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Un schéma aux volumes finis avec matrice signe pour les
systèmes non homogènes
Slah Sahmim
To cite this version:
Slah Sahmim. Un schéma aux volumes finis avec matrice signe pour les systèmes non homogènes.
Mathématiques [math]. Université Paris-Nord - Paris XIII, 2005. Français. �tel-00010000�
HAL Id: tel-00010000
https://tel.archives-ouvertes.fr/tel-00010000
Submitted on 30 Aug 2005
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‡Œ º  xi− 1
2
U (x, tn )dx.
xi− 1
2
n
).
F (U (xi+ 1 , t)) = φ(Uin , Ui+1
φ [email protected] E~<[email protected]{~}6E$FD6EVIJE?A?Ç Fz?AƒÐ{[email protected]‚X?)E„I [email protected] LÌ$E
Ô {~‚X?ALUDGH=CF ‡Œ º  ®MAE_ÅGHE$FD e
2
xi+ 1
2
¿&<>=C?)@ t Ì$=ƒÐ<[email protected]|GJ} E~FD@6E t E~D
n
Œ º. tn+1
º
n
n
−∆xi Uin + ∆tn φ(Uin , Ui+1
) + ∆xi Uin+1 − ∆tn φ(Ui−1
, Uin )
Z tn+1 Z x 1
i+ 2
Q(x, t, U )dx dt.
=
tn
xi− 1
ÖF‘<aE$?)D[email protected]}®{$Ì[email protected]|[email protected]|EžIHE+}6Ì ÊA{~ƒ´LÓ}6=?A}IÃL [email protected]ƒ„EÖ}|?AGHÅULFDE e
2
n
n
) − φ(Ui−1
, Uin ) + ∆tn Qni
Uin+1 = Uin − rn,i φ(Uin , Ui+1
= =CF LÓ<a=C}|{ r = ∆t º
∆x
Z
Z
L<)<[email protected]= ÇGJƒÐLUDGH=CFfMA?+D[email protected]|ƒÐE™}|[email protected]|Ì$E¿$MA{!&F)GJE™GJÌ~GC<&[email protected]
1
$
E
|
}
D
I
Q
Q(x, t, U )dxdt.
∆x ∆t
ÔZL„MA{_D[email protected]ƒ„GJº F&LUD6GJ=CF MA6? ?&?Ç Fz?Aƒ„{[email protected]|GJ‚z?AEžE$}|DÑIJE 3=CFAM)E$ƒ„E$FDfƒ;~ƒÐEªMAEžIÃLӃÐ{_DÊA=XMAEªMAE~}fÅ= IJ?Aƒ„E$} AFAGJ}
,[email protected]E!Ç)E~ƒÐ<)IJE¿&<[email protected](IJLÌ$IJL}6}|EžM)E$} Q }6Ì ÊA{~ƒ´LU}$¿&IJ
E ?A?Ç Fz?AƒÐ{[email protected]‚X?)E+} {$Ì[email protected]|GHD e
n
n,i
i
n
i
xi+ 1
tn+1
i
n
tn
2
xi− 1
2
1
n
n
φ(Uin , Ui+1
) + f (Uin )) −
) = (f (Ui+1
2
Q−
n
n
Q(Ui , Ui+1 ) = A(Ũ )
1
n
n
|Q(Uin , Ui+1
)|(Ui+1
− Uin ).
2
$E }‡DfIÃLvƒ´L1D@6GHÌ$EÖÌ[email protected]LUÌ~D{[email protected]}|DGH‚z?AE+MA? }6Ì Ê){$ƒÐL º
ËÖLFA}IHEÖ}6Ì ÊA{~ƒ´LvMAEªÀ=XE¿
MA{!AFAGHD(IJLƒ´L1D@6GHÌ$EMAEªÀf=XE¿)= @6E~<[email protected]{ }6E~FzD6E€I {_D LUDƒ„=^ÈCE~F vMA{_D[email protected]|ƒÐ[email protected](E~F63=FAÌ~D6GJ=CF MAE~}fM)E$?Ç {_D LUD6} U E_D U º Ũ
ŒP
Q
n
i
n
i+1
:
D
D I
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› ¢ž¯€µ•-¤– £~¤ÂZ– —™˜ š› œ Á =CFA}|GJMA{[email protected]=CFA}®?AF }|ÈX}‡DÍ$ƒ„E+MAEIJ=CGH} M)EÖBAGJIJLF„M&LFA}
E~Df‚z?AG¬} {$Ì[email protected],D e
Rm

∂F (W )
∂W


+
= Q(x, W )
∂t
∂x


W (x, 0) = W0 (x).
¿GH}6}|?AEÖMAEIÃL€ƒ„{$Ì$LFAGH‚X?)EÑMAE~}+?&?AGHMAE$}~¿
M&LFA}
Œ º?> R×]0, T [
=C?)} LGJ}|=CFA}®I ÊÈX<a=DÊAÍ~}6E3=CF)M&Lƒ„E$FD LIHEтz?AEÖIHEÖ<[email protected]|=CBAIJÍ~ƒÐEMAE+ÀGJE~ƒ´LF)F´Ì[email protected]@6E~}6<a=CFAM&LUFzD$¿
MA!{ AFAGZ<&[email protected] e
W (x) = W }|G x < 0, E~D W (x) = W }|G x > 0,
LMAƒ„E~D?AF)Ež}|=CIJ?DGJ=F L?)D=}6GJƒ„[email protected]|EF)=D{~E W (x, t) = H x º
t
,™IJ?A}<[email protected]|{$Ì$GH}6{~ƒÐE~FzD e
G =CFF)=DE a E_D a MAE~}+<>E~FzD6E$}ªMAEvÌ[email protected]$}+M)EMAGJ}|Ì$=CFD6GJFz?AGHD6{=C?MAEvM){~DE~FDE¿¬GHI­E!ÇGJ}|D6E
?AF =C<>{[email protected][email protected]FA=UD{ Rs D6E$I¬‚z?AE
0
L
0
L
R
R
}|G

