;JDP9-./ZH@24E>8H>462h0;<>8D2D0HJVY7-29A.>9 />4.?A/EH<>n0>9A;J2D7kN,/E/= ;JD`9-./=0?-/EW7-G2F289-.j7k9A>8P/132?%/6n0>8G=0H@/,=?-246//Q07,hFVK7-2H@OF;JDP B ! ! 1 ! .1 $ ! ! L ! L ` + * 1 U ! O L U ! .
O -
! ' ! 1 ! .1 $ ! ' ! L ! L ` ' + ' *71 U ! O L U ! . O 0
132? .1 >DQ
L
D@'/ED46/_> D/EN >=0=0?A2un;JGg>9A;J2D|9-2 9-./b7-2H@59A;J2D ;<7P;@O/ED hFV B1 U *"
! ` 1 ! U #" $ .1 O L U #" ! ` L ! U #" $ L '
# ./o!'/50Gg>8D0D_hx25DQ0>8?AVb462DQ;^9A;J2D7 >?-/T9-?A/>9A/QbhFV ;@D9-?A2FQ0546;@D0PcP.27k9D2FQ0/7g259A7-;@Q0/j9-.0/ 4E2G=059>9A;J2D>8HQ02Gg>8;@D>8DQhFVp;@G=27k;@D0Pf9-./dOu>HJ5/7g21 Q?A2=HJ/E9M9-.;@4qFD0/7-7-/7²>8DQ=0?-/7-7-50?A/>9²9-./7-/=x2;@D97:9-2hx/9-.0/ 7->G/O>8H@50/7²>7M9-.27-/,>89M9-./hx250DQ0>8?AV
,99A.0/j462>?A7-/7k9lH@/EO/EH ;<
U
tY²9-./`Q;<7A46?A/69-;<7-/Qc/5>9-;@2D7>?-/`7k2HJO/Qc57-;@D0P>s!'/N 9-2Do;^9A/E?>9A;J2D 7A4.0/EG/0# .0/;^9A/E?>9A;J2DK;<7%Q/6eD0/Q`;@DK9A/E?AGg7 281²9-.0/>4E2h0;<>8DKG>89-?A;@4E/7B
_ ` ! ' U ! L ! U O W ` ! ' U ! 1 ! U
O ` !
U ! L ! U
O ` !
U ! 1 ! U ' ~ ;@O/D O U O O ' U
î :¦íq>¨ °q.Ȳ¦."¦ ²¨
aF2HJ509-;@2D 21g9-.0/7k//5>89-;@2D7s;<7 2h9A>;JD0/Q >7]132HJH@2N'7EaF/E9k9-;@D0P
O
U
`
- ;@D 6Y>8Dv;@D0;J9-;<>8H
>=0=0?A2unF;@Gg>9A;J2Dc9-2 O ' U ;<7B9 O ' U
` _ 9)
' U + ' U VO U O O ' U Y¸N%.0;@H@/j9-.0/d/5>9-;@2Dp132? O U ;<7 2h9A>;JD0/QhFVK/EH@;JG;@D>9A;JD0PD O ' U
13?-2G /5>89-;@2D7 ,>DQ 6%tYOF;
+ _ 9) W O U ` U + U O U O O ' U + O ' U '
?95>9-;@2D ';@7Z7k2HJO/QK9-2KVF;J/H@QE
O U >8DQ O U O O ' U >8?A/50=Q0>9A/Q]>4E4E2?Q;JDP9A2 O U O U $ O U >DQ O ' U O ' U $ O ' U
+ _ 9)
W
O
U
Y0?-/7k=x/469-;@O/HJVF# ./;^9A/E?>9-;@2D`=0?A246/E/Q07 50D9-;@H9A.0/D02?AG
$ ' U + ' U VO U O O ' U $ / $ $ U + U VO U O O ' U $ /
N%.0/?-/ DR;@7²9A.0/%DF50Glhx/E?¸2150DqD2N%D7²;JD2D/ Q0;J?A/469-;@2DIYu;<7¸7k5C4E;J/D9AHJVR7kGg>HJHX[Dl9A.0;<7mN2?-qN,/,13250DQ 9A.>9 >?-/H@>89-;@O/%?A/Q54t9-;@2Dg;@Dg9A.0/'?A/7-;@Q5>8H21%5"57"9)- 9
Q 462G=>?-/QR9-29A.0/;@D0;J9-;<>8H?-/7k;<Q5>H013?A2G 9-.0/ 7-2H@59-;@2D>899A.0/=0?A/EOF;@257r9-;@G/Z7k9-/E=3m9-2h/)7k50Cg4E;J/D9Y.02N/O/?rGl54.gP?-/>9A/E?>4E465?A>46;@/7r>?-/'/>7-;JH@V 2h9A>;JD>h0H@/);J1m?-/5;J?A/Q %$;?a0* I#'a # .0//6C46;@/ED4EV>DQg>44650?>4EV2819-./\d50HJ9-;@P?A;@QG/E9-.02Q;<7¸e?7"9Q0/EG2D7k9-?>9A/QRhFVR4E2G=>?-;<7k2DN%;J9-. >7k/?