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g|SC"D5Y>=:D2FPC[<I?Ua"=6?)Q`=UA$AŠY>=:s7C"DlHÑsNIDGFG<>=:DUjzKPCuHR=RYIEÊ]=:DG=US­F2K5N>DGs•?"Q`=FP=USIKGECuS¡KG<ICuNIAY—\]=@NZKP=UY¡ES

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=USIQ`=UKFG<>=:DG=UES¤f.;<>=‚ESIEFGE?"A]Q`=UAAz\1CuNZSIYI?DGX6E$KƒFG<>=yQ:ED5Q:A=‚Ô

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;,<IE$KQ:?"Si\]=mES­FP=:D5^IDG=:FP=UYb?"K6FG<>=)s7CuA$AC–cKUÕ)ghSbKGCuH=TKGHR?"A$Aƒ^Z?DGFC"s“FG<>=TQ`=UAA \]CuNISZYI?DGX"j.FG<>=:D5=TEK

ç•è­é

−1

−0.5

0

0.5

1

−1

−0.5

0

0.5

1

t=0,

λ

=0.01, M=80,

µ

+

=

µ

−

=1

−1

−0.5

0

0.5

1

−1

−0.5

0

0.5

1

t=163.28

E

_area

=3.95%

−1

−0.5

0

0.5

1

−1

−0.5

0

0.5

1

t=39.4531

E

_area

=1.05%

−1

−0.5

0

0.5

1

−1

−0.5

0

0.5

1

t=109.765625

E

_area

=3.55%

çê1é

−1

−0.5

0

0.5

1

−1

−0.5

0

0.5

1

t=0,

λ

=0.01, M=80, (0.01 1)

−1

−0.5

0

0.5

1

−1

−0.5

0

0.5

1

t=85.1378

E

_area

=3.71%

−1

−0.5

0

0.5

1

−1

−0.5

0

0.5

1

t=14.5128

E

area

=2.07%

−1

−0.5

0

0.5

1

−1

−0.5

0

0.5

1

t=37.9503

E

_area

=3.68%

ç•ë–é

−1

−0.5

0

0.5

1

−1

−0.5

0

0.5

1

t=77.0128

E

_area

=3.72%

−1

−0.5

0

0.5

1

−1

−0.5

0

0.5

1

t=0

−1

−0.5

0

0.5

1

−1

−0.5

0

0.5

1

t=10.6074

E

_area

=1.53%

−1

−0.5

0

0.5

1

−1

−0.5

0

0.5

1

t=30.1222

E

_area

=3.44%

ç•ì§é

−1

−0.5

0

0.5

1

−1

−0.5

0

0.5

1

t=0

−1

−0.5

0

0.5

1

−1

−0.5

0

0.5

1

t=14.6779

E

_area

=1.00%

−1

−0.5

0

0.5

1

−1

−0.5

0

0.5

1

t=1.0761

E

_area

=0.51%

−1

−0.5

0

0.5

1

−1

−0.5

0

0.5

1

t=2.9288

E

_area

=0.11%

íŸîïuð>ñGòWóOôŠõ,ö>ò ë òU÷$÷ ê]ø ðIù ìZè ñGú èûƒì îü]ò:ñGòUù û’û îýRò ë`ø ýþZð û ò ì ðIÿ5îù>ï èyê ú ï"ñ5î ì6è ù ì‘èVë`ø ùIÿ ûGè ù û ÿGð>ñ è"ë ò û òUùIÿGî ø ù æ îù ç•è­étè ù ì çê]éŸè ù ì æ îù ç•ë–éyè ù ìÏç•ì§é öIî÷ò û ö>ò ê]ø ðIù ìZè ñGú îù è ÷$÷ ë:è ÿPòUÿ è þIþIñ øuè"ë ö>òUÿ ûPødû ö>òmòOðIî÷î ê ñ5îðIý ÿ ûGèû ò ,èdë îñ ë ÷ò eû ö>ò û î$ýòTÿ ë:è ÷ò[îÿ"ò:ñGú ì îü1ò:ñGòUù û õ,ö>ò ê1ø ðZù ìIè ñGúý ø "òUÿ è ÿ û ò:ñ è ÿ ø ñ û ö>òyñ èû î ø ù oï"ò û ÿW÷ è ñGï"ò:ñ 2ç•è­é õ,ö>ò2ñ5òUÿGðI÷ û! î û ö)òOð è ÷ îÿ ë`ø ÿGî û

ú"#$ % æ

“è ù ì & æ èûû îýò(') ') ó+* -,/. ó æ '0 æ * 213+.+34yè ù ì '5 æ 3 ó 76+4 õ,öIòRÿGö è þ]òRîÿ8 è ñ è9!eè ú:7ñ ø ý û ö>òRòOðIî÷î ê ñ5îðIýwÿ ûGèû ò‘ò"òUù ! î û ö('; æ 3 ó ¡çê]é õ,ö>ò ñGòUÿGðZ÷ û<! î û ö ì îü1ò:ñ5òUù û Oî$ÿ ë`ø ÿGî û ú æ æ è ù ì : æ èûWû îýRò=' '

