HAL Id: tel-00804116
https://tel.archives-ouvertes.fr/tel-00804116
Submitted on 24 Mar 2013
HAL is a multi-disciplinary open access
archive for the deposit and dissemination of
sci-entific research documents, whether they are
pub-lished or not. The documents may come from
teaching and research institutions in France or
abroad, or from public or private research centers.
L’archive ouverte pluridisciplinaire HAL, est
destinée au dépôt et à la diffusion de documents
scientifiques de niveau recherche, publiés ou non,
émanant des établissements d’enseignement et de
recherche français ou étrangers, des laboratoires
publics ou privés.
Walid Kheriji
To cite this version:
Walid Kheriji. Pressure correction schemes for compressible flows. Numerical Analysis [math.NA].
Université de Provence - Aix-Marseille I, 2011. English. �tel-00804116�
-.-%- /0 123456237890:; "<=>?6237890 03 15 2<7890
(,+A$ )+,'+%BA$ )$ 1B'C(1B'"D $& $' "!.+%1B'"D $
$-) <965?> EFG
!"#$
!"# !$%&'(#)&*#+,& ,&
!"#$%& $ '(%)*+$&,*#- $ .&!+$)"$
!" !$%!&' ( )*+,-)*+./012
3$4!5& ( *6*7821
#-.&'%-& &% .!"%&'"& "$)(/"&0&'%
+# !"#$ %&'()*) )& 123!4&0$#& 5166 7(%#& 8 +,-&./'0 /' 1.(('1-).2 /' 3('00).2 3.4( 5'0 ,647-).20 /' 278)'(90-.%'0 1.+3('00):5'0 )7?0 309? /0 34H:0 I 9#: ;+ <+=)&>?;@A3 J %K !"#$%$#& ()*+,- ./001#234# !53#6$7+8--9.: 8;.89<=10>&/9?2&01@&A ./001#234# !B3/?C /# 5D.9.: ,:"<(>/214 ,E/F&?/234#
F3"@1#3? 35+G,.H /I2#3%3(1?J$#3? 3A<+?&63#A&2$%3K#163? 3 ,E/F&?/234#
F3./0>/L@35,.G8; K#1J3AA34#<+?&63#A&2$%3K#163? 3 : 234#%32>LA3
-.-%- /0 123456237890:; "<=>?6237890 03 15 2<7890
(,+A$ )+,'+%BA$ )$ 1B'C(1B'"D $& $' "!.+%1B'"D $
$-) <965?> EFG
!"#$
!"# !$%&'(#)&*#+,& ,&
!"#$%& $ '(%)*+$&,*#- $ .&!+$)"$
!" !$%!&' ( )*+,-)*+./012
3$4!5& ( *6*7821
#-.&'%-& &% .!"%&'"& "$)(/"&0&'%
+# !"#$ %&'()*) )& 123!4&0$#& 5166 7(%#& 8 +,-&./'0 /' 1.(('1-).2 /' 3('00).2 3.4( 5'0 ,647-).20 /' 278)'(90-.%'0 1.+3('00):5'0 )7?0 309? /0 34H:0 I 9#: ;+ <+=)&>?;@A3 J %K !"#$%$#& ()*+,- ./001#234# !53#6$7+8--9.: 8;.89<=10>&/9?2&01@&A ./001#234# !B3/?C /# 5D.9.: ,:"<(>/214 ,E/F&?/234#
F3"@1#3? 35+G,.H /I2#3%3(1?J$#3? 3A<+?&63#A&2$%3K#163? 3 ,E/F&?/234#
F3./0>/L@35,.G8; K#1J3AA34#<+?&63#A&2$%3K#163? 3 : 234#%32>LA3
!"!#$%&'"()!&*& !"#$%&' #)%"* * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * + ,-./-.0"&'"1"%2". * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * 3 3 4%56$-..)78-70$%#$%6) 90:)-$;<#%1-.-='0#)%". * * * * * * * * * * * * * * * * * * * * * * * > 3* ?" )568) )#. /-5-. /-5- * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * @ 3*+ ? 6$-..'$- %$$- #)%". /-5- * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * 3 A 4%56$-..)78-90:)-$;<#%1-.-='0#)%". * * * * * * * * * * * * * * * * * * * * * * * * * * * * * B > C'8-$-='0#)%". * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * D
>* E/-&). $-#-1)"-#) -"-$FG7080" --='0#)%"0"&#/- %$$- #):-.%'$ -#-$5. * * * +H
>*+ I0..)"F#% #/-8)5)#)"#/-. /-5- * * * * * * * * * * * * * * * * * * * * * * * * * * * ++
>*3 9'5-$) 08#-.#. * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * +3
B 4%" 8'.)%"0"&6-$.6- #):-. * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * +A
@ ?IIC9J!KL<%5-$-.'8#.0..% )0#-&#% M")#-:%8'5- %":- #)%"%6-$0#%$.* * * * * * * * +B
+)$,*(#!
-.#!&&/#! 1##! (*1" &($22!#!3 & )!4!&51# 6$#1(#1,* 41"1,)$&* $"3 (718,)$&!917&
!* !"#$%&' #)%"* * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * 33 !*+ E/-6$-..'$- %$$- #)%". /-5- * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * 3> !*+* E)5-.-5);&). $-#)N0#)%" * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * 3> !*+*+ J). $-#-.60 -.0"&'"1"%2". * * * * * * * * * * * * * * * * * * * * * * * * * * * * * 3B !*+*3 J). $-#--='0#)%". * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * 3O !*3 9'5-$) 08-P6-$)5-"#.QR)-50""6$%78-5. * * * * * * * * * * * * * * * * * * * * * * * * * * A
"#$#% &'()*' ,-./012'/304 # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # 56
"#$#6 78'9:.!()*' ,-./0 # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # 5;
"#5 <-8'9(!40=)!'=>3-0)- >)0 # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # ?@
"#? A'= 3.)!'= # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # B%
!"#$%&' ((
)*+$+%'*%+%",,'&'-+ !'/'+0)& )/#&'++$12'3)4+56"&%( 71"&)%)#$ 8"9$'&:;%)<'+
'=>"%$)*+? ""#% "=-2'(. -!'=# # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # B? ""#6 C0)*0)>=(.=,='8=) # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # B? ""#$ &'4020).3-)>))' !>-0(-' D=!-0 '3.40 '= 0 -!'='102>-'2) # # # # # # # # # # # # # # # # BB ""#5 <= !413! !-) *040 # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # BE ""#5#% 7*0) *040 # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # BE ""#5#6 F)-!4>-0) # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # @% ""#5#$ G>))!=H -'-*03!4!-!= -*0) *040 # # # # # # # # # # # # # # # # # # # # # # # # # # # @5 ""#? G20)).20 '220 -!'=) *040 # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # ;I ""#?#% 7*0) *040 # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # ;I ""#?#6 F)-!4>-0) # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # ;% ""#?#$ G>))!=H -'-*03!4!-!= -*0) *040 # # # # # # # # # # # # # # # # # # # # # # # # # # # ;$ !"#$%&' (((
@* >* )*-$%$)*"22A +%"12' #&'++>&' )&&' %$)* + !'/' 0)& )/#&'++$12' 8"9$'&:;%)<'+
'=>"%$)*+ """#% "=-2'(. -!'=# # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # ;E """#6 C0)*0)>=(.=,='8=) # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # EI """#$ 7*0-!409!413! !-=.402! >3) *040 # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # E6 """#$#% &04!9(!) 20-0>3H'2!-*4 # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # E6 """#$#6 7*0J.33K(!) 20-0>3H'2!-*4 >=(!-)D2)-12'102-!0) # # # # # # # # # # # # # # # # # # E$ """#5 G20)).20 '220 -!'=) *040 # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # EE """#5#% &04!9(!) 20-0>3H'2!-*4 # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # EE """#5#6 L!) 20-0>3H'2!-*4 # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # %II """#5#$ G2'102-!0)'J-*0) *040# # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # %I%
"""#? <110=(!MN -*0C<A(!) 20-!O>-!'='J-*0(!))!1>-!'=-024# # # # # # # # # # # # # # # # # # %I$
"""#?#% 7*0-8'9(!40=)!'=>3 >)0 # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # %I$
"""#?#6 FM-0=)!'=-' -*0-*2009(!40=)!'=>3 >)0 # # # # # # # # # # # # # # # # # # # # # # # # %IB
!"#$%&' (B
)*+$+%'*%+%",,'&'- + !'/'+0)& )/#&'++$12'3)4+ 56"&% (( 7 C>2'&'=>"%$)*+?