x

WL ,
< aL


t



 x
x
x 
H
= Rs
, WL , WR ,
W (x, t) = H
=

t
t
t





x

 WR ,
> aR .
t
}6G a
L
<
}|G
x
< aR
t
Œ º ÀLU<A<>E~IJ=CF)}‚z?AE¾IJÌ[email protected]?AMA=ÐE~DÖÄ®E$FAÉzÊ&LUIJMA=C?)F;=CFDփ„=CFD@|{€M&LUFA} ¾ÖÄ ’)Œ y‚z?AEVIJE$}f{$‚z?&L1DGJ=FA}
MAE6)LGHFD (#‰E~F&LFDÓL ÅCE~Ì ?AF 3=CFAM E$F [email protected]|ƒÐE´MAE-ƒ´[email protected]Ì ÊAEU¿yMA=CF)F&LGJE~FDÓIJGJE~? ;?AFAE-}|=CIJ?DGJ=F
L?)D6=C}6GHƒÐGHIÃ[email protected] MA?Ð<)@6=CBAIHÍ$ƒ„EMAEfÀfGHE$ƒÐLFAF³¿C‚z?&LUFAM´IJLªM)GJ}6Ì~=CFDGHFX?)GHD{(GHFAGHD6GÃLIHE®MAEIJLª}|=CIJ?DGJ=F
{~DLGHD}|GHD?){$E vI [email protected]|=CGHD MAEªIÃLƒÐ[email protected]Ì ÊAE º
,[email protected]ª[email protected]|}$¿aFA=C?A}+E!Ç<[email protected]ƒÐ=CF)}ւz?AEÓFA=C?A}+Ì~=CFA}6GHMA{[email protected]|=CFA}ª?AFAEv}|=CIJ?DGJ=FL?9}6E~FA}ªMAE$}+MAGH}|[email protected] BA?)D6GJ=CFA}MA? <[email protected]|=CBAIHÍ$ƒ„E ‡Œ º > _¿)E$F ?)DGHIJGJ}6LFD(IÃL<)@6=C<a=C}6G,DGH=CF ii) MA?-DÊA{[email protected]Í~ƒÐE ‡Œ º  º Œ e
Z
∂R
[W.nt + f (W ) .nx ] dσ =
Z
R
Q (x, W ) dxdt.
‡Œ º = ÖF}|E-<[email protected]=<>=C}|º E-MAE-Ì$=ºFA}|D6º @[email protected] ?AF }|Ì ÊA{$ƒÐLFX?)ƒÐ{[email protected]‚z?AE-BAL}6{-?AF)GJ‚z?AE$ƒ„E$FDÓ}|[email protected]~}MAE$?XÇ
<[email protected]=<[email protected]{~D{~} ‡Œ  ®E~D Œ = Œ@
6 D "
$ I ËÑ{!AFAGJ}|}6=CFA}f<[email protected]?AF ƒ´LGHIJIJLKCE[email protected]ƒ„E e
∆t L ÅCE$Ì
− x ¿)IHE$} <&LU} MAEÑD6E$ƒ„<A} E~D M E~}6<&LUÌ$E
,
r =
∆t = t
− t E_D ∆x = x
@6E~}6<aE$Ì~D6∆x
GHÅE$ƒ„E$FD º
=C?)}fF)=D[email protected]=CFA} x IJE+Ì~E$FD@|E€M)EžIJLvÌ$E$IHIJ?AIHE ]x , x [ º
GJFA}|<[email protected]‰MAEIJLÖMA{$ƒÐ[email protected]|Ì ÊAEM)EÑÏ+=XMA?AFA=^ŠϪ=XMhORz¿XÏ+=M =’hE~D E~FÐ?)D6GJIHGJ}LUFzD ?AFAE<[email protected]~ƒÐGHÍ[email protected]|E
3=CGJ} IJE~} <[email protected]=<[email protected]{~D{~} Œ º  E~D Œ º = _¿X}[email protected]®IJEÑM)=Cƒ´LUGJFAE R =]x , x [×[t , t [ ¿=F-{$Ì[email protected]|GHD e
n
n
n
n+1
i− 12
i+ 12
n
i+ 21
i− 21
i
i+ 21
i− 12
=
n
n+1
n
n
Win+1 = Win − rn F Rs(0, Win , Wi+1
− F Rs(0, Wi−1
, Win + ∆tn Qni ,
Qni
E~}|Df?AFAE€L<A<)@6= ÇGJƒÐLUDGH=CF MAE
1
∆tn ∆x
Z
Q (x, W ) dxdt.
­=C?)@I {_ÅULIJ?&L1DGJ=FÖMAE F Rs(0, W , W ) ¿ I GJMA{~E™GJÌ~GE$}‡DyM {_ÅG,D[email protected]¬MAE‰Ì Lº IHÌ[email protected]¬IÃL }|=CIJ?DGJ=F
E_Ç)LÌ_DE Rs ¿C=C?vƒ<;$ƒ„E M GJFD@|=M)[email protected]|E ?AFÓ<[email protected]=BAIJÍ~ƒÐE M)EÀfGHE$ƒÐLFAFIHGJFA{[email protected]}6{ =?A}™Ì Ê[email protected]|Ì ÊA=CFA}
[email protected]|º E$Ì_DE$ƒ„E$Fº D?Aº F)EÑÅ[email protected](L<A<)@6=XÌ ÊA{$E+MAE+IJLV}6=IJ?)D6GJ=CFÐE_Ç)LÌ~D6E¿&E~F-E_Ç<AIJ=GHD LUFzD®IJE~} <[email protected]=<[email protected]{~D{~}
‡Œ  ®E_D Œ = ¾ÑGHFA}6G¿A=CF {$Ì[email protected]<[email protected]E_ÇE$ƒ„<AIJE+}|[email protected][email protected] x e
R
,
n
i
n
i+1
i+ 12
L <A<[email protected]|=^ÇGHƒ´LUD6GJ=CF´MAE Rs 0, W , W .
,­=C?)@V=CB)[email protected]Ì~E~D6D6E L<)<[email protected]= ÇGJƒÐLUDGH=CF³¿¬=CFR?)DGHIJGH}6EÓMAE FA=?)ÅCE$L? IÃL <[email protected]=<[email protected]{~D{ ‡Œ º = +}[email protected]€?AF
=C?)Å[email protected]|D π =]X , X [×]t , t + θ[ DE~I¬‚X?)E x ∈]X , X [ º
ƙF‘}|?A<A<a=C}LFD‚z?AE X , X E_D θ > 0 }|=CFDfÌ ÊA=CGJ}|GJ}(MAE+D6E$IJIH%E L½~=CF‘‚z?AE ∀t ∈]t , t + θ[ e
n
Wi+
1 =
n
i
2
θ
−
+
n
−
Rs
X − − xi+ 1
2
t − tn
i+ 12
n
−
n
i+1
+
+
n
, Win , Wi+1
n
!
= Win
E~DfE$F [email protected]=XMA?AGJ}6LFDIJE~}(MAGJ}‡D LFAÌ~E$} e dX
=CF‘L
−
E~D
Rs
X + − xi+ 1
2
t − tn
= |X − − xi+ 1 |
2
E~D
n
, Win , Wi+1
!
n
n
= Wi+1
,
dX + = |X + − xi+ 1 |
2
¿
ZX +
Z
−
n
+
n
n
n
W (x, tn + θ) dx = dX Wi +dX Wi+1 −θ F Wi+1 − F (Wi ) + Q (x, W ) dxdt.
πθ
X−
ÖF‘MA{ &FAG,D e
ŒR
:
D
D n
Wi+
1 =
2
dX −
1
+ dX +
I
ZX +
W (x, tn + θ) dx.
X−
¾ ÅE$̪IJEÌ ÊA=CG Ç e X = x E_D X = x ¿=CF =CB)D6GJE~FzD®?AFAE+<[email protected]|E$ƒ„GJÍ[email protected]ÑE!Ç<[email protected]~}6}6GH=CF KC{$F){[email protected]LUIJE¿
E$F 3=FAÌ~D6GJ=CF‘MAE θ ¿>MAEVIÃLÓ}6=CIH?)DGH=CF LU<A<[email protected]=XÌ ÊA{~E W GJ}|}6?AEM)EVÌ$E_D6D6EV<[email protected]|E$ƒ„GJÍ[email protected]|E€{_D L<aE¿>‚z?AE
FA=C?A}fL<)<>[email protected]=CF)}({~D LU<>EªMAEª<[email protected]|{$MAGHÌ~DGH=CF e
−
i
+
i+1
n
i+ 12
=
Œ ºP θ 1
n
n
n
bn 1 ,
(Win + Wi+1
)−
F (Wi+1
) − F (Win ) + θ Q
Wi+
1 =
i+ 2
2
2
∆x
bn 1 = Q
b xi , xi+1 , W n , W n
Q
i
i+1
i+
E$}‡D?)FAEVLU<A<[email protected]= ÇGJƒÐLUD6GJ=CF³¿5vMA{[email protected]MAEªƒÐLFAGJÍ[email protected] Ž|?)MAGJÌ~GJE$?)}6E¿&M)E
2
Z
1
θ∆x
Q (x, W ) dxdt.
πθ
F-<[email protected]|E$ƒ„[email protected]®Ì ÊA=CG ÇÐMA?-<&[email protected]ƒÐÍ_D@|E θ LV{_D{ÖM)EÖIJE<[email protected][email protected]ÑÌ$=Cƒ„ƒÐE?AFAE%[email protected]Ì~D6GJ=CFÐMA?-<&L}®MAE
DE~ƒÐ<A} ∆t ¿&E$F {$Ì[email protected]|GHÅULFD θ = α ∆t º
2
Ö[email protected]@6G,ÅCE LGHFA}6G ;IÃL;<[email protected]|E$ƒ„GJÍ[email protected]|E [email protected]ƒ„E MA? }|Ì ÊA{$ƒÐL À Î =CI,Å[email protected] ÀfGHE$ƒÐLFAF f=CF
Î =Cƒ„=CKCÍ~FAE e
n
n
i+ 21
n