æ ,>7. æ 6+4

'; ó 1? * . ó ’è ù ì '8 +.4 æ ó 14 õ,ö>ò6ÿ5ö è þ]òRîÿ;"ò:ñGú ë ÷ ø ÿPò ûPø—û ö>òRòOðIî÷î ê ñ5îðIý ÿ ûGèû ò èû '; +.4 @ ø)û ö>òRý ø"û î ø ùîÿ‚ýVð ë ö è ÿ û ò:ñ û ö è ù û ö>ò ë:è ÿPò ! î û öòOð è ÷Oî$ÿ ë`ø ÿGî û ú Tç•ë–é õ,ö>òñGòUÿ5ðI÷ û5! î û ö—òOð è ÷ îÿ ë`ø ÿGî û

ú)A : æ

gih/j

−0.5

−0.4

−0.3

−0.2

−0.1

0

0.1

0.2

0.3

0.4

0.5

−0.5

−0.4

−0.3

−0.2

−0.1

0

0.1

0.2

0.3

0.4

0.5

M=160, N

_b

=160,

µ

=(1,1)

t=0

E

_area

= 0.64%

t=68.125

gYkaj

−0.5

−0.4

−0.3

−0.2

−0.1

0

0.1

0.2

0.3

0.4

0.5

−0.5

−0.4

−0.3

−0.2

−0.1

0

0.1

0.2

0.3

0.4

0.5

M=160, N

_b

=160,

µ

+

=0.01,

µ

−

=1

t=0

E

_area

=0.59%

t=18.2807

lmonIp>qErs>t;uvwrxyrzUz ka{ p}|w~ h qE h€ ~wm‚ƒrqEr| €5€ mP„r+…=uvwr~ {+€†€ r~:‡}z {+€ mPˆ € vwrmP|wm € m h z‰xyrzPz kƒ{ pw|w~ h qE Š vwmUxv‹mUˆ € v>rxmoqxzorŒGŽ‘-s/’4…5uv>r“ˆp>qE” h xyr € r|wˆEm { |‹mPˆ<nImo•+r|0mP| g s>…7– j … gih/j uv>r“ˆ { zPmP~‹zUmP|>r;mPˆ € v>r xyrzPz kƒ{ pw|w~ h qE h€;— Ž&˜+™4‘ ef ’‹mP| € v>rx h ˆ†r { ”Lrš4p h z•mPˆEx { ˆEm € )› œAŽ›žŸŽ e …uv>rGqErz h€ mP•+r h qEr h xv h |>n+r h€—

Ž ˜+™4‘ef ’RmPˆ‘7˜[s?¡¢… gYkaj uv>r=ˆ { zPmP~0zPmP|>r“mPˆ € vwr=xyrzPz kƒ{ pw|w~ h q h€5—

Ž e ™4‘f ™/£RmPˆ

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R› œ Ž¤‘ e

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Ž e ™4‘f ™/£¦mPˆ

‘7’+§I¡¢…©¨¥n h mP| € v>r„ {+€ m { |‹mPˆL” h ˆ € rq Š v>r| € vwr;q h€ m {

› žª › œ mPˆJz

h

qn+rq…

gih/j

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

Evolution of the front, M=160, N

_b

=160,

µ

=(1,1)

t=0

t=14.6875

gYkaj

−0.25

0

−0.2

−0.15

−0.1

−0.05

0

0.05

0.1

0.15

0.2

0.25

0.05

0.1

0.15

0.2

0.25

0.3

0.35

t=0

t=3.75

t=6.25

t=9.3625

t=14.6875

Blow−up of tips, M=160, N

_b

=160,

µ

=(1,1)

Relative Error in Area

e(3.75)=0.35%

e(6.35)=0.044%

e(9.3625)=3.2%

e(14.6875)=1.92%

lmonIp>qErc’4t=uv>rGr• { zPp € m { | { ” € v>rcxyrzPz ka{ p}|w~ h qE)” { q“” { q«C¬ h „‡}zorc– Š m € v:rš?p h zD•4mUˆEx { ˆEm € )› œ Ž

ÁÂ;ÃEÄ>ÅGÆÅÂEÇwÈoÃÉFÃEÄwÅcÊyÅÈPÈDË¥ÌUÈPȉÍ+ÆÎ+ÏwÇwÎ+ÈPÈPÐ)Ï>ÅÑYÒ+ÆÓÔÎ+Ò+Õ}†ÅÆEÖ+ÅÏÌU×:ÃEÄ>ÅGÅyØÙaÅÆÌPÓGÅ×?ÃEÂÚÛÜ×:݉ÌPÍ>ÚÞ4É