"P#% "=-2'(. -!'=# # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # %%%
"P#6 C0)*0)>=(.=,='8=) # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # %%6
"P#$ <= !413! !-) *040 # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # %%$
!"#"$ %&'()* ,'-'.)/'-) '/',012343/ ''563-)7/3/(-&' 7,,' -)8'*76, '-',9* " " " ::;
!"#"# <3**)/0-7 -&'4)9)-)/-&'* &'9' " " " " " " " " " " " " " " " " " " " " " " " " " " " ::=
!"> ?@,'**6,' 7,,' -)7/* &'9' " " " " " " " " " " " " " " " " " " " " " " " " " " " " " " " " " " :$A
!">": %&'* &'9' " " " " " " " " " " " " " " " " " " " " " " " " " " " " " " " " " " " " " " " " :$A
!">"$ %&'()* ,'-'.)/'-) '/',012343/ ''563-)7/3/(-&' 7,,' -)8'*76, '-',9* " " " :$;
!">"# <3**)/0-7 -&'4)9)-)/-&'* &'9' " " " " " " " " " " " " " " " " " " " " " " " " " " " :$B
!"A C69',) 34-'*-* " " " " " " " " " " " " " " " " " " " " " " " " " " " " " " " " " " " " " " " " " " :#D !!"#" $%" &'"()!!*+,-."( /,+0%" %,("1"!",23(,4". ?": %&'*1*-'97E 7/*',83-)7/43F*3/()-*93-&'93-) 34@,7@',-)'* " " " " " " " " " " " " " " :#G ?"$ H746-)7/7E-&'I)'93// @,724'9 " " " " " " " " " " " " " " " " " " " " " " " " " " " " " " " " :>: ?"$": H7(*&7 .-62' " " " " " " " " " " " " " " " " " " " " " " " " " " " " " " " " " " " " " " :># ?"$"$ %F7JK6)(*&7 .-62' " " " " " " " " " " " " " " " " " " " " " " " " " " " " " " " " " " " :>> !!"#"3 !!"#" 5 60)11"+"4 4'3 +"0'8)0',!39 *+"332+" ,++" 0',!3 3 %"("3 )!4 ).. 3*""4 -)+,: 0+,*' ;,<3 =>? L": /-,7(6 -)7/" " " " " " " " " " " " " " " " " " " " " " " " " " " " " " " " " " " " " " " " " " " " :>=
L"$ %&'* &'9'*M0'/',34E7,9 " " " " " " " " " " " " " " " " " " " " " " " " " " " " " " " " " " " :>=
L"$": N'*&'*3/(6/./7F/* " " " " " " " " " " " " " " " " " " " " " " " " " " " " " " " " " " :>=
L"$"$ %&'* &'9'* " " " " " " " " " " " " " " " " " " " " " " " " " " " " " " " " " " " " " " " :A:
L"# %&'*-32)4)-1)**6'3/( 7/*'56'/ '* " " " " " " " " " " " " " " " " " " " " " " " " " " " " " " :A#
L"#": ? *-32)4)-1,'*64-E7,-&' 7/8' -)7/ " " " " " " " " " " " " " " " " " " " " " " " " " " :A#
L"#"$ O)* ,'-)P3-)7/7E-&' 7/8' -)7/-',9" " " " " " " " " " " " " " " " " " " " " " " " " " :A>
L"> H &'9'*3/(*-32)4)-1'*-)93-'* " " " " " " " " " " " " " " " " " " " " " " " " " " " " " " " " " :AG
L"A Q64','563-)7/*3/( *746-)7/*F)-&*&7 .* " " " " " " " " " " " " " " " " " " " " " " " " " " " :;D
L"; O)* 6**)7/3/(@',*@' -)8'* " " " " " " " " " " " " " " " " " " " " " " " " " " " " " " " " " " " :;D !!"#"@ A'3 +"0'8)0',! ,/0%" B'3 ,23 4'33'*)0',! 0"+(<'0% 0%" C @ 3 %"(" =DE R": /-,7(6 -)7/" " " " " " " " " " " " " " " " " " " " " " " " " " " " " " " " " " " " " " " " " " " " :;A R"$ O)* ,'-)P3-)7/7E-&'()**)@3-)7/-',9 " " " " " " " " " " " " " " " " " " " " " " " " " " " " " " :;; R"$": %&'-F7J()9'/*)7/34 3*' " " " " " " " " " " " " " " " " " " " " " " " " " " " " " " " " :;; R"$"$ QS-'/*)7/-7-&'-&,''J()9'/*)7/34 3*' " " " " " " " " " " " " " " " " " " " " " " " " :;= R"# ?*-,7/0 7/8',0'/ ','*64- " " " " " " " " " " " " " " " " " " " " " " " " " " " " " " " " " " " :GD 5'-.',1+)*%'"
!"#$%&' #)%"
!"# $%&' $(# )*&+%&,*- (. .!* !"#$#%# &' ()&$*+,*#' #$*! '# &' ./,'#0 1% 20)$,' /012345 !* #'"66
7*6-# %+ 0123 %9*& (66 &"#'# &*6(.*- .% "%:";":< &(-"(."%:= >#*- ": ":->#.&? %& ,*-" ":*= %& :(.>&(6
&(-"(."%:5 2)* "7 (66?= 0123 (&&"*# *@)*&."#*# (:- %:-> .# &*#*(& ! ": .!* -%,(": %+ :> 6*(& #(+*.?=
)&%.* ."%:(<(":#."%:";":<&(-"(."%:= %:.&%6 (:-)&%.* ."%:%+ :> 6*(&,(.*&"(6#(:-)&%.* ."%:(<(":#.
,(6" "%>#( .#5A: *##*:."(6)(&.%+ .!*#(+*.?(:(6?#"# %:#"#.# ": #.>-?":<.!* -"B*&*:.#".>(."%:# .!(.
(:> 6*(& &*( .%& (:+( *=+&%,:%&,(6%)*&(."%: %:-"."%:#.% #*9*&*( "-*:.#5C(:?D%$#%+":.*&*#.
":.!"# %:.*@.(&* %,)&*##"E6*=*".!*& ,%:%)!(#" /!?-&%<*: %,E>#."%:=-*D(<&(."%:%&-*.%:(."%:":
.!*&*( .%& %:.(":,*:. ": .!*6(.* )!(#*#%+ #*9*&*( "-*:. # *:(&""= *@)6%#"%:%+ <(#*%>#,"@.>&*# ":
":->#.&"(6*:9"&%:,*:.=5554%&,>6."F)!(#" /)&",(&?( "-*:.-*)&*##>&";(."%:=E>EE6":<)%%6#
<*:*&(.*-E?.!*":.*&( ."%:E*.$**:.!*,%6.*:#.&> .>&*#%+.!*9*##*6(:-.!* %&*(:-.!* %: &*.*D%%&%+.!*
%:.(":,*:.=%: * (<(":": #*9*&*( "-*:.#6(.*)!(#*#=55545
G>&(",!*&*"# .% %:.&"E>.*.%.!*-*9*6%),*:.%+( 6(##%+# !*,*#+%&.!* %,)>.(."%: %+ %,)&*#F
#"E6*D%$#5 !* %:#"-*&*- #?#.*,#%+ <%9*&:":<*H>(."%:# (&* %>)6*-(:- #.&%:<6?:%:6":*(&= (:-.!*
/":->#.&"(64 ())6" (."%:# ": 9"*$":9%69* %,)6*@ <*%,*.&? (:- D%$#= )%##"E6? %,E":":<H>(#"F#.*(-?
#.(.*#$".! H>" ' .&(:#"*:.)!(#*#=$".! #.&%:<)!?#" (6 )&%)*&."*#/": )(&." >6(&= -*:#".? %& %,)&*##"F
E"6".?4 %:.&(#.#5A %&-":<6?=.!* (6<%&".!,# (&*-*9*6%)*-#% (#.% &*(6";*( %,)&%,"#*E*.$**: .$%
,(": &*H>"&*,*:.# I )&*#*&9* .!* #.(E"6".? ": ( $"-* &(:<* %+ C( ! :>,E*&# (:- ":.&%-> * #>J "*:.
-* %>)6":< .% +( "6".(.* .!* &*#%6>."%: %+ -"# &*.* (6<*E&(" #?#.*,#5 K&*##>&* %&&* ."%: ,*.!%-# #**,
.% E* ( <%%- !%" * .% (--&*## .!*#* &*H>"&*,*:.#5 !"# 6(## %+ # !*,*# $(# 7&#. ":.&%-> *- ": .!*
+&(,*$%&'%+": %,)&*##"E6*D%$#(6%:<.",*(<%LM=NO℄=(:-#> !(6<%&".!,#(&*:%$H>".*
!"#$"%$&'()*+),$#-./*'##0*' "**' $+",# &'!'#)*' 2'##3"302)*+,$&' ",$'4$ "% "!3*'##+52'6"7#8
'(', $&"09& $&'+* )332+ )$+", $" "!3*'##+52':)(+'*;<$"='# '>0)$+",#!)? )2#" 5' $*) '@ 5) = $" $&'
2)$' #+4$+'#8 7+$& $&' #'!+,)2 7"*= "% A)*2"7),@ B!#@', CDE8 DF℄8 7&" @'('2"33'@ ), +$'*)$+(' )29";
*+$&!H$&'#"; )22'@IJK!'$&"@-+, 20@+,9),'22+3$+ "**' $"*#$'3%"*$&'3*'##0*'.L)$'*",83*'##0*'
"**' $+", '>0)$+",# )33')*'@+, ,0!'*+ )2# &'!'# 3*"3"#'@ 5? #'('*)2 *'#')* &'*#8 '##',$+)22?+, $&'
M,+$';("20!' %*)!'7"*=80#+,9 '+$&'* ) "22" )$'@ CF 8 NE8 OF8 FO8 OD8 EF℄ "* ) #$)99'*'@ )**),9'!',$
CP8ON8 O 8OO8 D8NQ8FR8 PD8 PO8 PQ8 P ℄"% 0,=,"7,#S +,$&' M*#$ )#'8#"!' "**' $+(') $+",#)*' $"5'
%"*'#'', $" )("+@ $&'0#0)2 "@@;'(', @' "032+,9"%$&' 3*'##0*'+, $&'2"7 T) & ,0!5'**'9+!'.<"!'
"% $&'#' )29"*+$&!#)*''##',$+)22?+!32+ +$8 #+, '$&'M,)2#$)9' "% )$+!'#$'3+,("2('#$&'0,=,"7,)$
$&'',@;"%;#$'3 $+!'2'('2S$&'',@;"%;#$'3#"20$+",+# $&',"5$)+,'@5?<IT/LK;2+='+$'*)$+('3*" '##'#
CPN8OO8NE8OF8FO8OD8EF℄.U&'"$&'*# &'!'#CON8O 8F 8D8NQ8PE8FR8PO8PQ8P ℄)*'3*'@+ $"*; "**' $"*
!'$&"@#87&'*'5)#+ )22?$7"#$'3#)*'3'*%"*!'@#'>0',$+)22?VM*#$)#'!+;'432+ +$@' "032'@3*'@+ $+",
"% $&' !"!',$0! "* ('2" +$?H),@ 3"##+52? ','*9?8 %"* ,",;5)*"$*"3+ 6"7#-),@8 #' ",@8) "**' $+",
#$'3 7&'*' $&' ',@;"% #$'3 3*'##0*' +# '()20)$'@ ),@ $&' !"!',$0! ),@ ('2" +$? )*' "**' $'@8 )#+,
3*"W' $+",!'$&"@#%"*+, "!3*'##+52'6"7#H#''CX8FP℄%"*$&'"*+9+,)23)3'*#8CEE℄%"*) "!3*'&',#+('
+,$*"@0 $+",),@ C R℄%"* )*'(+'7"%!"#$()*+),$#-.U&' J&)*) $'*+#$+ ;Y)#'@<32+$HJY<- # &'!' H#''
CFQ℄ %"*) *' ',$*'(+'7 "*CPP℄ %"*$&' #'!+,)2 3)3'*- 7)# @'('2"3'@ +, $&' M,+$';'2'!',$ ",$'4$),@
5'2",9#$"$&+#2)$$'* 2)##"%!'$&"@#.