n
n
αi+
1
αi+ 21

1

n
n
n
n
n
2
bn 1

rn F (Wi+1 ) − F (Wi ) +
∆tQ
 Wi+ 1 = (Wi + Wi+1 ) −
i+ 2
2
2
2
2

h i


 W n+1 = Win − rn F W n 1 − F W n 1 + ∆tQni .
i
i+
i−
2
Œ º@ 2
Á E~D|D[email protected]|ƒÐE MA?}6Ì Ê){$ƒÐL )À Î Lp{_D{;?)D6GJIHGJ}6{‘L ÅCE~Ì}6?AÌ~Ì${$} <[email protected]ÐMAE$} <)@6=CBAIHÍ$ƒ„E$}ÐFA=CF
ÊA=Cƒ„=CKCÍ~FAE$}ªŒ^ËjM&LFA}IHE+Ì L}MAEªƒÐLGJIHIÃLKCE~} ?)[email protected]|ƒÐE~} 3Å[email protected] Ä®E~F&’ º
D tXopi 1r³sZt Q CtL i k´xt‘w­i 1iAoH7kmt Zt Q kaxt α I
Á EÖ<&[email protected]ƒ„Í~D@|EÖLV<[email protected] =CBXŽ|E~Ì~D8G ¬MAEÖÌ~=CFD@[email protected]®IÃLžMAG >?A}|GJ=CFÐFX?)ƒÐ{[email protected]‚z?AEMA?-}6Ì Ê){$ƒÐLÌ$=Cƒ„ƒÐE
IJE+ƒ„=CFD@| E+I LF&LI,È}|E+ƒÐE~FA{$EªM&LUFA}IJEÖÌ$L}}6Ì LUIÃ[email protected]ÖM&LUFA}IJE+Ì Ê&L<)GHD@|E 2 º ,­[email protected]E_ÇE$ƒ„<AIJE+MALFA}
IJE„Ì L}€M ?AFAE´{~‚z?&LUDGH=CFp}|Ì [email protected]|EÓIJGJF){ [email protected]Ó}[email protected]žƒÐLGJIHIÃLKCEV?AFAG [email protected]ƒ„E„IHEÐÌ Ê)=CG,Ç MAE α = 1
’
n
i+ 21
n
i+ 21
6 D "
$ I @6E~MA=CFAFAEžE_Ç)LÌ_DE~ƒÐE~FzDfIHE+}6Ì ÊA{~ƒ´L„MAEžÔLmÇ [email protected]|= º
K-sZk+1t'Ai 1q3i k^t ­suUlZnXopi B D FG I
ËÖLFA} IJE(BA?D®MAEfI E_ÇXD6E$FA}|GJ=CF„MA?´}|Ì ÊA{$ƒÐL À Î L?„Ì LU} BAGHMAGJƒ„E$F)}6GJ=FAFAE$I¿=„GJI&E~}|D MAG „Ì~GJIJE
MAEªMA!{ [email protected] ?)F {~‚z?AGHÅULIHE$FDMAE+IÃLM)GJ}|DLFAÌ~E ¿&GHIdE~}|DfL<A<[email protected]? FA{~Ì$E$}|}[email protected]ªMAEª<[email protected]|=C<>=}[email protected](?)FAE
Å[email protected]E®M)E º À Î F)E LGH}LFD <AL} L<)<&L∆x
@L JD@|E Ì~E~D|DE MAGH}|DLFAÌ$E M&LFA}­IÃLf<AÊ&L}|E <[email protected]{~MAGJÌ_D[email protected]
MA? }6Ì ÊA{~ƒ´L
Á E$IJLRÌ$=FA}6GH}|DE R{~Ì[email protected]|[email protected] θ = α θ̄ = θ̄ E$}‡D´M){!&FAGf<&[email protected]„IÃLÅXGHD6E$}6}|E MAE‘À?A}LF)=mÅ cÅ[email protected]
&[email protected]|EŒ º Œ Ì$=ƒÐƒ„EÖ}6?)GHD e
n
i+ 21
t
tn+θ̄
tn+θ
Win
n
Wi+1
tn
X−
xi+ 1
Œ ºŒ θ̄ =
∆x
.
n
2Si+
1
X+
x
2
Á E$IJL„LUƒÐÍ~FAE Ó IJLFA=C?)ÅE$IHIJEªE_Ç<[email protected]|E$}6}|GJ=CF MAEªIÃLv<AÊAL}6Eª<[email protected]|{$MAGHÌ~D[email protected]MA? }6Ì ÊA{~ƒ´L e
2
n
Wi+
1
2
n
n
αi+
1 αi+ 12 ∆x n
1
n
n
n
n
2
b 1.
= (Wi + Wi+1 ) − n
F (Wi+1 ) − F (Wi ) +
Q
n
i+ 2
2
2Si+ 1
2 Si+
1
2
Œ ºR 2
D tXopi 1r³sZt I ,­=C?)@ÑIJE~}<[email protected]=CB)IJÍ$ƒ„E$}D@6LGHD6{$}M&LF)}ÑÌ$EžD@6L^ÅU LGHI0¿IJEªDº [email protected]|ƒÐEžFA=CF‘ÊA=Cƒ„=CKCÍ~FAE Q
Ì$=CFD6GJE$FD KC{$F){[email protected]LUIJE$ƒ„E$FD ?AFÓD[email protected]ƒ„EMAEMA{[email protected],ÅC{$E}6<&L1DGÃLUIJE M [email protected]|[email protected]žŒ ¾ÑGHFA}6G¿<>[email protected]‰IJE(}|ÈX}‡DÍ$ƒ„E
Œ
:
D
D I
MAE6)LGHFD (#‰E~F&LFD ¿yIÃL <[email protected]|E$ƒ„GJÍ[email protected] Ì~=CƒÐ<a=C}6LFDE-MA? D[email protected]ƒ„E }[email protected]Ì~E E~}|DÓFz?AIHIJE E_DvIÃL;}|E$Ì$=FAMAE
} {$Ì[email protected] Q = −gH dZ MA=CFD´?AFAEL<)<[email protected]= ÇGJƒÐLUDGH=CF }|DLFAM&[email protected]|M<[email protected]„IÃLR<)Ê&L}6E‘<[email protected]{~MAGJÌ_D[email protected]
dx
} {$Ì[email protected]
n
n
b n 1 = −g Hi + Hi+1 Zi+1 − Zi .
Q
i+ 2
2
∆x
ÔZ LÖMAGJ}‡D LFAÌ~E ∆x M)GJ}6<[email protected]LHDyM)=CFAÌ [email protected]|ƒÐE }|[email protected]|Ì$º E(MAE I {_D L<aE <[email protected]|{$MAGHÌ[email protected]~¿E~D <>[email protected]ƒ„E~D‰LGHFA}6G
I E_ÇXDE~FA}6GH=CF MA?‘}|Ì ÊA{$ƒÐL„L? Ì L}BAGHMAGJƒ„E$Î FA}|GJ=CF)FAE$I
ÔZL [email protected]ƒ„E+MA{!&F)GHDG,ÅCEªMA? }6Ì ÊA{~ƒ´L)À* M&LUFA}IHEÖÌ LU}KC{~FA{[email protected]³} {~Ì[email protected],DfMA=CFAÌ e