-;@/7281)>8D>HJV9-;<4E>H,>DQT=?-/O;@257kH@VWX?A/E=x2?-9-/QsDF50G/?-;<4E>H,?A/7-50HJ9A7l132?gQ?A2=0H@/697k=?-/>Q;@D0P]L2N'7E !/ENU?-/7k5H^97r>8?A/ 9-.0/Dg=0?A/7-/ED9-/QN%.;@4.g7-.02No.2Np9A.0/'G/69A.02Qg4E>8Dghx/'?A/>Q;JH@V>=0=0H@;J/Q9-2l7-;JG50H<>9-/ Q?A2=HJ/E9 G29-;@2D`2Dd7k5h7"9A?A>89-/7N%;J9-.j9A2=x2P?A>=0.0;<4Z>DQ>u2? N/E9k9A>h0;@HJ;J9"V;@D0.02G2P/ED/E;J9-;@/7
E- B= :,t-89!tg
# .0/Z>4E4E50?>46Vl2819-.0//7k9-;@G>89-/'21 I#F?'Y8/5>9-;@2DEX~%tY8;<7Q/G2D7k9-?>9A/QRhFVR4E2G=>8?A;@D0PB;^9,N%;^9A.9-.0/ 9A?-50/BH@24E>8H/?-?A2?,132? 9-.0/2?AQ;@D>?-VQ;*/E?A/ED9-;<>8H/5>9-;@2D 1 $ 1
` - O 1 N
U ` 1 U ' & # .0;<7%/5>9-;@2D`;<7%4.027-/EDj7k;@D4E/);J9A7>8D>HJV9-;<4E>H7k2HJ59A;J2D
1 N ` 1 U
U * U / O :(6
;<7m281x/6n0>469-H@V9-.0/'7A>8G/132?AG >7¸#²>8DD0/E?B7 } -ux46H@27-/Q132?-G 7k2HJ59A;J2Dl132?meH@G_9-.0;<4qFD0/7A7¸>9m9A.0/'46/D9A?-/ 21 >Df>8n;@7-VFGRG/E9-?A;@4>8H@HJV]7-=0?A/>Q0;JD0PdQ?A2=HJ/E92O/E?>j.02G2P/D0/E257B7k5h7"9A?A>89-/ 7-/E/gH<>9-/?M1F1
!
;<7 9A.0/DF50G/E?A;@4>8HI7-2H@59-;@2Dj21m/5>9A;J2D &:,2h9>8;@D0/Qj>99-;@GR/
!
57-;@D0Pg9-./B9-/EG=x2?>8HMQ0;@7A46?A/69A;@7A>9A;J2D 7A4.0/EG/%Q/7-4E?-;@h/Q>h2O/Y9-./EDR9-.0/ 9A?-5/,H@24E>H/E?A?-2?²;<7²9-./'Q;/E?A/ED46/ h/E9"N//EDD1
! Yu9A.0/%DF50G/E?A;@4>8H 7-2H@59-;@2Dj2h9A>;JD0/Qj>9%9-;@G/ ! Y>8DQ9A.0/l4E2?A?-/7k=x2DQ0;JD0P>8D>8H@V9A;@4>8HI7-2H@59-;@2D`132?%9-.0/7A>8G/;@D0;J9-;<>8H 4E2DQ;J9-;@2D 1 N
¦§,§8¨¹¤x¦0§¹,§(B¹,(R§(>.¤*¸(§,R§, B¦,
0
1
2
3
4
5
6
t
1
1.5
2
2.5
3
ELTE/TLE
TOL = 10
−1
TOL = 10
−2
TOL = 10
−3
TOL = 10
−4
TOL = 10
−5
TOL = 10
−6
×ÔUà>ØlÛaÑgòlÎßÖaÔÕÜÜ>éM gó©ÖaÜqó¸ ©éÂÜÛ3ÑjñGؤßÖaÔÕÜ1Ðá1âìßç3ßXéÂØ¤ÐlÓÖaÔÕÜ1ÐÜ>éOÖaÔÕÞqÑ1úGÖBú.ßÐ.äó ò
U6<Ex K .Et$9PE :
# .0/Z/6C46;@/ED46VR2819-.0/)\d50HJ9-;@P?A;@Q>8DQR9A;JG/6W[7k9-/E==0;JDPl=0?A246/Q050?-/7r>8?A/Q/G2D7k9-?>9A/QRhFV4E2D7-;<Q/E?A;JDP 9A.0/ =0?-2h0H@/EG_281xQ?-2=0H@/69¸7k=0?A/>Q;@D0PZ2D>Z.02?A;E2D9>8HF7k50h7"9A?A>89-/132?¸>=>8?-9-;<>8H@HJVBN,/69k9A;JDPB7-V7"9A/EG_N%.0/?-/ 9A.0//50;@H@;Jh0?A;@50G 4E2D9A>4t9>8DPH@/D 7 - >DQ`9-.0/l/'/4t97%281¸P?A>uOF;J9"V>8?A/D0/EPHJ/4t9-/Q
W4X
`
-:t 82? 9A.0/Z=>8?-9-;<4650H<>8? 4E>7k/)281:>7k=?-/>Q;@D0P>8n;@7-VFGRG/E9-?A;@4ZQ?-2=0H@/69Y4E/ED9-?A/Q;JD0;J9-;<>8H@H@V>9 EDCOTF=
`
-'
OT-'
tY 9A.0/BeH@G 9-.;@4qFD0/7-7%>89%9-.0/Q?A2=HJ/E9%4E/ED9-?A/Y 1 U <tY0;<7%P;JO/EDs} -MhFV