I, $&+# 7"*=8 +!32+ +$;+,;$+!' @+# *'$+Z)$+",# )*' )@@*'##'@%"* $&'+*H*'2)$+('- #+!32+ +$? +, (+'7 "% $&'
$&'"*'$+ )2#$0@+'#S&"7'('*8,",;+$'*)$+('3*'##0*' "**' $+",# &'!'#)*'"0*!)+, ", '*,%"*3*) $+ )2
"!30$)$+",#.[' ",#+@'*&'*'#$)99'*'@\+,\#3) '@+# *'$+Z)$+",#87+$&$&')+!$"50+2@# &'!'#7&+ &
)*' #$)52' ),@ ) 0*)$' )$ )22 T) & ,0!5'*# ),@8 +, 3)*$+ 02)*8 5"+2 @"7, $" ) 0#0)2 )29"*+$&! %"*
+, "!3*'##+52'6"7#H"*8!"*'9','*)22?8%"*$&')#?!3$"$+ !"@'2"%(),+#&+,9T) &,0!5'*6"7#
CEO℄-7&',$&'T) &,0!5'*$',@#$"Z'*".U&+#2)#$*'>0+*'!',$)2#"+!32+'#$&)$8+%7'+!32'!',$037+,@+,9
$' &,+>0'# H),@ 7' 7+22 &)(' $" %"* #$)5+2+$? *')#",#-8 037+,@+,9 !)? &)(' $" 5' 3'*%"*!'@ %"* ') &
'>0)$+",#'3)*)$'2?),@7+$&*'#3' $$"$&'!)$'*+)2('2" +$?",2?.U&+#+#+, ",$*)@+ $+",7+$&$&'!"#$
"!!", #$*)$'9?)@"3$'@ %"* &?3'*5"2+ #?#$'!#87&'*' 037+,@+,9 +# 50+2$%*"! $&' 7)('#$*0 $0*' "%
$&'#?#$'!H#'' !"CFX8O℄%"*#0*('?#),@CDO8DD8NO℄%"*),)2?#+#"%$&'#'# &'!'#)$2"7T) &,0!5'*-8
),@?+'2@#)29"*+$&!#7&+ &)*'0#'@+,3*) $+ 'H#''8 !"8$&'#"; )22'@B]<T%)!+2?"%# &'!'#CED8E ℄-8
50$ #)* '2? #$0@+'@ %*"! ) $&'"*'$+ )2 3"+,$ "% (+'7. ^,' "% "0* !)+, ", '*,# &'*' 7+22 $&0# 5' $"
5*+,98)#%)*)#3"##+52'8$&'"*'$+ )2)*90!',$##033"*$+,9"0*,0!'*+ )2@'('2"3!',$#.L'$0#M*#$*' )22
) H3"##+52'- "!!", #='2'$", "% ",('*9', ' #$0@+'# +, $&' M,+$' ("20!' ",$'4$ CNF℄. U&' 3*""% !)?
0#0)22?5'@' "!3"#'@+,$" $&*''#$'3#V
(i)
U&'M*#$#$'3+#$"9'$$&''4+#$', '),@#"!'#$%&'%&'#$+!)$'#",$&')33*"4+!)$'#"20$+",8"*8 +,"$&'*7"*@#8$" "5$)+,#$)5+2+$?*'#02$#%"*$&'# &'!'.(ii)
:'4$8 03 $" $&' '4$*) $+", "% ) #05#'>0', '8 "!3) $,'## )*90!',$# ?+'2@ $&' '4+#$', ' "% )H3"##+52?7')=-2+!+$$")#'>0', '"%@+# *'$'#"20$+",#"5$)+,'@7+$&)#'>0', '"%@+# *'$+Z)$+",#
$&'#3) ' #$'3),@8 %"*0,#$')@? 3*"52'!#8 $&'$+!' #$'3"% 7&+ & $',@ $" Z'*".B$$&+# 3"+,$8 #
$%&'%&'#$+!)$'#!)?+!32?#"!'*'902)*+$?"%$&'2+!+$.
(iii)
_+,)22?8 $&'%) $ $&)$$&'2+!+$+#)#"20$+",$" H)7')=%"*!- "%$&' ",$+,0"0#3*"52'!+#3*"(', 5?3)##+,9$"$&'2+!+$+,H)7')=%"*!02)$+","%-$&'# &'!'.!"#$%&"!'(%)**#+,-%,$%"%./0)%(1#$% !)&"%**-'(%304-%"56#!7%*!"8+(%"%9+0#-!/*.#$%"%0(-:0#-!/
!; #$% !)&(%#% &"!<"0) *%%)*!+# !; "%0 $. ,+% #! #$% (0 7 !; !/#"!( =6#%&
(i)
> !; *&0 % #"0/*(0#%*!; #$% +/7/!?/@ $%/ % ?% !'#0-/0 !/4%"<%/ % !; *%9+%/ % !; ,-* "%#% *!(+#-!/*=6#%&
(i)
> -/ 0 *%/*%#!! ?%07 #! 0((!? #$% &0**0<% #! #$% (-)-# -/ #$% * $%)% =6#%&
(iii)
>A B$%"% -* #$+* /! $!&% 0# #$%&"%*%/#*#0<%#!&"!4%#$% !/4%"<%/ %!;#$%* $%)%*-/ #$%<%/%"0( 0*%*= !"! %C %&#;!"#$%'0"!#"!&-
4-* !+* 304-%"56#!7%*%9+0#-!/*. *%% DEF. FG. HF℄ ;!" #$%!"%#- 0( 0/0(1*-* !; #$% !/#-/+!+* &"!(%) 0/, DJF.FK.FL℄;!"* $%)% !/4%"<%/ %0/0(1*-*-/#$%*-)&(-M%, 0*%!;#$%*#%0,16#!7%*&"!'(%)>.0/,!+" #$%!"%#- 0(0/0(1*%*0"%#$%//% %**0"-(1*!)%?$0#-/ !)&(%#%AN!?%4%".-/'!#$#$%'0"!#"!&- 0/,#$% /!/5'0"!#"!&- 0*%*.0/,0#(%0*#;!")!*#40"-0/#*!;#$%* $%)%*.?%,!<%##$%;!((!?-/<"%*+(#*O
(i)
P% *$!? #$0# #$% ,-* "%#% *!(+#-!/ *0#-*M%* ,-* "%#% 0/0(!<+%* !; #$% %*#-)0#%* 7/!?/ -/ #$% !/#-/+!+* 0*%O &!*-#-4-#1 !;#$%,%/*-#1 0/,.-/ #$%/!/5'0"!#"!&- 0*%.!;#$%-/#%"/0(%/%"<1. ,% "%0*% !; #$% #!#0( %/%"<1. 0/,. ;!" #$% 4-* !+* '0"!#"!&- Q!?*. !/#"!( !; #$% 4%(! -#1 -/ #$%L
2
(H
1
)
/!")AB$%*%%*#-)0#%*0((!?#!&"!4%#$%%C-*#%/ %!;0#(%0*#!/%*!(+#-!/#!#$%* $%)%. '1#!&!(!<- 0(,%<"%%0"<+)%/#*A(ii)
#$%%&' () *+" &(-".)"( " &/*+" ' +"0" ('*.&()"(&$)+(&.0'.?%#$%/*$!?#$0##$%(-)-#*!;*%9+%/ %* !; *!(+#-!/*0"% ?%07 *!(+#-!/*#! #$% !/#-/+!+*&"!'(%). ?$- $ )01 '%*%%/=0/, -*
"%;%"%,#!$%"%0;#%">0*0 !/*-*#%/ 1&"!&%"#1!;#$%* $%)%*A
-/0((1. ?% !/;!"# #$%*% #$%!"%#- 0( %C&%"-)%/#* '1 /+)%"- 0( #%*#*. &%";!")%, ?-#$ #$% !&%/5*!+" %
*!;#?0"%R6R6DST℄. ,%4%(!&%,0# RU63!/#$%'0*-*!; #$%*!;#?0"% !)&!/%/#(-'"0"10/,&"!<"0))-/<
%/4-"!/)%/#V8WRXY36DHZ℄A
B$-* &0&%" -* !"<0/-:%,0* ;!((!?*A P% M"*# -/#"!,+ % #$% !/*-,%"%, *&0 % ,-* "%#-:0#-!/* =6% #-!/ J>A
B$%/ ?%#+"/ #! #$% '0"!#"!&- 304-%"56#!7%* %9+0#-!/* =6% #-!/ Z>. #! #$% [ !)&(%#%[ 304-%"56#!7%*
%9+0#-!/* =6% #-!/ S>. 0/,. M/0((1. #! #$% 8+(%" %9+0#-!/* =6% #-!/ E>@ ;!" %0 $ 0*%. ?% &"%*%/# #$%
* $%)%*.*+))0"-:%#$%#$%!"%#- 0("%*+(#*0/,#$%/+)%"- 0(#%*#*A
R/*%4%"0(#$%!"%#- 0(,%4%(!&)%/#*.?%0"%(%0,#!+*%0,%"-4%,;!")!;0,-* "%#%M/-#%4!(+)% !/4% #-!/
!&%"0#!"=;!"-/*#0/ %.#1&- 0((1.0 !/4% #-!/!&%"0#!";!"#$%7-/%#- %/%"<1.&!**-'(1?-#$"%*-,+0(#%")*.