n
n
αi+
1 αi+ 12 ∆x n

1

n
n
n
n
n
2
b


 Wi+ 21 = 2 (Wi + Wi+1 ) − 2S n 1 F (Wi+1 ) − F (Wi ) + 2 S n 1 Qi+ 12
i+
i+
2
2


h i


 W n+1 = W n − rn F W n 1 − F W n 1 + ∆tQn .
i
i
i
i+
i−
2
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n
wi+1
+ win − αi+
− 2u
1 sgn(c) wi+1 − wi
2
|c| n
n
n
1 − sgn(c)αi+
wi+1
− u + 1 + sgn(c)αi+
(win − u)
1
1
2
2
2
n
n
−u .
≤ |c| 1 + αi+ 1 max wi+k
=
2
k=0,1
QP
D D
1q q3w™t­sopi ,Zqo9so
hw2au qckmq) AK A)% /1#z\+*J7 \ X7_#J ] /1b,c0/1b)\Ö\_#X01b& 7 \ N
i) α
≥ 1 $ ∀(i, n) ∈ Z × N
α
+α
≤ 1 $ ∀(i, n) ∈ Z × N,
ii) r|c|
2
*J7+\^] 2&% „ ! ) %_! 7ž*J7 'a!cbd]_`'&7;#&„JC0v#±*J/^]6*
n
i+ 12
n
i+ 21
n
i− 21
min win ≤ min win+1 ≤ max win+1 ≤ max win .
i∈Z
i∈Z
i∈Z
i∈Z
mtXs'>t
Ô¬Ež}|Ì ÊA{$ƒÐLÓMAE+ÅC=CIH?AƒÐE~}'&FAGJ}  º . ® } {$Ì[email protected] e
n
n
win+1 = β0 win + β−1 wi−1
+ β1 wi+1
,
=
β−1 =
E~D
β1 =
r|c| rc n
n
1 + αi−
=
sgn(c) + αi−
,
1 sgn(c)
1
2
2
2
2
r|c| n
n
β0 = 1 −
αi+ 1 + αi−
,
1
2
2
2
r|c| rc n
n
−1 + αi+
sgn(c)
=
−
sgn(c)
+
α
.
1
i+ 21
2
2
2
­=C?)@®‚z?AEÖIHE}|Ì ÊA{$ƒÐLMAEÅ=CIJ?)ƒÐE~}&FAGH} Å{[email protected]|G&EÖIHE<)@6GJF)Ì$GJ<aEfMA? ƒÐL1ÇGJƒ?Aƒ ¿CGHIa}|? ÓD(M L Å[email protected]
E_D β + β + β = 1.
β ≥ 0,
β ≥ 0,
β ≥0
Á E€‚z?AG¬E~}|[email protected]G³<[email protected]IJE$}(Ì~=CFAMAG,DGH=CFA} (i) E~D (ii) º
,
−1
1
0
−1
0
1
g & k q3t su+1lnXopi B D F G Zi u´x3t CiAu α tu_k s t Q ~u k^i k^t
hw2au qckmq) AK A A /1#z\+*J7 \ X7_#J ] /1b,c0/1b)\Ö\_#X01b& 7 \ N
i) ∃ γ ≥ 1 7_*5$#&7 ∀(i, n) ∈ Z × N,
α
= γ$
$
n
i+ 12
ii) r |c| γ ≤ 1 ∀(i, n) ∈ Z × N ! ) 7+\^] 2&% „
7 \_:´/1bd/1 /1bd7
[email protected]
n
i+ 21
DDI
1D
D mtXs'>t
™Æ F ?DGJIHGJ}6LFDÑI =C<a{[email protected][email protected] H ¿>IJEV}|Ì ÊA{$ƒÐL-MAEÅC=IJ?Aƒ„E$}%AFAGJ}À Î } {$Ì[email protected]|GHD+M)EvIÃL´ƒÐLFAGHÍ[email protected]
}6?AG,ÅULFDE e
w
= H w ,w ,w
,
L ÅCE$Ì
H w ,w ,w
=β w +β w +β w ,
º
= β , β E_D β }|=CFDÑM){!&FAGH}M&Lº FA} º 0  º Œ º
Ë L<[email protected]Í~} (i) MAEfIÃL+<[email protected]|=C<a=C}6G,DGJ=F 0  Œ _¿C=F´L α ≥ 1 ¿ ∀(i, n) ∈ Z × N ¿}6?)<A<>=}6=CFA}®[email protected]}
‚z? GHI³E_ÇGH}|DE γ ≥ 1 ¿)D6E$I³‚z?AEª<[email protected] D=?)D (i, n) ∈ Z × N α = γ ¿AMALFA}(Ì$EªÌ$L} β , β E_D
β M)E~ÅXGJE~FAFAE$FD
n+1
i
n
i−1
0
−1
n
i
n
i−1
n
i+1
0
n
i
n
i
n
i+1
−1
n
i−1
1
n
i+1
1
n
i+ 21
n
i+ 21
0
−1
1
β−1 =
E~D
rc
r|c|
(1 + γsgn(c)) =
(sgn(c) + γ) ,
2
2
β0 = 1 − r|c|γ,
r|c|
rc
(−1 + γsgn(c)) =
(− sgn(c) + γ) .
2
2
β0 , β−1
β1
β−1
β1
r|c|γ ≤ 1,
β1 =
Ô¬Á Ež}|Ì ÊA{$ƒÐL)À Î E$}|Dfƒ„=CFA=D6=CFAE+}|G
E_º D }|=CFD<a=C}‡DG 3}(=? Fz?AIH} º
=Cƒ„ƒÐE γ ≥ 1 ¿&[email protected]} E~D }6=CFDf<>=}6GHD6G 3}
Ì~E€‚z?AG¬E~}|[email protected]G³<&[email protected]Ì$=CF)MAGHD6GJ=CF
β ≥ 0 }|G¬E$}|Df}|E$?AIHE$ƒ„E$FDf}6G
º
M = ‘[email protected]{~}6?AI,D LUD
1
  ?&9<C
=CGHE$FD f : R → R ?)FAE 3=CF)Ì~DGH=CFMAE Ì~IÃL}|}6E C FA=CFWIJGJF){ [email protected]¿
Ì$=CF)MAGHD6GJ=CF-GHFAGHD6GÃLIHEMA? <[email protected]|=CBAIJÍ~ƒÐE  º Œ º
,­=C?)@ γ ≥ 1 ¿ ÖF MA{!AFAGHDfI E$F)}6E$ƒVBAIJE+}|?AGHÅULFD e
1
1
Q †
5 462 4
(ii)
¿
$?$4DG
1
w0 ∈ L∞ (R)
{_D LFD IJL
X = {w ∈ R/|w| ≤ γ||w0 ||L∞ (R) }
E~Df=CF <a=C}6E A = max |f (w)| º
Ô¬E®}6Ì Ê){$ƒÐL)À Î <[email protected]­IJE <[email protected]=BAIJÍ~ƒÐE FA=CFIHGJFA{[email protected]|E™ÊA=Cƒ„=CKCÍ~FAE  º Œ _¿m} {$Ì[email protected]®IJLƒÐLFAGHÍ[email protected]
}6?AG,ÅULFDE e
w∈X