1=U:N `
,
132?F 7 -Y +
N%.0/?-/
>DQ
`
-'
~ ,>8?A/7-=0?-/>Q;@D0P462D7"9>8D9A7Y>R?A/7-50H^9'O/E?A;^e/Q`hFV II/H@>.`>DQj\]>8?AGl5?B7}t /EnF=x/E?A;@GR/D9A7N%.0;<4.f1325DQ
-'
& 4
4
-'
~M# .0/K>n;@7-VFGGR/E9-?A;@4H@50h0?A;<4E>9A;J2Df/5>9A;J2D7B132?9-.0;<7 =0?A2hHJ/G N/?-/7-50h7-/50/D9AHJVK7k2HJO/QDF50G/?-;<4E>[email protected] >8?-9B>8DQX?rH@/EVs}+u:N%.02g;JDFO/7k9-;@P>9A/Q9-.0/ ;@DL50/D46/d21Z9A.0/d=0?A/4E50?A7-2?ReH@G 9-.0;<4qFD0/7A7D132j2D9-.0/]7-=0?A/>Q;JDP?>9A/j=>8?>8G/69A/E?7
>8DQ
;JD /5>9-;@2D] + t@/E?A/Y9-.0;<7L2Nz;<7 7"9A5Q;@/QKhV7-2H@OF;JDP9-./)H@50h0?A;<4E>9A;J2D/5>89-;@2D7 k>8DQP":r132? 9A.0/=>8?-9-;<465H@>? 4>7-/B281 ;
`
-'
- - 0YV OG5
`
=O~ tYF9A.0/;@D0;^9A;@>HMQ?A2=0H@/69%=0?A28eHJ/B;<7 P;@O/ED`hFV
1 6
`
5D
+
~6-,
/ *
O-*
O ,
N%.0/?-/T;<79A.0/?A>Q;J57219-.0/s132F289A=0?-;@D9`219-./Q?A2=HJ/E9`2D9-.0/f7-50?k1>46/s2819A.0/c7k50h7"9A?A>89-/ 'D /5;JH@;Jh?-;@50G 462D9>469l>8D0PHJ/21
'
8
N,>74.027-/EDT7-2j9-.>9.1 U <
>9RH@>?-P/9-;@G/7:!'29-/9-.>89Y >=>8?-9 13?-2G|9-.0/;@D0;J9-;<>8HI=?-2eHJ/BD02g>8nF;<7-VGG/69A?-Vg;<7%;@G=27k/QK;@DK9-.0/DF50G/E?A;<4E>8HI7-2H@59A;J2D7E
# .0/132HJH@2N%;@D0P 7-/50/ED46/21%eP5?-/7/EH@54E;@Q0>89-/9-.0/`/'/469R21%9-.0/`\d50HJ9-;@P?A;@Qp=>8?>8G/69A/E?72DT9-.0/ 4E2DFO/?-P/ED4E/21 7-2H@59-;@2D7BhVs=0H@289k9A;JDPd9-.0/g?A/EH<>9A;JO/?-/7k;<Q5>H@7>899-.0/eD0/7k9lP?A;<Q H@/EO/EHXYIN%.0;<4.f>?-/ Q/EeD0/QR>7m9-./'?A>89-;@2B281?-/7k;<Q5>H0>9¸9-./%eD0/7"9rP?-;<QRHJ/O/EH0>81¶9-/?>BP;@O/DDF50Gh/?r281xeD0/'P?-;<Q'W[46V46H@/7 4E2G=>8?A/Q9-29A.0/;JD;^9A;@>HI?-/7k;<Q5>H>9%9-./h/P;@D0D0;@D0P281²9-.0/B9A;JG/7k9-/E=M
# .0// x/4t9Z21m9-.0/DF50Gh/?21 W4EV46H@/7 " " '>89;JD9A/E?AGR/Q;<>9-/P?-;<QjH@/EO/H@7<- W9 2Dj9-.0/ ?A/EH<>9A;JO/g?-/7k;<Q5>H@7YIN%.0/?-/g9A.0/eD0/7k9P?-;<QHJ/O/EH;@7~ ,n~ ,];/dW
`
î :¦íq>¨ °q.Ȳ¦."¦ ²¨
0
1
2
3
4
5
6
7
8
9
10
V−cycles on finest level
10
−10
10
−8
10
−6
10
−4
10
−2
10
0
10
2
10
4
Relative Residuals
0 V−cycle
1 V−cycle
2 V−cycles
3 V−cycles
4 V−cycles
5 V−cycles
©"¤"¥³¥±u ¢E°u»"m³ ²º¶¤[µ¤k«¬" ²©A¥ ¥±ur¯¬¢E¥"X»%"£u¯©A¥¸®tX¯¬£«¬"´6"«¬ Mt¢Z¥±ur¤"t¢E´6"X®t"¢u¤" ±u¯¬ ¥Xtµ ³F t«¬°¥¯¬t¢u -§I±uu¢u" ¥®t¯¬£%«¬"´6"«< 6¯¬ I6Ct·gI6Ct·©¢£
µ I6D§
0
1
2
3
4
5
6
7
8
9
10
V−cycles on finest level
10
−10
10
−8
10
−6
10
−4
10
−2
10