!'#0-/%,;"!)#$%M/-#%4!(+)%,-* "%#-:0#-!/!;#$% !/4% #-!/!;#$%4%(! -#1 !)&!/%/#*>@0/0'*#"0 #
&"%*%/#0#-!/!;*+ $ !)&+#0#-!/*-*<-4%/-/#$%Y&&%/,-CA
B$-*!"<0/-:0#-!/ (!*%(1;!((!?* #$%#$%*-*!/%O '0"!#"!&- Q!?*0"% 0,"%**%,-/ #$%M"*# #?! $0&#%"*
=/+)%"- 0(#%*#*.;! +**%,!/#$%-/4-* -, 0*%.#$%/#$%!"1>.#$%/0* $%)%;!"304-%"56#!7%*%9+0#-!/*-*
&"%*%/#%,0/,-#**#0'-(-#1-*&"!4%/@M/0((1.?%*$!?$!?#!0,0&#-##! !)&+#%,-* !/#-/+!+**!(+#-!/
!;8+(%"%9+0#-!/*A
!"#$"# %&' (&)&*+&#
W%##$% !)&+#0#-!/0(,!)0-/
Ω
'%0/!&%/&!(1<!/0(*+'*%#!;R
d
.
d
≤ 3
. 0/,M
'%0&0"#-#-!/!;Ω
.*+&&!*%,#!'%"%<+(0"-/#$%+*+0(*%/*%!;#$%M/-#%%(%)%/#(-#%"0#+"%=")!DG℄>AB$% %((*)01'%O
5 ;!" 0<%/%"0( ,!)0-/
Ω
. %-#$%" !/4%C9+0,"-(0#%"0(*=d = 2
> !"$%C0$%,"0 =d = 3
> !"*-)&(- %*.! "#$ % &#'%() *+, -#.)&%$(,/ #" 0+( + %$, +23,$34%),/ )#$'%4 *# % ##$&()%*, %5(/6 $, *%)74,/
8
d = 2
9#$$, *%)7.4%$3%$%44,4,3(3,&/8d = 3
98*+,"% ,/#"0+( +6#" #.$/,6%$,*+,)%4/#), ,//%$(42)#$'%4*# % ##$&()%*,%5(/9:
;2
E
%)&E(K)
0,&,)#*,*+,/,*#"%44(d
−1)
!"% ,/σ
#"*+,',/+%)&#"*+,,4,',)*K
∈ M
$,/3, *(<,42:=+,/,*#",&7,/() 4.&,&()*+,-#.)&%$2#"
Ω
(/&,)#*,&-2E
ext
%)&*+,/,*#"()*,$)%4#),/8 !"!E \E
ext
9(/ &,)#*,& -2
E
int
> %"% ,σ
∈ E
int
/,3%$%*()7 *+, ,44/K
%)&L
(/&,)#*,& -2σ = K
|L
: =+,#.*0%$&)#$'%4<, *#$*#%"% ,
σ
#"K
(/&,)#*,&-2n
K,σ
:?#$K
∈ M
%)&σ
∈ E
60,&,)#*,-2|K|
*+,',%/.$,#"
K
%)&-2|σ|
*+,(d
− 1)
!',%/.$,#"*+,"% ,σ
:?#$1
≤ i ≤ d
60,&,)#*,-2E
(i)
⊂ E
*+,/.-/,*#"*+, "% ,/#"E
0+( +%$,3,$3,)&( .4%$*#*+,i
th
.)(* <, *#$#"*+, %)#)( %4-%/(/#"R
d
:=+,/3% ,&(/ $,*(@%*(#) (//*%77,$,&6./()7,(*+,$*+,A%$B,$!C)&D,448ACD9/ +,',EFG6 FH℄6#$)#)!
#)"#$'()74#0!#$&,$J)(*,,4,',)*%33$#5('%*(#)/6)%',42*+,K%))% +,$%)&=.$,B,4,',)*8K=9EHL℄
"#$M.%&$(4%*,$%4#$+,5%+,&$( ',/+,/6#$ *+,D$#.@,(5!K%<(%$*8DK9,4,',)*ENN℄"#$/('34( (%4',/+,/:
?#$ %44*+,/,/3% , &(/ $,*(@%*(#)/6*+,&,7$,,/#" "$,,&#'"#$ *+,3$,//.$,6*+,&,)/(*2 %)&*+,()*,$)%4
,),$72%$,%//# (%*,&*#*+, ,44/#" *+,',/+
M
6 %)&%$,&,)#*,&-2Op
K
, ρ
K
, e
K
, K
∈ M
.
P,*./*+,)*.$)*# *+,&,7$,,/#""$,,&#'"#$*+,<,4# (*2:
! !""! $%&'()&%* #$ +&,)-%./' !0.!&1&(/ $,*(@%*(#)/ Q=+,&,7$,,/#" "$,,&#'"#$*+,<,4#!
(*(,/%$,4# %*,&%* *+, ,)*,$#"*+,"% ,/#"*+,',/+6 %)&0, +##/,*+,<,$/(#)#"*+,,4,',)*
0+,$, *+,2 $,3$,/,)**+, %<,$%7, #" *+, <,4# (*2 *+$#.7+ % "% ,: =+, /,* #" &,7$,,/ #" "$,,&#'
$,%&/O
{u
σ,i
, σ
∈ E, 1 ≤ i ≤ d}.
! 23+&(/ $,*(@%*(#)Q=+,&,7$,,/#""$,,&#'"#$*+,
i
th
#'3#),)*#"*+,<,4# (*26&,J),&%**+, ,)*$,/#"*+,"% ,σ
∈ E
(i)
6%$,&,)#*,& -2Ou
σ,i
, σ
∈ E
(i)
, 1
≤ i ≤ d
.
?#$*+,&,J)(*(#)#"*+,/ +,',/60,),,&%&.%4 ',/+0+( + (/&,J),&%/"#44#0/:
! !""! $%&'()&%* #$+&,)-%./' !0.!&1&(/ $,*(@%*(#)/Q?#$*+,K=#$DK&(/ $,*(@%*(#)6*+,
&.%4 ',/+ (/ *+,/%',"#$ %44 *+,<,4# (*2 #'3#),)*/:R+,)
K
∈ M
(/ %/('34,56% $, *%)74,/#$% .-#(&6 "#$
σ
∈ E(K)
6 0, &,J),D
K,σ
%/ *+, #), 0(*+ -%/(/σ
%)& 0(*+ <,$*,5 *+, '%//,)*,$#"
K
: R,*+./#-*%()%3%$*(*(#)#"K
()m
/.-!<#4.',/60+,$,m
(/*+,).'-,$/#""% ,/#"*+,',/+6,% +/.-!<#4.',+%<()7*+,/%',',%/.$,
|D
K,σ
| = |K|/m
:R,,5*,)&*+(/&,J)(*(#)*#7,),$%4M.%&$%)74,/%)& +,5%+,&$%6-2/.33#/()7 *+%* 0,+%<,-.(4*%3%$*(*(#)/*(44 #",M.%4!
<#4.',/.-! ,44/6%)&0(*+*+,/%', #)), *(<(*(,/>)#*,*+%**+(/(/#" #.$/,%40%2/3#//(-4,6-.*
*+%*/. +%<#4.',
D
K,σ
'%2-,)#4#)7,$% #),6/() ,6("K
(/"%$"$#'%3%44,4#7$%'6(*'%2)#*-, 3#//(-4,*# -.(4*% #),+%<()7
σ
%/ -%/(/6 *+,#33#/(*, <,$*,542()7 ()K
%)&%<#4.',,M.%4*#
|K|/m
:=+,<#4.',D
K,σ
(/$,",$$,&*# %/*+,+%4"!&(%'#)& ,44%//# (%*,&*#K
%)&σ
: ?#$σ
∈ E
int
6σ = K
|L
60,)#0&,J),*+,&(%'#)& ,44D
σ
%//# (%*,&*#σ
-2D
σ
= D
K,σ
∪ D
L,σ
>!" !"# %&'"()'"*+ , -*% '.& /01 # .&2&3 '.& !4)5 2&#. !&6&+!# *+ '.& *26*+&+' *7 '.&
8&5* "'9: -*% &) .*7'.&23 "'#!&;+"'"*+ !"<&%#7%*2 '.&=>*%1=*+&*+59?9'.& .*" & *7'.&
.)57 !")2*+! &553@." .37*%
K
∈ M
)+!σ
∈ E(K)
3"#+*@'.&%& ')+A5&*7?)#"#σ
)+!*72&)#4%&|D
K,σ
|
&B4)5'*.)57 '.&2&)#4%&*7K
:C&!&+*'&?9
|D
σ
|
'.&2&)#4%&*7'.&!4)5 &55|D
σ
|
3)+!?9ε = D
σ
|D
σ
′
'.&7) &#&6)%)'"+A'@*!")2*+! &55#D
σ
)+!D
σ
′
D#&&-"A4%&EF:D
σ
D
σ
′
σ
′
= K|M
K
L
M
|σ|
σ
=
K
|L
ε = D
σ
|D
σ
′
!"#E,G%"2)5)+! !4)52&#.*%'.&=)++) .&% >4%&H)+!1%*4(&"I =)8")%'&5&2&+'#:
!"#$%&''()*& )+%",%"$( .+/(&%01,"2&' &34+,("5'
>.&)!!%&##&!6%*?5&2"+'."##& '"*+%&)!# J
∂
t
ρ + div(ρ u) = 0,
DE)F∂
t
(ρ u) + div(ρ u
⊗ u) + ∇p − div(τ (u)) = 0,
DE?Fρ = ℘(p),
DE F@.&%&
t
#')+!#7*% '.&'"2&3ρ
3u
)+!p
)%&'.& !&+#"'938&5* "'93)+!6%&##4%&"+ '.&K*@3τ(u)
#')+!# 7*%'.& #.&)% #'%&##'&+#*%3 )+! '.&74+ '"*+℘
"#'.& &B4)'"*+*7 #')'&: >.&6%*?5&2 "# #466*#&! '* ?& 6*#&!*8&%Ω
×(0, T )
3@.&%&(0, T )
"#);+"'&'"2&"+'&%8)5:>."##9#'&224#'?ሹ&2&+'&!?9#4"')?5& ?*4+!)%9 *+!"'"*+#3)+!"+"'")5 *+!"'"*+#7*%ρ
)+!u
3'.&"+"'")5 *+!"'"*+7*%ρ
?&"+A#466*#&!6*#"'"8&: >.& 5*#4%&%&5)'"*+7*%τ
(u)
"#)##42&!'*?&Jτ(u) = µ(∇u + ∇
t
u)
−
2µ
3
divu I,
@.&%&
µ
#')+!# 7*% ) +*+ +&A)'"8& 6)%)2&'&%3 6*##"?59 !&6&+!"+A *+x
: C.&+ '.& 8"# *4# '&%2τ(u)
8)+"#.'.	#'&2DEF?& *2&#.96&%?*5" :!"#$%&"'(#")*
E
c
#+",-'"#- "'"/0*E
c
=
1
2
ρ
|u|
2
123,-'0#+"-''"/4/(&$ #(5 67)8)*u
*-"9&%:35#"/ 5(/;39 (;4(%-#-('% (543/#-39&"/-<3#-<"%3'&$%-'0 6738=∂
t
E
c
+ div
E
c
u
+ ∇p
· u = div
τ
(u)
· u.