w n
0
n
αi+
1
1 n
n
n
2
f (wi+1
) − f (win ) ,
w
+
w
−
1 =
i+1
i
n
i+ 2
2
2Si+ 1
2


w n+1 = w n − r f (w n 1 ) − f (w n 1 ) .
i
i
i+
i−
2
OR
2
 º?> D D
¬Ô E}6Ì ÊA{~ƒ´L-MAE€ÅC=CIH?AƒÐE~}%&FAGH} 0 º?> E$}‡DªÌ$=CFA}|[email protected]‡ÅL1DG <&[email protected]ÑÌ$=FA}|[email protected]?AÌ_DGJ=FE~D+IJL3=CFAÌ_DGJ=F;MA?
?&?Ç Fz?Aƒ„{[email protected]|GJ‚z?AE+E$}‡DÑM)=CFAFA{~Ež<[email protected]
n
n
n
n
g(win , wi+1
) = f (wi+
1 ) = f ϕ(wi , wi+1 ) ,
L ÅCE$Ì e
2
n
ϕ(win , wi+1
)
n
αi+
1
1 n
n
n
wi+1 + wi − n 2 f (wi+1
) − f (win ) .
=
2
2Si+ 1
­=C?)@ w = w ¿d=CFpL ϕ(w , wº ) = w ;³GHI } E$F)}6?AG,D€‚z?AE
}6?AG,DEªIJE+}|Ì ÊA{$ƒÐLÓE$}|DfÌ~=CFA}6GH}|DLFD
2
,
n
i+1
n
i
n
i
n
i
g (win , win ) = f (win )
n
i
1 q q3w™t­sopi ,Zqo9so
ÖF‘FA=UDE a⊥b = min(a, b) E~D a>b = max(a, b) º
hw2au qckmq) KA A b \_#m''&/^\^7Ö5$#&7 f 7 \_ #bd7 /1bd]_0/1b&X7€]_*,1\ \^7
#b \_ bd7V] /1b)\_ b&0$ *J/1\+\^/1#z\+*J7 \ X7_#J ] /1b,c0/1b)\Ö\_#X01b& 7 \ N
S
S
$
1)
≤α
≤γ
∀(i, n) ∈ Z × N $
|f (a )|
n
i+ 21
0 n
i+ 21
2)r ≤
1
γA
7 ];N a
n
i
7_™5$#h7 f X7
0
n
i+ 21
n
i+ 21
f 0 (ani+ 1 )
2
$
h
i
n
n
n
n
∈ wi+
⊥w
,
w
>w
1
i− 1
i+ 1
i− 1
2
2
2
2
7_*Z5$#&7
n
n
n
n
f wi+
− f wi−
= f 0 (ani ) wi+
1
1
1 − w
1
i−
7_
2
2
2
2
n
n
ani+ 1 ∈ wi+1
⊥win , wi+1
>win ,
7_*5$#&7
*J7+\^] 2&% „
C1
¿³E~Dž<&[email protected]
2
n
f (wi+1
)
−
f (win )
=f
0
ani+ 1
2
n
wi+1
− win ,
\$J\ ~c *J7('a_cbd]_`'&7+# „JC0v#±*J/^]6* ! ) min win ≤ min win+1 ≤ max win+1 ≤ max win .
i∈Z
i∈Z
i∈Z
m tXs'>t
h
f E$}‡DÖÌ~=CFDGJFz?AEVE~DÖM){[email protected],ÅULBAIJEª}|[email protected] w ⊥w , w
}6E~ƒÐE~FzD6}*AFAGJ}~¿&FA=C?A}MA=CF)FAEªI E_ÇGJ}‡DE$F)Ì$EªMAE a º
.C’
n
i− 21
n
i+ 21
n
i
i∈Z
n
n
>wi+
1
i− 21
2
i
;aIHE€D6ÊA{[email protected]|Í$ƒ„EVMAE~}ÖLÌ~Ì[email protected]|=CGJ}
DDI
1D
D ËÑ Efƒ;~ƒÐE(}|[email protected] w ⊥wº , w >w ¿IHED6ÊA{[email protected]Í$ƒ„EMAE$} LÌ~Ì[email protected]=GJ}6}|E$ƒ„E$FD} & FAGH}™FA=C?A} MA=CF)FAE
I E_ÇGJ}‡DE~FAÌ$EªMAE a I³} E~FA}6?AG,Dтz?AE
α
1
 º  w +w
−
w −w .
=
f a
w
n
i
n
i+1
n
i+ 12
n
i+ 21
ƙF‘<a=C}LUFzD
2
n
i
n
i+1
n
i+1
n
δi+
1
2
=CF =CB)DGHE$FD
=
n
i+ 12
n
2Si+
1
2
n
αi+
1
2
n
Si+
1
2
Ö@fIJLÌ$=CF)MAGHD6GJ=CF
n
i
n
wi+
1 =
n
i+ 21
n
i+1
n
i
f 0 ani+ 1 ,
2
1
1
n
n
n
win +
wi+1
.
1 + δi+
1 − δi+
1
1
2
2
2
2
MAEªIÃL<)@6=C<a=C}6G,DGH=CF 0 º  º . ®MA=CF)FAE
2
(1)
0
n
1 ≤ δi+
≤γ
1
¿AGJId} E$FA}|?AGHDf‚z?AE
2
n
0
n
|wi+
1 | ≤ γkw kL∞ (R) ≤ γkw kL∞ (R) ≤ γkw0 kL∞ (R) .