0
10
2
10
4
Relative Residuals
65x65
129x129
257x257
513x513
"¤[¥o³d¥X±u!¢k ¥p®tX¯¬£ «¬"´6"«jt¢ ¥X± ¤"t¢E´6"X®t"¢u¤" ±¯¬ ¥tµZ³ t«¬°¥¯¬t¢u "M¯¥±
$# ©¢u£%
µ I6D & §
0
1
2
3
4
5
6
7
8
9
10
V−cycles on finest level
10
−10
10
−8
10
−6
10
−4
10
−2
10
0
10
2
10
4
Relative Residuals
∆
t=10
−9
∆
t=5*10
−9
∆
t=10
−8
∆
t=5*10
−8
¤'ð"¤[¥]³¥¯¬»%s ¥X"¡ t¢z¥±uf¤"t¢6´6"X®t"¢u¤" ±u¯¬ ¥tµ¸³ Xt«¬°¥¯¬t¢u &"M¯¥X±
' (#
©¢u£r¥±u¢k ¥ ®tX¯¬£«¬"´6"«<
u¯¬ I6Ct·lI6Ct·§
10
3
10
4
10
5
10
6
10
7
Number of Unknowns
1
10
100
1000
CPU time (s)
65x65
129x129
257x257
513x513
1025x1025
£)¤:k¡8"¢u£u"¢u¤"³,¦¥¯¬»%© ©I³J°u¢u¤[¥¯¬t¢r³¥±u ¥¥X©«¢°u»8kI³0°u¢ª¢u*"M¢u -+"M¯¥±-,/.10
I6D &2 §
×ÔÕà1ØlÛaÑ43lòá65#â5ú á67â5úá8(â'9ÑüÑÓÖÜéÞTؤÙJÖaÔÕà>ÛaÔUäùÚ.ßSÛ6ß>ÞqÑÖaÑ5ÛaçÈÜ1ÐùÖaålÑ8ÓjÜ1ÐGè,ÑÛaà>ÑjФÓÑålÔUçLÖaÜÛ&õÜ>éç&Ü>ÙUØgÖaÔÕÜ1Фçjòìá:â'9 älÑjÚ#ÑÐ.älÑФÓjÑÜ>éý;Ë ÖaÔÕÞqÑìéÂÜÛ3ßÖ¹õgÚlÔUÓjß>Ù#ÖaÔÕÞqÑ3çLÖaÑjÚß>çìß!éÂØ¤ÐlÓÖaÔÕÜ1ÐÜ>éÖaålÑÖaÜÖ6ß>ÙÐGؤÞÈï#Ñ5ÛÜ>éØ¤Ð gÐlÜBëìФçjòg×ÙÕÜBë ÓjÜ1ФälÔJÖaÔÕÜ1Ðlç<9GÑjè,Ü1ÙÕØgÖaÔUÜ>Ð8Ü>éßÚ.ßÛ&ÖaÔUßÙUÙJõqëÑÖ&ÖaÔÕФàXägÛaÜ>Ú¤ÙÕÑÖëìÔJÖaå4=?>?@ ú)A/@
)B+C úEDGF1@
BC IHgú)J
@
KB 3ß>Фä á6LNMâ@á:
ÿ
¦§,§8¨¹¤x¦0§¹,§(B¹,(R§(>.¤*¸(§,R§, B¦,
57-;@D0P]>9lH@/>7"92D0/ W4EVF4EHJ/>899-./;JD9A/E?AGR/Q;<>9-/gH@/EO/EH,>7B9A.0/?A/EH<>9-;@O/g?A/7-;<Q5>8H<7>8?A/g?A/Q54E/QfhFV >HJG27k9 9"N2g2?AQ0/E?721²Gg>PD0;J9-5Q0/l462GR=>8?A/Qg9A2g9-./4E>7-/BN%.0/E?A/D02g;@D9-/E?AG/Q;<>9A/'W[46V46H@/B;@7'57k/Q H^9A.0250P.KG2?-/)9-.>8D`2D0/ 'W[46V4EHJ/Q2F/7 D289%;JG=0?A2O/)D029-;<46/>[email protected]/4E2DFO/?-P/ED4E/281:9-.0/7-2H@59A;J2D >89%9-.0/BeD/7k9'HJ/O/EHh5957-;JDP>R.0;@P./E?'2?AQ/? ;JD9-/?-=x2H<>9A;J2D`7A4.0/GR/7-54.d>7 h0;<465h0;@4B;@D9-/?-=x2H<>9A;J2D 132?'9-./805HJH¸\d50HJ9-;@P?A;<Qj;JD9-/?-=x2H<>9A;J2Dd;JD7"9A/>Q]281rh0;@HJ;@D0/>8?;@D9-/?-=x2H<>9A;J2DdG>uV`;@D46?A/>7k/B9A.0/lhx/ED0/Ee09 2157-;JD0PG2?A/;@D9-/E?AG/Q;<>9A/ W4EVF4EHJ/7E¸# .0/`;@D0;^9A;@>H ;JD4E?-/>7-/;@Dp?A/EH<>9A;JO/?A/7-;@Q5>8H<7l;<7Q50/K9-2s9-.0/ 1>4t99A.>9>899-.0/h/P;@D0;JDPd281 9-.0/K7k2HJ59A;J2Df=0?-24E/7A7B9-.0/;JD0;J9-;<>8HP50/7-7YIN%.0;<4.T462?-?A/7-=2DQ07Z9-2]9-.0/ 7-2H@59-;@2DU>9g9A.0/d=0?A/EOF;@257R9A;JG/]7k9-/E=MY7->89-;<7"e/7 ~ & /6n0>4t9-H@Vp7-;JD46/d;J9A7g?A;JP.9kW.>8DQo7k;<Q/]Q2F/7D0289 4.>D0P/:# .F57Y²>81¶9-/?