6>8 2+-%/"93#-('-%/"5"/"&#(3%#+",-'"#- "'"/0*)393' "1 !"# $% '(? &"@'" #+"5$' #-('P
: 5/(;(0, +
∞)
#(R
: 3%34/-;-#-<"(5s
7→ ℘(s)/s
2
: ?+"/"℘ = ℘
−1
A #+-% B$3'#-#*-%(5#"' 399"& #+""93%#- 4(#"'#-391!"#H
)"#+"5$' #-(' &"@'"& )*H(s) = sP(s), ∀s ∈
(0, +
∞)
A-#;3*"3%-9*)" +" ,"&#+3#ρ
H
′
(ρ)
−H(ρ) = ℘(ρ)
A#+"/"5(/":)*35(/;39 (;4$#3#-('&"#3-9"& -' #+"344"'&-C6%""DB$3#-('6EF88:;$9#-49*-'06738)*H
′
(ρ)
*-"9&%=∂
t
H(ρ)
+ div
H(ρ) u
+ p div(u) = 0.
6E8!"#$%&"'(#")*
S
#+"B$3'#-#*S = E
c
+
H(ρ)
1G$;;-'0 6>83'&6E8:?"0"#=∂
t
S + div
(
S + p) u
− div
τ
(u) u
=
−τ (u) : ∇u.
6F82+-%%+(?%#+3#:-'#+"+*4"/)(9- 3%":
S
-%3'"'#/(4*(5#+"%*%#";:3'& 3'"'#/(4*%(9$#-('#(678-% #+$%/"B$-/"&#(%3#-%5*=Z
T
0
Z
Ω
−S∂
t
ϕ
− (S + p) u · ∇ϕ
dxδt
−
Z
Ω
S(x, 0) ϕ(x, 0) dx ≤ 0, ∀ϕ ∈ C
∞
c
Ω
× [0, T )
, ϕ
≥ 0.
6H82+"':5(/;399*:-5?"%$44(%"#+3##+"<"9( -#*-%4/"% /-)"&#(I"/(3##+")($'&3/*6#+"'(/;39<"9( -#*:
-' #+" +*4"/)(9- 3%"8: -'#"0/3#-'0 6F8 *-"9&%: %-' " #+" <-% ($% &-%%-43#-(' #"/;
τ
(u) : ∇u
-% '('J '"03#-<"=d
dt
Z
Ω
1
2
ρ
|u|
2
+
H(ρ)
dx
≤ 0.
6 8 G-' "#+"5$' #-('P
-%-' /"3%-'0:K'"B$39-#*6 84/(<-&"%3'"%#-;3#"(5#+"%(9$#-('1L"%#$&*#?(% +";"%5(/#+"'$;"/- 39%(9$#-('(5G*%#";678?+- +&-M"/%)*#+"#-;"&-% /"#-I3#-('=
#+" @/%#('"-% -;49- -#:3'& #+" %" ('& ('"-%3'('J-#"/3#-<"4/"%%$/"J (//" #-('% +";" -'#/(&$ "& -'
N>O℄12+-%93##"/390(/-#+;63'&:)*3'"3%*"C#"'%-(':39%(#+"@/%#('"8?3%%+(?'-'N>O℄#(+3<"3#9"3%#
('"%(9$#-(':#(4/(<-&"%(9$#-('%%3#-%5*-'0
ρ > 0
63'&%(p > 0
83'&#()"$' ('&-#-('399*%#3)9":-'#+" %"'%" #+3# -#% 6#+"-/8 %(9$#-('6%8%3#-%@"% 3&-% /"#"3'39(0$"(5 K'"B$39-#*6 81 2+"/"%$9#% 4/"%"'#"&-'#+-%%" #-(' (;49";"'##+-%?(/,-' %"<"/39&-/" #-('%1Q(/#+"-;49- -#% +";"=
J L"@/%#&"/-<"&-% /"#"3'39(0$"%(56>83'&6E8:#+"@/%#69( 398)393' ""B$3#-(': !"! #+"&-% /"#"
,-'"#- "'"/0*)393' ":)"-'0()#3-'"&('&$39 "99%:3'&#+"%" ('&('": !"! #+""93%#- 4(#"'#-39
)393' ":('4/-;39 "99%1
2+"%""B$3#-('%3/"$%"& 3@/%##-;"#(()#3-'#+"%#3)-9-#*(5#+"% +";")*3%-;49"-'#"0/3#-('
-'%43 "6 !"! %$;;3#-('(<"/#+"4/-;393'&&$39 ('#/(9<(9$;"%81
J G" ('&: -' ('" %43 " &-;"'%-(' 3'& 5(/ #+" +*4"/)(9- 3%": ?" 4/(<" #+3# #+" 9-;-# (5 3'*
('<"/0"'# %"B$"' " (5 %(9$#-('% #( #+" % +";" -% 3 ?"3, %(9$#-(' #( #+" 4/()9"; 6-' 53 #: %3J
!"#$%%&'($))"#*+,+-.%"/"+,#+-.0") 2.+.3"#.+" .#.2*&$#0.%$)+" (,+.#+"$%4$%$# .)'5.)-,5
+-$+ )6 -$%"/"+$%),)$+")7.) +-..#+2,(&"#.86$%"+&9:;<
!,2+-. (2.))62. ,22. +",#) -./.' +-.2.)6%+)$2. .)).#+"$%%&+-. )$/.=+-. ) -./. ")6# ,#0"+",#$%%&
)+$4%.'$#0 +-.($))$*.+,+-.%"/"+"#+-.) -./.)-,5)+-$+' "# $).,> ,#?.2*.# .'+-.(2.0" +.0$#0
.#0 ,> )+.(?.%, "+".)#. .))$2"%&+.#0+, +-.)$/.>6# +",#'$#0 +-$++-.%"/"+"))+"%% $5.$3),%6+",#+,
+-.(2,4%./')$+")>&"#*+-..#+2,(&"#.86$%"+&<
@6/.2" $%+.)+)'(.2>,2/.05"+-+-.(2.))62. ,22. +",#) -./.' ,#>,2++-.). +-.,2.+" $%2.)6%+)<