Ë = w ∈ X ∀(i, n) ∈ Z × N E~DL?A}|}6G a ∈ X E_DM)=CFAÌ f (a ) ≤ A º
ƙF‘GHF1Ž|E$Ì_D LFDfI {$KLUIJGHD6{  º  ®M&LFA}I {~DL<aE+Ì[email protected]|@6E$Ì_D[email protected]=CF [email protected]L
2
n
i+ 21
L ÅCE$Ì e
0
n
i
n
i
 º = n
n
n
n
, win , wi+1
win+1 = H wi−1
= β0 win + β1 wi+1
+ β−1 wi−1
,
r
n
n
β0 = 1 − |f 0 (ani )| δi+
,
1 + δ
1
i− 2
2
2
r
r
n
n
0
β1 = |f 0 (ani )| sgn (f 0 (ani )) δi+
f 0 ani+ 1
− 1 = |f 0 (ani )| δi+
,
1 sgn
1 − sgn (f )
2
2
2
2
2
r
r
n
n
0
β−1 = |f 0 (ani )| sgn (f 0 (ani )) δi−
f 0 (ani− 1 ) + 1 = |f 0 (ani )| δi−
.
1 sgn
1 + sgn (f )
2
2
2
2
2
,
n
αi+
1
2
r
β 0 β1
β−1
β0 + β−1 + β1 = 1
­=C?)@žƒ„=CFD@|[email protected]+IHEv<[email protected]Ì$GH<>EM)?RƒÐL1ÇGJƒ?)ƒ‘¿aGJI }|? ÓDžMAEvD@6=?)Å[email protected]žIHE$}ªÌ$=FAMAGHD6GJ=CF)}Ö}[email protected]
º Ö@=CF L
E~D MAE+D6E$IJIHE+}[email protected]‡DE+‚z?AE ¿ E~D }|=CGJE~FD<>=C}|GHD6G 3} E_D
 º P β ≥0
⇐⇒
|δ | ≥ sgn (f ) ,
 º @ β ≥0
⇐⇒
δ
≥ − sgn (f ) ,
M = ³¿‰}6=C?)}ÐIJL9Ì~=CFAMAG,DGH=CF ∀(i, n) ∈ Z × N α ≥ S ¿‰=CFWL}|}[email protected]|E β ≥ 0 ¿
0
n
i+ 21
1
0
n
i− 21
−1
n
i+ 21
n
i+ 21
1
f 0 ani+ 1
2
β−1 ≥ 0
~E D β ≥ 0 ⇐⇒ r |f (a )| δ
2
Ö@f}|=C?A}IJE~}(Ì$=CF)MAGHD6GJ=CFA}(M)EžIJL<[email protected]=<>=C}|GHD6GJ=CF  º  º . ®=F‘L
0
0
n
i
n
i+ 12
n
+ δi−
1
r 0 n n
n
|f (ai )| δi+ 1 + δi−
≤ rAγ ≤ 1.
1
2
2
2
)Œ
.
2
≤ 1.
D D
tXo oRt A AK + /1bd]_0/1b #4 #J b&#H´%_!5$#&7#;\^] 2&% „ \^] 2Xc $07_b&bd7 mtXs'>t
,­=C?)@D6=C?)D w ∈ X =CF L
n
g win , wi+1
− f (w)
=
n
f ϕ win , wi+1
7 \_Ö] /1b&cb&#&7 `',"
− f (w)
n
−w
ϕ win , wi+1
≤ A
n
αi+
1
1
0
n
2
A 1 + n f ai+ 1
≤
2
2
Si+ 1
2
!
n
max wi+j
−w
j=0,1
n
≤ C max wi+j
−w ,
j=0,1
=
C=
A (γ + 1)
2
D Xt opi 1r³sZt
ËÖLFA}fIJE+Ì$L}= e f a
0
n
i+ 21
=0
¿&I {~D LU<>E+<[email protected]|{$MAGHÌ[email protected](MAE_ÅXGJE$FD
n
wi+
1 =
2
g & k q3t
1 n
n
wi + wi+1
.
2
hw2au qckmq) AK A $ b\_#m''&/^\^7 5$#&7 f X7®#bv\_ bd7] /1b)\_ b& /1#z\ *J7 \X7_#J] /1b,0"
0/1b)\+\_#X01b& 7 \;N
S
$ ∀i ∈ Z$ n ∈ N 7 ] γ̄ ∈ [1, γ]$ γ̄ 7 \_(] /1b)\_0b& 7%$
1) α
= γ̄
0
n
i+ 21
n
i+ 21
f 0 (ani+ 1 )
2
2)
rγA ≤ 1 7+\^] 2&% „ ! ) 7 \_:´/1bd/1 /1bd7 mtXs'>t
FAE Ì~=CFAMAG,DGH=CF }|? „}LFDE <[email protected];@|E$}|<>E~Ì~D[email protected]ÃL <)@6E$ƒ„GJÍ[email protected] Ì~=CFAMAG,DGH=CF MAE IJLW<[email protected]|=C<a=C}6G,DGJ=F
0 º  º . ®E~DfMAE)Ç[email protected]?)F γ̄ ∈ [1, γ] DE~I¬‚z?AE
S
 º R .
α
= γ̄
n
i+ 12
n
i+ 21
f 0 ani+ 1