>de?A7k9R4EVF4EHJ/Y²Q0/7-=0;^9A/K>d?A/Q054t9A;J2DT281%9A.0/?A/7-;@Q05>8H<7132?/5>9-;@2D X~ ( tY:9-.0/ ?A/7-;@Q05>8H<7 281m/5>89-;@2D ~ & ,4E>D`2D0H@V;@D46?A/>7k/0# .0/Gg>8PD;^9A5Q/281²9-.;@7;JD0;J9-;<>8HM;JD4E?-/>7-/B;JDd?A/EH<>9A;JO/ ?A/7-;@Q05>8H<7 ;@7Z46?A;^9A;@4>8H9A2g9-.0/7k5446/7-7 21²9A.0/7-2H@59-;@2Dj=0?A246/Q050?-/>DQK9-.0/?-/E132?A/Y>H<>8?AP/? D5Glhx/E?'21 =0?A/7-G2F289-.;JD0P]7-N,/E/E=7;@7=x/E?-132?AGR/Qs132?9A.0/ge?7k9'Wk+V46H@/P ' ` %BN%.0;@HJ/9"N2d=0?A/7-G2F289-.;JD0P 7-N//E=7,2D0HJV>?-/Z1325DQK9-2Rh/7-5C46;@/ED9 132?,9-./Z132HJH@2N%;@D0PW4EV46H@/7B '"
;<7%7-/69 9-2g~?-/P>8?QH@/7A7r21 9A.0/46V46H@/BDF50Glhx/E?
# .0/r/ x/4t9M219A.0/rP?-;<QB7k;E/r>89I9-.0/reD0/7k9MH@/EO/EH8;<7I/EnF=HJ2?-/QD0/EnF9I132?²>,e0n/QBDF50Glhx/E?M281;JD9A/E?AGR/Q;<>9-/ W4EVF4EHJ/7 " "
`
>8DQP
!
`
- 9 V8²;JP50?A/h'7-.02N'7'9-.>89Z132?Z>8H@H²P?A;@Q07YN%;J9-.]Wz?>8D0P;JDP 13?A2G| k~ ,n~ , :9-2Dn tY89-.0/)?A>89-;@2l219-.0/)?-/7k;<Q5>H@7,>1¶9A/E?9-.0/B462G=0HJ/E9-;@2D21I9-./ W4EVF4EHJ/ ;<7'/ x/4t9-;@O/HJV`;@DQ/E=x/EDQ/ED9Z21m9-.0/leD0/7k9P?-;<Q`HJ/O/EHX+H@27-/E?Z>8D>HJV7-;@7%21m9-.0/RQ0>9>?-/O/>H@7%9-.>89'9-.0;<7 ?>9A;J2`9A>q/7)>`O>8H@50/g21 >=0=0?A2unF;@Gg>9A/EH@V>8~ -0 8²;JP50?-/.34B7-.02N'7Z9-.>9 ¶enF/Q )9A;JG/7k9-/E=c7-;*/.>7 H@;^9-9-H@//'/4t9B2D 9-.0/462DFO/E?AP/D46/l.0;@7k9-2?-Vj50D9A;JHr>9-.0?A/7-.02H<Q]O>HJ50/
#
` c- 9;@7)?A/>4.0/Q # .0/;@D0;J9-;<>8Hr;@D46?A/>7k/R;@Ds?A/EH<>9A;JO/?A/7-;@Q05>8H<7)h/462G/77"9A/E/E=x/E?>DQ>8hx2O/9-.0;<7)9-.0?A/7-.02H@QsO>8H@50/R9-.0/ 7-2H@59-;@2Ds=0?A2F4E/Q5?-/l1>8;@H@7I# .0;<7ZH@;@GR;J9-;@D0PjO>8H@50/R2819A.0/R9-;@G/7k9-/=sQ/=/DQ07B2D hx289-.s9-./RG/7k.f7k;E/ >DQ .2N >QO>D46/Q]9A.0/g7k2HJ59A;J2Ds;<7BI>7ZeD0/?>8DQ]eD0/E?P?-;<Q07B>8?A/R57k/QYC
#
Q/4E?-/>7-/7B>8DQs>9 H<>9A/E?9-;@GR/7EY
#
;@D4E?-/>7-/7M!'29-/K9-.>89l9-./`4>8H<4650H<>9A;J2D7l>?-/132?lO/E?AVs/>8?AHJV9A;JG/7N%.0/?-/9-.0/ 7-=0?A/>Q0;JD0PG289-;@2DK;@7,G27k9%>4t9A;JO/Z>DQg9A.0/Q/EGg>8DQ07,2D9A.0/)Gl50HJ9-;@P?A;@QK7k2HJO/E? >8?A/G27k9%7k/O/E?A/%'D >Q0Q;J9-;@2D>Hr>QO>8D9>8P/R9A.0/E?A/6132?-/281,9-./>Q0>=9-;@O/R9-;@G/6W[7"9A/E=0=;JD0P]7A4.0/GR/Q0/7A46?A;Jhx/Q=0?A/EOF;@257-HJVd;<7 9A.>9Z9-.0/9A;JG/EW7k9-/= 7k/HJ/4t9A/Qj;<7Z>HJN >uV7%7kGg>8H@H@/E?'9A.>8Dj9A.0;<7H@;JG;J9-;@D0P9A;JG/R7"9A/E=]>7D02?A/7k9A>?k921m9-.0/ 7-2H@59-;@2D`=0?A246/Q050?-/B;<7 /EO/E?%D0/46/7A7A>8?AV