A. 72)+ )6//$2"B. "# +-") ). +",# +-. ,4+$"#.0 +-.,2.+" $% 2.)6%+) 9C. +",#) D<E $#0 D<F< ;< 5-" - $2.
0.+$"%.0"#G-$(+.2F,>+-")0, 6/.#+<H-.#5.)-,52.)6%+),>$#6/.2" $%+.)+9C. +",#D<F<0;'.I+2$ +.0
>2,/$/,2. ,/(2.-.#)"?.)+60&$%),$002.))"#*$#.I+.#)",#,>+-.) -./.+,+5, (-$).J,5)'(2.).#+.0
"#G-$(+.2E,>+-")0, 6/.#+<
!" #$ %&'(% %* + ,-&-
!"!# $%& ' %&)& K.+ 6) ,#)"0.2 $ 6#">,2/ ($2+"+",#
0 = t
0
< t
1
< . . . < t
N
= T
,> +-. +"/. "#+.2?$%(0, T )
' $#0 %.+δt = t
n+1
− t
n
>,2n = 0, 1, . . . , N
− 1
4.+-. ,#)+$#++"/.)+.(< A.4.*"#5"+-+-. 0") 2.+"B$+",#,> +-./$)) 4$%$# ..86$+",#9E$;< !,24,+-+-.LMG$#0NH ,2GN 0") 2.+"B$+",#)'%.+6) 0.#,+. 4&u
n+1
σ
· n
K,σ
+-.,6+5$20#,2/$% ?.%, "+& +, +-. >$ .σ
,>K
'5-" - ") ,/(6+.0'>,2+-.NH$#0GN.%./.#+)'4&+$3"#*+-."##.2(2,06 +,>+-.?.%, "+&$+ +-.>$ .5"+-+-. ,6+5$20#,2/$% ?. +,29$) "/(%".0 4& +-. #,+$+",#; $#0 5-" - ") *"?.#' >,2 +-. LMG ) -./.' 4& +-. ?$%6.,> +-. ,/(,#.#+ ,> +-.?.%, "+& $+ +-. .#+.2,> +-. >$ . 96( +, $ -$#*. ,> )"*#;< H-.0") 2.+. .86$+",#)$2.,4+$"#.04&$#6(5"#07#"+.?,%6/.0") 2.+"B$+",#$#02.$0=∀K ∈ M,
|K|
δt
(ρ
n+1
K
− ρ
n
K
) +
X
σ∈E(K)
F
K,σ
= 0,
5"+-F
K,σ
=
|σ| u
n+1
σ
· n
K,σ
ρ
e
n+1
σ
,
9O; $#0 5-.2.ρ
e
n+1
σ
") +-. 6(5"#0 $((2,I"/$+",# ,>ρ
n+1
$+ +-. >$ .σ
5"+- 2.)(. + +,u
n+1
σ
· n
K,σ
< H-") $((2,I"/$+",# .#)62.)+-$+ρ
n+1
> 0
">ρ
n
> 0
$#0 "> +-. 0.#)"+&") (2.) 2"4.0 +, $ (,)"+"?. ?$%6. $+ "#J,54,6#0$2".)< !,24,+-LMG$#0NH,2 GN0") 2.+"B$+",#)'>,21
≤ i ≤ d
$#0σ
∈ E
(i)
' 5.0.#,+.4&(divτ (u
n+1
))
σ,i
$#$((2,I"/$+",#,>+-.i
+- ,/(,#.#+,>+-.?") ,6)+.2/$)), "$+.0+,σ
'$#05.0.#,+.4&(∇p
n
)
σ,i
+-.i
+- ,/(,#.#+,>+-.0") 2.+.(2.))62.*2$0".#+$+ +-.>$ .σ
<A"+- +-.). #,+$+",#)'5.$2.$4%.+, 52"+.+-.>,%%,5"#**.#.2$%>,2/ ,>+-.$((2,I"/$+",#,>+-./,/.#+6/4$%$# ..86$+",#=|D
σ
|
δt
(ρ
n+1
σ
u
n+1
σ,i
− ρ
n
σ
u
n
σ,i
) +
X
ε∈E(D
σ
)
F
σ,ε
n+1
u
n+1
ε,i
+
|D
σ
|(∇p
n+1
)
σ,i
− |D
σ
|(divτ(u
n+1
))
σ,i
= 0,
9P; >,2
1
≤ i ≤ d
'$#0>,2σ
∈ E \ E
D
"#+-. $).,>+-.NH,2GN0") 2.+"B$+",#)'$#0σ
∈ E
(i)
\ E
D
"#+-. $). ,>+-.LMG) -./.<Q#+-")2.%$+",#'ρ
n+1
σ
$#0ρ
n
σ
)+$#0>,2$#$((2,I"/$+",#,>+-.0.#)"+&,#+-.>$ .σ
!" "#$%t
n+1
!&'
t
n
(%)*% "#,%-. /01# 1 $2)"&3" 4% 3&52)%' 0#"1 "1%2*)"(%!$'%&)#".
ρ
e
σ
2)%' #&"1%$!))4!-!& %67
F
n+1
σ,ε
#)"1%'#) (%"%$!))829"1(32:1"1%'2!-5! %ε
32"0!('D
σ
7!&'u
n+1
ε,i
)"!&') 53(!&!**(39#$!"#3&35u
n+1
i
3&ε
01# 1$!.4% 13)%&%#"1%( %&"(%'3(2*0#&';<1%=&#"%%-%$%&"'#) (%"#>!"#3& 35"1%
i
?"1 3$*3&%&"35"1%*(%))2(%:(!'#%&""%($(%!')@|D
σ
|(∇p
n+1
)
σ,i
=
−
X
M∈M
Z
M
p
n+1
divϕ
(i)
σ
dx,
0#"1ϕ
(i)
σ
(%!')ϕ
(i)
σ
= ϕ
σ
e
(i)
7 01%(%ϕ
σ
#) "1% =&#"% %-%$%&" )1!*% 52& "#3& !))3 #!"%' "3σ
!&'e
(i)
)"!&')53("1%
i
th
,% "3(35"1% !&3&# !-4!)#)35
R
d
; A#& %"1%*(%))2(%#)*#% %0#)% 3&)"!&"72)#&:"1%
'%=&#"#3& 35"1%B<3(CB)1!*%52& "#3&)7!&%!). 3$*2"!"#3&.#%-')53(!&#&"%(&!-5! %
σ = K
|L
@|D
σ
|(∇p
n+1
)
σ,i
=
|σ| (p
n+1
L
− p
n+1
K
) n
K,σ
· e
(i)
,
!&'753(!&%9"%(&!-5! %
σ
∈ E(K) ∩ E
ext
\ E
D
@|D
σ
|(∇p
n+1
)
σ,i
=
−|σ| p
n
K
n
K,σ
· e
(i)
.
<1%)% %9*(%))#3&) 3#& #'% 01# 1"1%'#) (%"%:(!'#%&"#&"1%DEC'#) (%"#>!"#3&;
<1%=&#"%%-%$%&"'#) (%"#>!"#3&35"1%,#) 32)"%($
(divτ (u
n+1
))
σ,i
7!))3 #!"%'"3σ
!&'"3"1% 3$*3&%&"i
7 (%!')@|D
σ
|(divτ(u
n+1
))
σ,i
=
−µ
X
K∈M
Z
K
∇u
n+1
· ∇ϕ
(i)
σ
−
µ
3
X
K∈M
Z
K
div u
n+1
div ϕ
(i)
σ
.
<1%DEC'#) (%"#>!"#3&35 "1#))!$%,#) 32)"%($#)'%"!#-%' #&FG℄;
<1% $!#& $3"#,!"#3& "3 #$*-%$%&" ! =&#"% ,3-2$% !**(39#$!"#3& 53( "1% =()" "03"%($) #& / 6 #) "3
34"!#&!'#) (%"%%I2#,!-%&"35"1% J#&%"# %&%(:.4!-!& %/)%% &%9")% "#3&6;K3("1#) (%)2-""3 4%,!-#'7
"1% &% %))!(. 3&'#"#3&#) "1!" "1% 3&,% "#3& 3*%(!"3( ,!&#)1%) 53( ! 3&)"!&" ,%-3 #".7 !"! "1!" "1%
53--30#&:'#) (%"%$!))4!-!& %3,%("1%'#!$3&' %--)#))!"#)=%'FL7GM℄@
∀σ ∈ E
int
,
|D
σ
|
δt
(ρ
n+1
σ
− ρ
n
σ
) +
X
ε∈E(D
σ
)
F
σ,ε
n+1
= 0.
/N6<1#):3,%(&)"1% 13# %53("1%'%=&#"#3&35"1%'%&)#".!**(39#$!"#3&
ρ
σ
!&'"1%$!))829%)F
σ,ε
;<1%'%&)#".
ρ
σ
#)'%=&%' 4.!0%#:1"%'!,%(!:%@∀σ ∈ E
int
7σ = K
|L
7|D
σ
| ρ
σ
=
|D
K,σ
| ρ
K
+
|D
L,σ
| ρ
L
!&'∀σ ∈ E
ext
\ E
D
7σ
∈ E(K)
7ρ
σ
= ρ
K
; K3(!'2!- %':%ε
#& -2'%' #& "1%*(#$!- %--K
7 "1% 829F
σ,ε
#) 3$*2"%'!)!-#&%!( 3$4#&!"#3&/0#"1 3&)"!&" 3%O #%&")7 !"! #&'%*%&'%&"35"1%%':%!&'"1% %--635"1%$!))829%)"1(32:1"1%5! %)35
K
7 !"! "1%I2!&"#"#%)(F
n+1
K,σ
)
σ∈E(K)
!**%!(#&:#&"1%'#) (%"%$!)) 4!-!& %/P6;Q%'3&3":#,%1%(%"1#))%"35 3%O #%&")7!&'(%5%("3FL7R 7GS℄53(!'%"!#-%' 3&)"(2 "#3&35"1#)!**(39#$!"#3&;
!"!# $%&'(% '&'*+, #-.-& '/ '.-0(% 12('&(%-. %3'&(%(,-&3 0(-#%.%(,
Q% 4%:#& 4. '%(#,#&: ! '#) (%"% J#&%"# %&%(:. 4!-!& % %I2!"#3&; T%"
δ
up
4% ! 3%O #%&" '%=&%' 4.
δ
up
= 1
!"#$%
δ
up
= 0
&$ '() )$')+)% #,)-.&'('(&,$/'#'&/$0'()1/1)$'21 3#4#$ ))52#'&/$+)#%, 6
|D
σ
|
δt
(ρ
n+1
σ
u
n+1
σ,i
− ρ
n
σ
u
n
σ,i
) +
X
ε=D
σ
|D
σ′
1
2
F
n+1
σ,ε
(u
n+1
σ,i
+ u
n+1
σ
′
,i
)
+ δ
up
X
ε=D
σ
|D
σ′
1
2
|F
n+1
σ,ε
| (u
n+1
σ,i
− u
n+1
σ
′
,i
) +
|D
σ
| (∇p
n+1
)
σ,i
− |D
σ
|(divτ(u
n+1
))
σ,i
= 0.
7#8&$9 '() &$$)+ :+/%2 ' /; '(&, )52#'&/$ <&'( '() /++),:/$%&$9 =)4/ &'> 2$8$/<$0 !"!
u
n+1
σ,i
0 >&)4%,T
conv
σ,i
+ T
up
σ,i
+ T
p,τ
σ,i
= 0
0<&'(6T
conv
σ,i
=
h |D
σ
|
δt
ρ
n+1
σ
u
n+1
σ,i
− ρ
n
σ
u
n
σ,i
+
X
ε=D
σ
|D
σ′
1
2
F
n+1
σ,ε
(u
n+1
σ,i
+ u
n+1
σ
′
,i
)
i
u
n+1
σ,i
,
T
σ,i
up
=
δ
up
h
X
ε=D
σ
|D
σ′
1
2
|F
n+1
σ,ε
| (u
n+1
σ,i
− u
n+1
σ
′
,i
)
i
u
n+1
σ,i
,
T
σ,i
p,τ
=
|D
σ
| (∇p
n+1
)
σ,i
u
n+1
σ,i
− |D
σ
|(divτ(u
n+1
))
σ,i
u
n+1
σ,i
.