Ë©Å¢Ù°ÈoÒ+ÆÆÅϟÃEÄ>ÅRÅÖ+ÒIÈPÇ>ÃEÌoÒIןÙ}ÆEÒ4ÊyÅ“Ò+Ñ<ÃEÄ>ÅBÊyÅÈPÈLՃÒIÇw×wÏwÎÆÐ0Ë<ÌoÃEÄßÅà?ÇwÎ+ÈLÖÌPÂEÊyÒIÂÌoÃáÐâ ã äåâæçä ¿ É

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äÞ+éêÉêÎ+×wÏ0ëäéFÉ°Ë<Ä>ÅÆEœé‹äíìWîLïßðñ†ò_ó äíìiô=ïßõñ†ò_ö÷ɃÌPÂ¥ÃEÄwϠÙ}ÎÃEÌUÎ+È Â†Ã†ÅÙ(ÂEÌoøÅ+ÚJù:ÅG†ÅœÃEÄwÎÃ

ÊyÅÈPÈÌPÂÏwÌoÖÌPÏwÌU×>Í=Î+×}ÏÅÖ+Å×?ÃEÇwÎ+ÈPÈoÐRË¥ÌPÈUȃÕwÆEÅÎúBÇ>ÙFÚLûÄ>ōÂÌPӓÇ}ÈPÎÃEÌoÒI×BÌPÂ<Ï>ÒI×>Å;Ë¥ÌPÃEÄÎ ¿üý Õ?Ð ¿üý Í+ÆÌPÏ

Î+×wÏ ¿üý ÓcÎÆEú+ÅÏBكÒIÌP×?ÃEÂþÒI×ÃEÄ>Å;ÊyÅÈPȃՃÒIÇw×wÏwÎÆÐ+ÚÝÌoÍ>ÚwÞBìYÕañ©ÌUÂLÃEÄ>Å8Õ}ÈoÒËJÿáÇ>ÙRÙ}ÈoÒ+ÃLÒ+щÃEÄ>Å;Ç>ÙwكÅÆLÃEÌoÙFÚ

ûÄ>Å5ÎÆEÅÎGÊÄwÎ+×>Í+ÅÂLË¥ÄwÌUÊÄRÌP©Î+ÈP†ÒGÓÎÆEú+ÅÏRÒI×¢ÃEÄ>Å}ÍIÇ>ÆÅ+ÉÎÆÅÆÅÈPÎÃEÌoÖ+ÅÈoÐcÂÓÎ+ÈPÈWÉÈoÅÂE©ÃEÄwÎ+× À É>ÑYÒ+Æ

ÂEÇwÊÄR΍ÙwÆÒ+Õ}ÈoÅÓ Ë<ÌoÃEÄcÃEÄwÅ<ÃEÌoÙ°ÂHÏ>ÅÅكÅ×wÌP×w͓†ÒӓÇ}ÊÄRÌP×ՃÒ+ÃEÄ¢ÏwÌoÆEÅÊyÃEÌPÒI×wÂÚÛ ×¢ÝÌoÍ>Ú ü É4Ë©ÅÙ}ÈoÒ+ÆÆÅÏ¢ÃEÄ>Å

ÂEÌPÓ=ÇwÈPÎÃEÌPÒI×Ë¥ÌoÃEĦÏwÌ ƒÅÆEÅ×?ÃHÖÌPÂEÊyÒIÂEÌoà ÐGâ ã:ä ýý>¿ Î+×wÏcâæ)ä ¿ ÉË©ÅÂEÅÅÂEÌUÓÌPÈPÎÆCՃÅÄwÎ9Ö4ÌPÒ+ÆþÎ+©ÃEÄ>Å5ÊÎ+†Å

Ò+ÑÃEÄ>Å8Åà4ÇwÎ+ÈêÖÌPÂEÊyÒIÂEÌoà ЦհÇ>ÃJÑYÎ+ÂEÆÅÆ<ÓGÒ+ÃEÌoÒI׋Î+ÂJËþœÏwÌPÂEÊÇw†ÅÏRÑdÒ+ÆDØ>Î+ÓGÙ}ÈoÅ

¿

Ú

ìiÎ/ñ

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

Evolution of the front, M=160, N

_b

=160,

µ

=(0.01,1)

t=0

t=2.3392

ìYÕañ

−0.25

0

−0.2

−0.15

−0.1

−0.05

0

0.05

0.1

0.15

0.2

0.25

0.05

0.1

0.15

0.2

0.25

0.3

0.35

t=0

t=0.5235

t=0.8773

t=1.1600

t=1.4691

t=1.8214

t=2.3392

Relative error in Area

e(0.5235)=1.34%

e(0.8773)=0.20%

e(1.16)=1.47%

e(1.4691)=2.73%

e(1.8214)=4.82%

e(2.3392)=6.68%

Blow−up of tips, M=160, N

_b

=160,

µ

=(0.01,1)

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