[DcHJ;@P.9219A.0/>8hx2O/YM>8H@H7-50h7-/50/D9?A/7-50HJ9A7N,/E?A/2h9>8;@D0/Qf57k;@D0PP 'W[46V4EHJ/7>9/>4.fHJ/O/H H^9A.0250P.46H@/>?-H@Vl2O/E?-W4>859A;J257EY9A.0;@7.>72D0H@V>l7kGg>8H@H>Q0Q;J9-;@2D>H462G=059A>89-;@2D>H4627"9,7k;@D46/GR27"9 21:9A.0/462G=0509A>9A;J2D>8H9A;JG/;<7'7-=/D9'=x/E?-132?AG;JDP4EV46H@/7%>9 9A.0/BeD0/7"9'H@/EO/EHX
8²;JP50?-/Q 7-.02N'7Z.02Nb9-./g+
*v9A;JG/132?>g9"VF=0;<4E>H:9A;JG/7"9A/E=Q/E=x/EDQ7Z2D P?-;<Q]Q/D7k;J9"Vd>8DQ Q/G2D7k9-?>9A/7g9A.>9j2D/ 281l9-.0/G27k9K;@GR=x2?-9A>D9K=x289-/D9-;<>8HB>QO>8D9>8P/721\d50HJ9-;@P?A;@QzGR/E9-.02Q07Y D>G/EH@VT9-.>9j+
* 9-;@GR/ Ou>?-;@/7HJ;@D0/>?-H@VA;/9-.0/ 7-H@2=x/d281B9-.0/dH@;@D0/];@D 8:;@P5?-/ 3Q N%;J9-. 9A.0/]9A289A>HDF50Gh/?2815D0qFD02N%D7Y;<7>4.;J/O/QUhFVo9-.0/7k2HJO/E?b7KD029-/Q>h2O/YrN%./ED >HJH@;J/Q9-2 /O/E?-W;JD4E?-/>7-;JDP462G=0509A>9A;J2D>8Hx=x2N/?Y9-.0;<713/>9A50?A/Z21I9-.0/B7-2H@O/E?;@7=>?k9A;@4E50H@>?-H@V;JG=x2?-9A>D9,N%.0/ED 7-Gg>8H@H^W[7-4>8H@/Z=.0/ED02G/ED>RD0//QK9-2hx/7k;@Gl50H<>9A/Q`/6C4E;J/D9AHJVK>8DQj>44650?>9A/EH@V
8²;JP50?-/ =>: ;@HJH@57k9-?>9-/7'.02NS9A.0/l9A;JG/R7"9A/E=]O>8?A;@/7'>7>1350D4t9-;@2Ds281b1V2Y7k.2N%;JD0P9-.>89)>1¶9A/E?9-.0/ 7-2H@59-;@2Dd.>7'7-/69-9-H@/Q]Q2N%D`9A.0/9A;JG/6W[7k9-/E==0;JDP7-4./EG//ED>8h0H@/7'7-50h7k9A>D9-;<>8HM/EC46;@/ED4EVKP>8;@D7 9-2h/ ?A/>HJ;<7k/Q>9,H@>89-/?9A;JG/7 >DQg9A.>9,9-.0/)O>8H@50/21 1V2'.>72D0H@Vg>lGg>8?AP;@D>H;@DL50/D46/);JD9-.0/B7-/ED7-/'9A.>9 H<>8?AP/?F1V2ZO>8H@50/7'=/?-G;J9'H@>?-P/E?9-;@GR/7k9-/=7 9A2Rhx/57-/Q`D0/>8? 9-.//50;@HJ;@h0?A;J50G 7"9>9-/
# .0/l>4E465?A>46V21²9A.0/l7-2H@59-;@2D7%;@7Z462D7k;<Q/?-/QKD/6nF938:;@P5?-/ 3h ;@DFO/7"9A;JP>9-/7 9-.//'/469281¸P?A;<Q 7-;*/2DT9-./`Q?A2=HJ/E9l9-.;@4qFD0/7-7.1=U:Nl132?7-;JG50H@>89-;@2D7N%.0/?-/X132 `
-'
î :¦íq>¨ °q.Ȳ¦."¦ ²¨
1
10
100
1000
10000
Number of time step
10
−10
10
−9
10
−8
10
−7
10
−6
10
−5
10
−4
∆
t
h* = 0.05
h* = 0.04
h* = 0.02
h* = 0.01
h* = 0.005
©O"¤"¥:³0¡uk¤"°u Xt «¬»p¥X±u¯¬¤Xª¢u" t¢¥X¯¬»% ¥X"¡j[´6t«¬°¥¯¬t¢§OI±uu¢u" ¥®tX¯¬£«¬[´6k«:¯¬ - 2 ¯<§t§ C'1C§
0
2.5e−06
5e−06
7.5e−06
1e−05
t
0
1
2
3
4
5
h
0
(t)
33x33
65x65
129x129
257x257
513x513
1025x1025
) B°u«¥¯¬®tX¯¬£ t«¬°¥X¯¬t¢ _³U¥X±w£uXt¡«¬"¥ ¥±u¯¬¤ªE¢u" X x©A¥¥±uM¤"k¢6¥XI³©²¡©¥¯©«¬«µ "[¥¥¯¬¢u® £Xt¡u«¬[¥)© B©R³J°u¢u¤[¥X¯¬t¢d³r¥¯¬»%t M©¢u£j®t¯¬£ £k¢ X¯¥¶µ ":¯¥±