?)11#-@-A0#::4&)%/$'()%2#41),(0 >&)4%,6
T
σ,i
conv
=
1
2
|D
σ
|
δt
h
ρ
n+1
σ
(u
n+1
σ,i
)
2
− ρ
n
σ
(u
n
σ,i
)
2
i
+
1
2
X
ε=D
σ
|D
σ′
F
σ,ε
n+1
u
n+1
σ,i
u
n+1
σ
′
,i
+
|D
σ
|
2 δt
ρ
n
σ
u
n+1
σ,i
− u
n
σ,i
2
.
?)'2, %)B$)R
n+1
σ,i
3>'(),21/;T
up
σ,i
#$%'()4#,' ')+1/;T
conv
σ,i
6R
n+1
σ,i
=
1
2
|D
σ
|
δt
ρ
n
σ
u
n+1
σ,i
− u
n
σ,i
2
+ δ
up
h
X
ε=D
σ
|D
σ′
1
2
|F
n+1
σ,ε
| (u
n+1
σ,i
− u
n+1
σ
′
,i
)
i
u
n+1
σ,i
.
CD".&'('(&, $/'#'&/$0<)'(2,/3'#&$'();/44/<&$9+)4#'&/$6
1
2
|D
σ
|
δt
h
ρ
n+1
σ
(u
n+1
σ,i
)
2
− ρ
n
σ
(u
n
σ,i
)
2
i
+
1
2
X
ε=D
σ
|D
σ′
F
σ,ε
n+1
u
n+1
σ,i
u
n+1
σ
′
,i
+
|D
σ
| (∇p
n+1
)
σ,i
u
n+1
σ,i
− |D
σ
|(divτ(u
n+1
))
σ,i
u
n+1
σ,i
=
−R
n+1
σ,i
.
CC" .)+) /9$&E)#' '() 4);'F(#$%,&%)#%&, +)') 8&$)'& )$)+9>3#4#$ )0 !"! #+)#,/$#34)%&, +)'&E#'&/$ /;G52#'&/$ A"0<&'(# /$,)+=#'&=)B$&')=/421)%&, +)'&E#'&/$/;'()8&$)'& )$)+9> /$=) '&/$')+1,-7()
+&9('F(#$%,&%) /$,&,',&$ #$21)+& #4+),&%2#40'(),&9$/;<(& (<&443),'2%&)%
4#')+-.) $/< '2+$ '/ '() )4#,'& :/')$' 3#4#$ )- H24'&:4>&$9 '() %&, +)') 1#,, 3#4#$ ) )52#'&/$ @" 3>
H
′
(ρ
K
)
#$%&$=/8&$9?)11# -@-C>&)4%,0∀K ∈ M
6|K|
δt
(
H(ρ
n+1
K
)
− H(ρ
n
K
)) +
X
σ∈E(K)
|σ|
H(ρ
n+1
σ
) + p
K
u
n+1
σ
· n
K,σ
=
−R
n+1
K
,
CA" <&'(6R
n+1
K
=
1
2
|K|
δt
H
′′
(ρ
n,n+1
K
)(ρ
n+1
K
− ρ
n
K
)
2
−
1
2
X
σ∈E(K)
|σ| u
n+1
σ
· n
K,σ
H
′′
(ρ
n+1
σ
)(ρ
n+1
σ
− ρ
n+1
K
)
2
,
"#$%$ &#$ '()*&+&,
ρ
n,n+1
K
∈ [min(ρ
n+1
K
, ρ
n
K
), max(ρ
n+1
K
, ρ
n
K
)]
)*-. /0% )*, /) $σ
∈ E(K)
.ρ
n+1
σ
∈
[min(ρ
n+1
σ
, ρ
n+1
K
), max(ρ
n+1
σ
, ρ
n+1
K
)]
23'()&+0* 4 56 +7 ) 8*+&$ 90:(;$ -+7 %$&+<)&+0* 0/ &#$ 4*0* 0*7$%9)&+9$6 $:)7&+ =0&$*&+): >):)* $ 4?6.
"+&# )*0*=07+&+9$%$7+-():&$%;.&#)*@7&0&#$/) & &#)&&#$/(* &+0*
H
+7 0*9$A)*-&#)& )*(="+*-)==%0A+;)&+0*0/&#$-$*7+&,+7(7$-+* &#$;)77>):)* $2
B#$7&)>+:+&,0/&#$7 #$;$+7&#$*0>&)+*$->,7(;;+*CD
(i)
3'()&+0*4 609$%&#$ 0;=0*$*&7i
)*- &#$/) $7σ
∈ E
/0%&#$ EB 0%FE-+7 %$&+<)&+0*7.)*-09$%
i
)*-σ
∈ E
(i)
/0%&#$GHF7 #$;$.
(ii)
3'()&+0*4 5609$%K
∈ M
.(iii)
)*-.8*)::,.&#$&"00>&)+*$-%$:)&+0*72I$&(77(==07$&#)&&#$9$:0 +&,9)*+7#$7)&&#$>0(*-)%,.)*-:$&(7&#$*+*90@$&#%$$)%C(;$*&72J+%7&.
&#$-+7 %$&$C%)-+$*&)*--+9$%C$* $0=$%)&0%7)%$-():"+&#%$7=$ &&0&#$
L
2
+**$%=%0-( &.+*&#$7$*7$ &#)& DX
i,E
|D
σ
| (∇p
n+1
)
σ,i
u
n+1
σ,i
+
X
K∈M
p
K
X
σ∈E(K)
|σ| u
n+1
σ
· n
K,σ
= 0,
"#$%$ &#$*0&)&+0*P
i,E
;$)*7&#)& "$7(;09$%&#$ 0;=0*$*&+*-$Ai
)*- 0*σ
∈ E
/0%&#$ EB)*-FE-+7 %$&+<)&+0*7.)*-0*
i
)*-σ
∈ E
(i)
/0%&#$GHF7 #$;$2K$ 0*-."$7(==07$&#)&47$$K$ &+0*L6D
X
i,E
|D
σ
| (divτ(u
n+1
))
σ,i
u
n+1
σ,i
≥ 0.
B#+%-.%$0%-$%+*C&#$7(;;)&+0*7,+$:-7./0%&#$=)%&0/&#$%$;)+*-$%0/&#$;0;$*&(;>):)* $$'()&+0*
)770 +)&$-&0 &#$(="+*-+*CD
X
i,E
T
σ,i
up
= δ
up
X
i, ¯
E (ε=D
σ
|D
σ′
)
1
2
|F
n+1
σ,ε
| (u
n+1
σ,i
− u
n+1
σ
′
,i
)
2
≥ 0,
"#$%$&#$*0&)&+0*P
i, ¯
E (ε=D
σ
|D
σ′
)
;$)*7&#)&"$=$%/0%;&#$7(;09$%
i
)*-&#$/) $70/&#$-():;$7#)770 +)&$-&0 &#$ 0;=0*$*&
i
0/ &#$9$:0 +&,.)*- &#)&. /0% )/) $ε
+* &#$7(;. &#$&"0)-M) $*&-():$::7)%$-$*0&$- >,
D
σ
)*-D
′
σ
2J+*)::,.7+* $&#$ 0*7$%9)&+9$N(A$79)*+7#+* &#$7(;;)&+0*. "$&#(7C$&D
1
2
X
i,E
|D
σ
|
δt
h
ρ
n+1
σ
(u
n+1
σ,i
)
2
− ρ
n
σ
(u
n
σ,i
)
2
i
+
X
K∈M
|K|
δt
(
H(ρ
n+1
K
)
− H(ρ
n
K
))
≤ 0,
4 ?6"#+ #+7)-+7 %$&$)*):0C($&04O62
!"! $%&&'() *+ *,- .'/'*'( *,-& ,-/- 0"1 %&-2
P$/0 (7+*&#+77$ &+0*0*&#$+*9+7 +- Q/0%;0/R%0>:$;4 6.)*-7#0"&#)&.+/)7$'($* $0/70:(&+0*7
+7 0*&%0::$- +* 7(+&)>:$ *0%;7 )*- 0*9$%C$7&0 ) :+;+&. &#+7 :)&&$% *$ $77)%+:,7)&+78$7 ) 4=)%& 0/ &#$6
"$)@/0%;(:)&+0*0/&#$ 0*&+*(0(7=%0>:$;2
I$&
(
M
(m)
, δt
(m)
)
m∈N
>$)7$'($* $0/;$7#$7)*-&+;$7&$=7.7( #&#)&&#$&+;$7&$=δt
(m)
)*-&#$7+<$h
(m)
0/&#$;$7#M
(m)
.-$8*$- >,Dh
(m)
=
sup
K∈M
(m)
-+);(K),
!"# $%!&$'(
m
→ ∞
) *!ρ
(m)
+p
(m)
'"#u
(m)
,! -! ($./ 0$" 102!" ,3 -! ( -!5! 60 - -! 5!(-M
(m)
'"# -! 05! ( !7δt
(m)
+ $&+ 5$&! 7&! 0(!.3 (7!'80"1+ ' 9: 2!&(0$" $; -! ( -!5! 6-0 - 5'3 ,! $, '0"!# ,3 '80"1 -!
<=>2'&0'" +$".3$"!-$&0%$" '.( &07!$;5!(-!(+(/77$(0"1 -' -!2!& 0 '. $57$"!" $; -!2!.$ 0 3
? -! #!1&!!$; ;&!!#$5$; 6-0 - '&! .$ ' !#$" -! $7'"#,$ $5 ,$/"#'&0!(@2'"0(-!(+ '"# -' -!