.
2
DDI
1D
D ËÖLFA}fÌ$EªÌ$L}I {~D LU<>E+<[email protected]|{$MAGHÌ[email protected]MAE_ÅGHE$FD e
n
wi+
1 =
1
1
n
(1 + γ̄ sgn (f 0 )) win + (1 − γ̄ sgn (f 0 )) wi+1
2
2
E~DfI {~DL<aE+Ì[email protected]|@6E$Ì_D[email protected]} {~Ì[email protected]|GHD}6=?A}[email protected]ƒ„E+Ì$=FA}[email protected]|ÅULUDG,ÅCEª}6?)GHÅULFDE
2
º Œ^’ 0 º ŒCŒ 0
n
n
win+1 = win − r f wi+
− f wi−
,
1
1
2
2
n
n
= H wi−1
, win , wi+1
,
L ÅCE$Ì H E~}|DfI =Cº <a{[email protected][email protected](MA{!AFAG¬M&LFA}  º = º
Ô¬EÑ}6Ì Ê){$ƒÐL 0 Œ^’ ™E$}‡D(ƒÐ=FA=D=FAE}|GdI =C<a{[email protected][email protected] H E$}|D Ì[email protected]=GJ}6}6LFD <&[email protected]®@LU<A<>[email protected]|D+VÌ Ê&LÌ~?AFAE
MAEªÌ$E~}fÅ[email protected]|GÃLBAIHE w , w E_D w º
n
i
n
i−1
n
i+1
r
∂H
n
n
n
w
,
w
,
w
= (sgn (f 0 ) + γ̄) |f 0 (ani )| ,
i−1
i
i+1
n
∂wi−1
2
∂H
r
n
n
wi−1
, win , wi+1
= − (sgn (f 0 ) − γ̄) |f 0 (ani )|,
n
∂wi+1
2
∂H
n
n
= 1 − rγ̄f 0 (ani ).
wi−1
, win , wi+1
n
∂wi
rγ̄A ≤ 1
H
¬Ô ELUGHD(‚z?AE γ̄ ≥ 1 E_D
¿AI =C<a{[email protected]L1DE$?)@ Î $E }‡DfÌ[email protected]=GJ}6}6LFD<&Lº @@LU<A<>[email protected]|D*vÌ Ê&LÌ~?AFAEžMAE
Ì$E~}fÅ[email protected]|GÃLBAIHE$}~¿)GJId} E$FA}|?AGHDf‚z?AEžIJE+}6Ì Ê){$ƒÐL>)À* ~E }|DƒÐ=FA=D=FAE
Z) l ) n#:7Xopt KA A /1c w ∈ L (R)$³\^/1#z\ *J7 \Ñ] /1b,c0/1b)\-X% ‰b&07 \-b)\ *,('a / '&/^\_c0/1b
! ‘*, \^/1*.#X0/1bWb&#´%_!5$#h7LX/1b&bd% 7+')´*J7Ó\^] 2&% „ ] /1b 7_ X7)7_\Ð* #b&5$#&7
\^/1*.#0/1bp7_b& / 'a5$#&7 mtXs'>t
Ô¬E }|Ì ÊA{$ƒÐL 0 º > dE$}‡Dy?)F€}|Ì ÊA{$ƒÐL [email protected]=CGH}¬<>=Cº GHFº D}¬Ì$=FA}6 [email protected]|ÅULUD8G AÌ~=CFA}6GH}|DLFD­ƒ„=CFA=UD=CFAE `ϪÀ RŒ ¿
}6=C?)}(IJE$}®Ì$=CFAM)GHDGH=CFA} MAE+Ià Lž<[email protected]=<>=C}|º GHD6GJ=CF 0   _¿XGJIa} E$FA}|?AGHD‚z?AE+Ì~EÖ}6Ì Ê){$ƒÐLÓE~}|D(Ì~=CFA}|GJ}|DLFD
L ÅCE$̪D=?)DEªÌ$=FAMAGHD6GJ=CF-M [email protected]=C<)GJE
0
∞
S C ֆ 9A† E 7:9<CJADCJ=2 4 7:9 ‰$C Aц 4D5 _T G7)† C 69<GKAD?$=7CK;9
ÔZL Ì~=CFAMAG,DGH=CF;MAEÓ}|DLBAGHIJGHD6{ (2) MAEvIÃL <[email protected]|=C<a=C}6G,DGJ=F 0 º  º . fE_ÇGJKCEv‚z?AE |f º (a )| }|=CGHD%&FAG º
FAEªÌ~=CFAMAG,DGJ=F-}6? „}LUFzD6EV<[email protected](Ì~E$IÃL¿AE$}‡D‚z?AE a E~D w
@6E~}|D6E$FDÑ[email protected]|FA{$}
1
1
0
n
i
. .
n
i+ 21
n
i
D D
ÖË LFA}fÌ$Eª‚z?AG¬<[email protected]|{$Ì~Í$MAEU¿aÌ~E$IJL„Lv{_D{ª=CB)D6E$Fz?³¿&E$F GJƒ„<a=C}LFD(‚z?AE w ∈ X ¿A= X E~}|DMA{!&F)G
L? M){$BA?)DÑMAEVÌ~E~D|DEV}|E$Ì_DGJ=F‘ÅXGÃLÓ?AF;<&[email protected]Lƒ„Í~º [email protected] γ ≥ 1 º f=C?A}ÅC=C?AIH=CFA}GJÌ$G¬IH E~Å[email protected]ÑI GHFAMA{_D[email protected]
ƒÐGHF&[email protected]=XMA?AG,DEv<&[email protected]ªÌ~EÐ<&[email protected]Lƒ„Í[email protected] γ ,­=C?)@€Ì~E$IÃL¿¬[email protected]|E$ƒÐ[email protected]‚z?AEӂz? ?AFAE´Ì~=CFAMAG,DGH=CF
}6? „}6LFDEV<[email protected](‚z?AE w @6E~}|D6E€[email protected]){+E$}|Df‚z?AE w ∈ w ⊥w , w >w . Ö@
n
i+ 21
n
i+ 21
n
i+ 21
n
wi+
1 =
n
1 + δi+
1 ≥ 0
E_D
n
1 − δi+
1 ≥ 0
2
n
δi+
1
2
n
i+1
n
i
n
i+1
1
1
n
n
n
n
1 + δi+
w
+
1
−
δ
wi+1
,
1
i
i+ 21
2
2
2
?AFAEªÌ$=FAMAGHD6GJ=CF-}|? „}LFD6EVE$}‡DfMA=CFAÌ
2
n
i
≤1
2
n
αi+
1
2
⇐⇒
¿AGJI³} E~FA}6?)GHDf‚z?AE
n
Si+
1
2 .
≤
f 0 ani+ 1
Á E$IJL€L ÅCE~ÌÑIJLªÌ~=CFAMAG,DGH=CF (1) MAEIJLª<)@6=C<a=C}6G,DGH=CF 0 º  º . yMA?´<)@6GJF)Ì$GJ<aE(MA?Ѓ´L1ÇGHƒ?Aƒ GHƒÐ<a=C}|E
‚z?AE α = S º¬» GHF&LIHE$ƒ„E$FD+=CF9MA{~MA?AGHDžIHEv}6Ì ÊA{~ƒ´L )À* Î 3Ì E$}|D€IJEv}|Ì ÊA{$ƒÐL
f a
)*
À Î E~F LGJ}6LFD®[email protected]‡ÅCE$F)[email protected] IHEÑ}6GHKCFAEMAE f M&LFA} I {_D L<aEÑ<)@6{$M)GJÌ[email protected] ‚z?AGd} {~Ì[email protected]|GHD }|=C?A}(IJL
[email protected]ƒ„E 
2
n
i+ 21
}6=CG,D
n
i+ 21
n
i+ 12
0
0
1 n
1
n
n
n