DD u§
10
−7
10
−6
10
−5
10
−4
10
−3
10
−2
t
1
10
h
0
(t)
h* = 0.05
h* = 0.04
h* = 0.02
h* = 0.01
h* = 0.005
¤' B°u«¥¯¬®tX¯¬£ Xt«¬°¥¯¬t¢u ³ ¥±u £uXt¡«¬"¥ ¥±u¯¬¤ªE¢u" X ²©A¥I¥X±ur¤""¢E¥X¸³©¡©¥¯©«¬«µ?"[¥¥¯¬¢u® £Xt¡u«¬[¥© ©M³J°u¢u¤[¥X¯¬t¢³¥¯¬»%6't©¢£¡u"¤k°X Xt u«¬»T¥±u¯¬¤ªE¢u" X §1I±u¢u" ¥I®t¯¬£«¬[´6"«¯¬ 2 ¯<§t§,C',C§
¦§,§8¨¹¤x¦0§¹,§(B¹,(R§(>.¤*¸(§,R§, B¦,
óß>ïlÙÕÑ6òOóå¤Ñ·ÑüÑjÓ5ÖÈÜéÚgÛaÑjÓjØgÛaç&Ü>ÛÈø¤ÙÕÞÖaålÔÕÓTGФÑç&ç J
Ü1Ðç&ÚlÛaÑjß1älÔÕÐlàîÛ6ßSÖaÑÚ¤ßÛ6ßÞqÑÖaÑÛaç<9ø¤Ð¤ÑjçLÖTà>ÛaÔUäÙÕÑjè,ÑjÙÔUç ÿ C Ý ÿ C ú¤ëìÔJÖaå- @ &21ò
132
$%/7A4E>HJ/Q
-0- -:8 -0@, & -0&0 -0-6 - &~ -0@, -0&~ (
-0-~ - + -0@, -0& &0 -0-0 -:-:( -0@+%( -0& ,0
-0- - - -0@+%( -0& , &
;@DL8²;JP50?-/=4tYM132?7-2H@59A;J2D72h9A>;JD0/QN%;^9A.p>jeD0/7"9P?A;@QfH@/EO/EH21~ (un~ (YVR
W
`
- 93>8DQ >Q0>8=9A;JO/9A;JG/j7"9A/E=0=0;@D0P²9;@HJH@57k9-?>9-/79A.>9g>1¶9A/E?>8DT;@D0;J9-;<>8H =0.>7k/9-.0/jQ?A2=0H@/699-.0;<4qFD0/7A71 U N 2h/VF7'9-.0//6n=x/4t9A/Qd=x2N,/E?H<>uNb?-/H@>89-;@2D7-.0;@=P1 U N
`
c7
,
YN%./E?A/9-.0/R462F/6C4E;J/D97
>8DQ
>?-/ ;@Dc#:>h0HJ/dM!'29-/9A.>9YM2N%;@D0P`9-2j9-./9-;@GR/K7-4>8H@;JD0Pj/GR=HJ2V/Qs;JDf9A.0/g=0?A/7-/ED9l7"9A5QVYI;JDc2?AQ0/E?)9-2 4E2G=>8?A/%9A.0/7-/?A/7-50HJ9A7N%;^9A.9-.27-/281ma4.FN,>?k9T'>8DQ ?rH@/EVj}+89A.0/
O>8H@50/7Gl57k9 h/ZGl50HJ9-;@=0H@;J/QhFV 9A.0/1>469-2?
-'
- , 9
,
# .0/
Ou>HJ5/7'>?-/B13250DQ`9-2ghx/l4EHJ27k/B9A2g9-.0/l/6n=/4t9A/QjO>8H@50/281K-~0YM} -uY>8DQ 9A.0/O>8H@50/721
>8H<7-2l462GR=>8?A/'N/HJHxN%;J9-.9A.0/O>HJ50/7
`
-'
&0132
`
-'
-: ¸>DQD-0& + 1V2
`
-'
- - 2h9A>;JD0/QhFV}+u
M8$:Z-: E #x :¶:$j 4! 6Xt l-#x*E
+2GR=>8?A;@7-2D;<7Gg>Q0/'h/E9"N//EDDF50G/?-;<4E>H=0?A/Q;<4t9A;J2D7>DQ4E2?A?-/7k=x2DQ0;JD0P>D>8H@V9-;<4E>H7-2H@59-;@2D7r132? 9A.0/g4E>7-/R281>8nF;<7-VGG/69A?-;<4R7k=0?A/>Q;@D0PL2N 281,>`462G=0HJ/E9-/HJV]N,/69-9-;@D0PjQ?-2=0H@/69YI4E/ED9-?A/Q ;JD;^9A;@>HJH@V]>9 DCOGF= `
<-'
=O-'
6Y2O/E?r>.2?A;*2D9A>H07k50h7"9A?A>89-/'N%.0/DP?>uOF;^9>9A;J2D>8H/ x/4t9A7,Q2GR;@D>89-/'2O/E?7-50?-1>46/ 9A/ED7-;J2Dj2D0/7%7k2R9-.>9%9-./ 2DQKD5Glhx/E?Y
W4X
Y0;<7%H<>8?AP/[D`9A.0;@74E>7-/)9-.0/H@50h0?A;<4E>9A;J2DK/5>9A;J2D7,132? 9A.0/RQ0?-2=0HJ/E9'9-.;@4qFD0/7-7 1 4>8D]hx/R?-/Q546/Q`9-29-.0/132H@HJ2N%;@D0P7-/4E2DQWX2?AQ/?'=IQ/x132?)>n;@7-VFGGR/E9-?A;@4 L2N
!