5!'(/&! $; -!;' !( 0(!A/'. $ 9)B$ -!#0( &! !/"8"$6"(+ 6!'(($ 0' !70! !60(! $"( '" ;/" 0$"(
$" 05!0" !&2'.('"#$"7&05'. $&#/'. 5!(-!(+($ -!#!"(0 3
ρ
(m)
+ -!7&!((/&!p
(m)
'"# -!2!.$ 0 3u
(m)
'&!#!C"!#'.5$( !2!&36-!&!$"Ω
× (0, T )
,3Dρ
(m)
(x, t) =
N −1
X
n=0
X
K∈M
(ρ
(m)
)
n
K
X
K
X
(n,n+1)
,
p
(m)
(x, t) =
N −1
X
n=0
X
K∈M
(p
(m)
)
n
K
X
K
X
(n,n+1)
,
u
(m)
(x, t) =
N −1
X
n=0
X
σ∈E
(u
(m)
)
n
σ
X
D
σ
X
(n,n+1)
,
6-!&!X
K
+X
D
σ
'"#X
(n,n+1)
( '"#;$& -! -'&' !&0( 0 ;/" 0$" $;K
+D
σ
'"# -!0" !&2'.(t
n
, t
n+1
)
&!(7! 02!.3)
E!(/77$(!'/"0;$&5 $" &$.$" -! &'"(.' !(0"(7' !'"# 05!$; -!(!A/!" !$;($./ 0$"(+6-0 -6!
"$6( ' !)F$& #0( &! !;/" 0$"
q
'"#v
#!C"!# $" -!7&05'. '"# #/'.5!(-+ &!(7! 02!.3+ 6!#!C"! ' #0( &! !L
1
(0, T ); BV(Ω)
"$&5,3Dkqk
T ,x,BV
=
N
X
n=0
δt
X
σ∈E, σ=K|L
|q
n
L
− q
K
n
|,
kvk
T ,x,BV
=
N
X
n=0
δt
X
ε∈ ¯
E, σ=D
σ
|D
′
σ
|v
n
σ
′
− v
σ
n
|,
'"#'#0( &! !L
1
Ω; BV((0, T ))
"$&5,3Dkqk
T ,t,BV
=
X
K∈M
h
K
N −1
X
n=0
|q
n+1
K
− q
K
n
|,
kvk
T ,t,BV
=
X
σ∈E
h
σ
N −1
X
n=0
|v
n+1
σ
− v
n
σ
|,
6-!&!+ ;$&
σ = K
|L
+h
σ
= (h
K
+ h
L
)/2
) E! (/77$(! -!;$..$60"1 /"0;$&5 ,$/"#( $; -!(!A/!" ! $;($./ 0$"(60 -&!(7! $ -!(! 6$"$&5(D
kρ
(m)
k
T ,x,BV
+
kp
(m)
k
T ,x,BV
+
ku
(m)
k
T ,x,BV
≤ C,
∀m ∈ N,
?9G@ '"#Dku
(m)
k
T ,t,BV
≤ C, ∀m ∈ N.
?9H@ =6!'8($./ 0$" $ -! $" 0"/$/(7&$,.!5(' 0(C!(+;$&'"3ϕ
∈ C
∞
c
[0, T )
× Ω
D−
Z
Ω×(0,T )
h
ρ ∂
t
ϕ + ρ u ∂
x
ϕ
i
dxδt
−
Z
Ω
ρ(x, 0) ϕ(x, 0) dx = 0,
?9I'@−
Z
Ω×(0,T )
h
ρ u ∂
t
ϕ + (ρ u
2
+ p) ∂
x
ϕ
i
dxδt
−
Z
Ω
ρ(x, 0) u(x, 0) ϕ(x, 0) dx = 0,
?9I,@ρ = ℘(p).
?9I @J$ ! -' -!(! &!.' 0$"('&!"$ (/K 0!" $ #!C"! '6!'8($./ 0$" $ -!7&$,.!5+(0" ! -!3#$"$
#$%&'&#$() (#*&+,-(.#, './' '.-0/1-(/'&(2-%30 '.-4&5&'#+ /(-67-$ - #+(#47'&#$('# '.-
%&( 1-'-81#34-5*'.&(&584&-(*4##(-40(8-/9&$:*'./' !"# !"%" &%'( "# !" )*+! #!& ,#*,.& .&('.-1-(74'
,-/1-(--9&$:;<' &(('/'-%&$ '.-+#44#,&$:'.-#1-5;
!"#$"% &'&( =-'
Ω
3- /$ #8-$ 3#7$%-% &$'-1>/4 #+R
; =-'(
M
(m)
, δt
(m)
)
m∈N
3- / (-67-$ - #+ 5-(.-( /$% '&5-('-8(* (7 . './'h
(m)
/$%δt
(m)
'-$% '# ?-1# /(m
'-$%( '# &$2$&'0; =-'ρ
(m)
, p
(m)
, u
(m)
m∈N
3-'.- #11-(8#$%&$:(-67-$ -#+(#47'&#$(;@-(788#(-'./''.&((-67-$ -(/'&(2-(A BC/$%A DC/$% #$>-1:-(
&$
L
r
(0, T )
× Ω
3
*+#11
≤ r < ∞
*'#(¯
ρ, ¯
p, ¯
u)
∈ L
∞
(0, T )
× Ω
3
;
E.-$'.-4&5&'
(¯
ρ, ¯
p, ¯
u)
(/'&(2-('.-(0('-5A FC/$%'.--$'1#80 #$%&'&#$ABC;!""# E.-8/((/:-'#'.-4&5&'&$'.--67/'&#$(#+'.-( .-5-&('- .$& /4*37'&$>#9-(1/'.-1('/$%/1%
/1:75-$'(;
G3'/&$&$: '.- -$'1#80 #$%&'&#$ &( 5#1- &$'1& /'-; @- $--% '# 8/(('# '.- 4&5&' &$ '.- 9&$-'& -$-1:0
3/4/$ -A C/$%&$'.--4/('& 8#'-$'&/43/4/$ -A !C(&574'/$-#7(40;E#'.&(8718#(-*+#1
ϕ
∈ C
∞
c
[0, T )
×
Ω
*,-%-2$-',#&$'-18#4/'-(H #$-&( %-2$-% #>-1'.-%7/4 -44(/$% &(7(-%/(/'-('+7$ '&#$ +#1A C/$%'.-(- #$%#$-&(%-2$-%#>-1'.-81&5/4 -44(*/$%&(7(-%/(/'-('+7$ '&#$+#1A !C;@-'.-$8/(('#
'.-4&5&'&$'.-I%&J-1-$'&/4'-15(I#+'.-(-%&( 1-'--67/'&#$(*/$%%&(1-:/1%'.-$#$K$-:/'&>-1-(&%7/4(
A/''.-4-+'K./$%(&%-C;L81#34-5&(8#(-% 30'.-1-(&%7/4/((# &/'-%'#'.-78,&$%&$:*,.& . 1-/%(H
R
σ
=
h
X
ε=D
σ
|D
σ′
1
2
|F
n+1
σ,ε
| (u
n+1
σ,i
− u
n+1
σ
′
,i
)
i
u
n+1
σ,i
,
!" #$%&'(!)*+$' $'&-!.!)+!/ 0)(%# !'!#-'#')!)*$)+#) "% 1+'#$ #$'&#%2341%#-&2%3 2.#$ #
'#3 56%&%%! & "'& 2%#% ! 1)(-%#) "'7-&')!#%23
−µ ∆u u
+'#$ !-3%2' 18'& )&'#5µ
#%!"'!(#)9%2) &#$%&: %&#%:/;%#-&!)+ )3: 2%#$'&#%23#)
µ
|∇u|
2
4'!#$%&%!&%)*"''6-#')!&/<)2
ψ
2%(-1 2*-! #')!+'#$ )3: #&-::)2#42%3 2.'!(#$ #
−µ u ∆u − µ|∇u|
2
=
−div(µu ∇u)
4+%(%#=Z
T
0
Z
Ω
−µ u ∆u − µ|∇u|
2
i
ψ dxδt =
Z
T
0
Z
Ω
µu ∇u
· ∇ψ dxδt ≤ C
ψ
kuk
L
∞
kuk
W
1,1
µ,
!"#$%2%*)2%4'*
kuk
L
∞
!"kuk
W
1,1
2%6)-!"%"4#$%"'7%2%! %6%#+%%!−µ u ∆u
!"µ
|∇u|
2
6%$ 8%&1'.%
µ
/>%#-2!'!( ##$%"'& 2%#%1%8%14#$'& )3:-# #')!&-((%&#&#$ #R
σ
6%$ 8%& ##$%1'3'# & "'&&': #')!#%23? !"! "'& 2%#%%@-'8 1%!#)*
µ
|∇u|
2
A4#$%&'(!)*+$' $'&(- 2 !#%%"/0$%& 3% 2(-3%!#'&-&%"'!
"'7%2%!#+ 5'!#$%!)!B6 2)#2):' &%=#$%C8'& )-&#%23C
R
σ
'& )3:%!& #%"'!#$%'!#%2! 1%!%2(56 1 ! %65 C"'&&': #')!#%23C?&%%D% #')!E/FA/
#"$%&' ()*+,-&+.+/ -0" -&%,1.%-"12
G!#$% &&-3:#')!&)*#$%#$%)2%3/H/F4+% !&$ 2:%!?FIA !"?FEA/G!"%%"4#):2)8%#$ ##$%1'3'#'&
+% .&)1-#')!4'#'&&-J '%!##) $ 8%=
lim
m→+∞
h
(m)
h
kρ
(m)
k
T ,x,BV
+
kp
(m)
k
T ,x,BV
+
ku
(m)
k
T ,x,BV
i
= 0.
G! ""'#')!4#$'&%&#'3 #%3 56%:2)8%!? !"!)#&-::)&%"A65 ""'!(#)#$%& $%3% !-3%2' 1"'7-&')!
& 1%" 65
(h
(m)
)
β
4 +'#$