1
!
` WYX
!
!
61
Q
!
1
!
O < -
;@D`9-/E?AGg7 281 FY9-.0/?>Q0;@>HIQ;@7k9A>D46/)13?A2G|9A.0/46/D9-?A/)21:9A.0/7-50h7k9-?>9-/
# .0/;<Q/>HJ;<7-/Qj4E>7k/N%.0/?-/9-.0/;@D0;J9-;<>8H²Q?-2=0H@/69=0?A28eHJ/;<7';@Dj9-.0/132?AG 21¸>g(Z;@?A>4Q/EHJ9A>g1350D4t9A;J2DIY P;JO/ED`hV
1 O ` - ` - 132? 7 ->DQ ~
U
1
` 2bO <
N%.0/?-/2v;<79-.0/BQ;@G/ED7-;J2D0H@/7A7O2H@50G/21I9-.0/BQ?A2=0H@/69YF.>7h//ED`>8D>HJV7-/Qg=0?A/EOF;@257-H@VR;@Ds}~uX# .0/ H<>9-9-/E?Z7"9A5QVK7-.02N,/Q9-.>89'/5>89-;@2D N:- 46250H<Qhx/l7-2H@O/Q`>8D>8H@V9A;@4>8H@HJVhV`Q/6eD0;JDP9-.0/7-;JG;@H@>?-;J9"V
O>8?A;<>8h0H@/
`
Q
&
Y09-2VF;J/H@Q
1
`
-'
(6
2
WYX
*
&
-'
(, ,
+
/
&
'
<~
8²;JP50?-/G& >:¸462G=>8?A/7¸DF50G/?-;<4E>H0=0?A/Q;<4t9A;J2D7r21x9A.0/ZQ0?-2=0HJ/E9r9-.;@4qFD0/7-7>9;J9A7,46/D9A?-/Y:1 U <tYF>7> 1350D4t9-;@2Dd21M9A;JG/N%;J9-.`9-./l7k;@G;JH<>8?A;J9"V7k2HJ509-;@2D N~ 132? 9A.0/l4E>7k/
WYX
`
- & &R>8DQ82
`
-'
î :¦íq>¨ °q.Ȳ¦."¦ ²¨
10
−8
10
−7
10
−6
10
−5
10
−4
10
−3
t
1
10
h
0
(t)
Similarity Solution
Numerical Solution
©²Xt¡«¬"¥`¥X±u¯¬¤Xª¢u" s©A¥d¥±uf¤""¢6¥Xc³© ¤"t»%¡u«¬[¥"«µ1"[¥¥¯¬¢u®²£Xt¡u«¬[¥
&HA¤kt»%¡u©X¯¬ Xt¢ 8[¥6"""¢ ¥±ul¢°»%k¯¬¤k©«m¡uX"£u¯¬¤[¥¯¬t¢u ©¢u£j¥±u ¯¬»%¯¬«©X¯¥¶µ% t«¬°¥¯¬t¢ SC[§
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
x
0
0.5
1
1.5
2
2.5
3
h
t=3x10
−6
t=4x10
−5
t=3x10
−4
t=9x10
−4
)t X º< X"¤[¥X¯¬t¢©« ¥±u¯¬¤ªE¢u" X ¡uX<u«¬k R©¤"Xt X ©i¤"t»%¡u«¬[¥"«µ "[¥¥¯¬¢u®b£ut¡u«¬[¥5HR¤"t»%¡©X¯¬ t¢ 8[¥6""k¢)¢°u»%"¯¬¤-©«¡uX"£u¯¬¤[¥¯¬t¢u µ»8t«¬ ©¢u£ ¥±u¸ X¯¬»%¯¬«©¯¥¶µ Xt«¬°¥X¯¬t¢ SC§
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
x
0
0.5
1
1.5
2
2.5
3
3.5
4
h
t=2.6x10
−7
t=6.9x10
−6
t=8.2x10
−5
t=7.1x10
−4
¤'Xt X º¶ X"¤[¥X¯¬t¢u©«²¥X±¯¬¤ª¢k K'¡u<«¬" ³Jt ©¤"t»%¡u«¬[¥"«µ "[¥¥X¯¬¢u®£uXt¡«¬"¥,-´6"'©¤"¯¬X¤"°u«© ¥Xt¡8t®t©¡±EµgH%¤"t»%¡©X¯¬ t¢ 8"¥6"k"¢U¢E°u»%"X¯¬¤k©« ¡uX"£u¯¬¤[¥¯¬t¢u µ»8t«¬ v©¢£ ¥±u X¯¬»%¯¬«©X¯¥¶µ Xt«¬°¥¯¬t¢ I[§