f (wi+1
) − f (win )
wi+ 21 = 2 wi + wi+1 −
2 f 0 (ani+ 1 )
2 

w n+1 = w n − r f w n 1 − f w n 1 ,
i
i
i+
i−
2
2

1
1 n
n
n
n

wi+
− win
wi + wi+1
− sgn f 0 (ani+ 1 ) wi+1
1 =
2
2
2
2

w n+1 = w n − r f w n 1 − f w n 1
.
D Xt opi 1r³sZt
G =CF }6?)<A<>=}6E-‚z?AE f [email protected]|MAE´?)F }6GJKFAE´Ì~=CFA}|DLFDE_Dv‚z?AE I =CFR{$Ì[email protected]|GHD€IJEÐ}6Ì ÊA{~ƒ´L À Î
}6=C?)}[email protected]ƒ„E
= H w ,w ,w
,
w
I [email protected]|@6E$?)@fM)Eª[email protected]=CF)Ì [email protected]+E~}|DMA=CFAFA{~Eª<&[email protected][email protected](ÀfL^Å[email protected]|D(E_DÑϪ=XMAIJE }6ÉzG™ŒRQRŒ `ϪÀ RŒ i
i
i+ 2
i− 2
0
n
i−1
n+1
i
n
i
n
i+1
L ÅCE$Ì B (w, r) = 1
2r
Á =Cƒ„ƒÐE
E~D
∂
∂w
ET (w)(x, t) = −τ
B (w, r)
+ ◦(τ 2 )
∂x
∂x
2
j=1
X
j=−1
j2
∂H
1
2
(w, w, w) − f 0 (w)
n
∂wi+j
2
º
∂H
r
(w, w, w) = (1 − sgn (f 0 )) (f 0 (w))
n
∂wi+1
2
∂H
r
(w, w, w) = (1 + sgn (f 0 )) (f 0 (w)) ,
n
∂wi−1
2
.
>
DDI
1D
D [email protected]}
B (w, r) =
1
1
r |f 0 (w)| − f 0 (w)2
2
2r
2
=
|f 0 (w)|
(1 − r |f 0 (w)|) .
2r
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IÃLvÌ$=FAMAGHD6GJ=CF-M)Ež}‡D LB)GJIJG,D{E$}|DfGHF3{[email protected]|GJE$?)@6E 1 IJE+}6Ì Ê){$ƒÐL„E~}|DfM [email protected]@|E 1
 7)† [email protected] 4 ; EHG9? ;DCK5
r³sZi)kmq t 9ht#Uu
Ô {~‚X?ALUDGH=CFÐM)EÑÄ )@[email protected]} @6E$<)@6{$}|E$FDE?AFAEfIJ=CG)MAEÌ~=CFA}|[email protected]|ÅULUD6GJ=CF„FA=CF„IJGHFA{ [email protected]®Ì$=CƒB)GJFA{~EL ÅE$Ì
?AFAE+<&[email protected]|DGHEÖFA=CF-Ê)=CƒÐ=KCÍ$FAEÑMAG >?A}6G,ÅCEU¿)ƒ„=MA{~IJGH}LFD M)Eփ´LUFAGJÍ[email protected]Ö}|GJƒ„<AIJ8G &{$Ef?AF-{$Ì~=C?AIHE$ƒ„E$FD
E~Df‚z?AG¬} {$Ì[email protected],D e
1
1
[email protected];
∂w
∂ w2
∂2w
+
( )=ν 2,
∂t
∂x 2
∂x
ν
= w E$}‡D?AFAEÖÅ[email protected]|GÃLBAIHºEÑÌ~=CFA}[email protected]|ÅULUD6GHÅCEU¿ E$}‡Df?AF Ì$=XE„Ì$GHE$FDMAEÖÅXGJ}|Ì$=C}|GHD{ º ÖF Ì$=CF)}6GJM)Í[email protected]+IHE
Ì L}= ν DE$F)M Å[email protected]} 0 ËÖLFA}fÌ$EªÌ LU}fI {~‚z?&LUDGH=CF-MAE_ÅGHE$FD e
2
∂w ∂( w2 )
+
= 0,
∂t
∂x
E~Df=CF <[email protected]|E$FAM‘Ì$=Cƒ„ƒÐEÑÌ$=CFAM)GHDGH=CF-GJF)GHDGJLIJE


0 }|G 0 ≤ x < 0.2,






1 }|G 0.2 ≤ x < 0.4,
w (x) =


1 − 5(x − 0.4) }|G 0.4 ≤ x < 0.6,






}6GHFA=CF .
0
Á E~}|D‘?AF <)@6=CBAIHÍ$ƒ„EMAE ÀfGHE$ƒÐLFAFDE~}|D6{R<&[email protected] » º ËÑE #‰?)ÈX}|D Ë #+’ > 0¿(MA=FzD IJL }|=CIJ?DGJ=F
%
Ì$=CFD6GJE$FD+?)FAE=CFAM)[email protected]@6{ LÌ~D6GJ=CF;‚z?AG­}69E [email protected]ƒ„E ´<&[email protected]|D[email protected]ÑMAEVIÃLÐ<)@6E$ƒ„GJÍ[email protected]€MAGJ}|Ì$=CFDGHFz?AGHD6{
E~D´‚X?)GÑÌ$=ƒÐ<[email protected]|E$}|}6E;IJL ~=CFAE;[email protected] x = 0.4 E_D x = 0.6 ¿ E_D ?AF)E =CFAM)E;MAE Ì Ê)=̑‚X?)GÑ}6E
<[email protected]=<&LKC*E +<&[email protected]‡D[email protected] x = 0.6 L ÅCE~̏?)FAEfÅXG,DE$}|}6E{$KCLIJ2E σ = 1 º ÖFÐÌ~=CƒÐ<[email protected](IHE$}™@6{~}6?AI,D LUD6}
Fz» ?AƒÐ{[email protected]‚X?)E$}‰MA=CFAF){$}‰<&»®[email protected]™º IHE(}6Ì ÊA{~ƒ´"L )*À >Î E_D IHE }6Ì Ê){$ƒÐL Î 2ÈB)@6GJM)EÔLmÇ [email protected]|= −ÔZL17Ç @[email protected]Ì ÊA}Ì$=CFA½~?v<&[email protected] º ËÑÁ +E » #‰?)ÈX}|D Ë #֒ )E_D IJLf}|=CIJ?)D6GJ=CFžDÊA{~=CÎ @6GH‚z?AE L?Xº ÇV<ƒ ;$ƒ„E$}GJF)}|D LUFzD6}
ÔpÅUL?)D 1 <[email protected]€}6Ì ÊA {~ƒ´EL À ‘º E_Dªº ’ „<[email protected]ÑIHE€}|Ì ÊA{$ƒÐL
t = 0.1s E_D t = 0.6s ÔZL
ÊÈB)@6GJM)E;E~D´=CFƒ„=CFD@6E‘[email protected]´Ì[email protected] M [email protected]@[email protected] pI GHFA}|DLFDÐD +’ =U} ,‰?)GJ}6‚z?AE IJE }|Ì ÊA{$ƒÐL
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0
D D
)À Î ´M)E~ÅXGJE~FzDIHEÑ}|Ì ÊA{$ƒÐLvMAE+ÔZL1Ç7 RE$FAM)@6= <>=C?)@ IJE<&[email protected]LUƒÐÍ_D@6EMAE+Ì[email protected] α = 1
E~DfIJEª}|Ì ÊA{$ƒÐLÐÊÈ[email protected]ª{$KLUIJE$ƒ„E$FD<[email protected]IJEª<&[email protected]Lƒ„Í[email protected] θ = 0 ¿&FA=C?)}LUIJIJ=FA}L„Ì Ê[email protected]~}
Ì$=C?)@6BaE$}fMAEª@6{$<[email protected]|D6GHDGH=CF MAE α E~D θ º
ËÑEª<AIJ?)} IJE~} <&[email protected]ƒÐÍ_D@|E$} α E~D θ Ì[email protected]$FD(IJLMAG>?A}6GH=CF Fz?Aƒ„{[email protected]|GJ‚z?AE¿)[email protected]~ƒ´[email protected]|‚z?AE
‚z?AEVÌ~E$IJIHEªÌ$GE$}|D{$IHE~Å{$EV<[email protected]ª}|Ì ÊA{$ƒÐEL )*À Î Ð I [email protected]|=CGHDM ?AFAE€=CFAMAEžMAE€@[email protected]|{!LÌ_DGJ=F
E~DÑL?Aº }|}6GGJƒ„<>[email protected]‡D LFD6EÖ<>[email protected]IHEª}6Ì ÊA{~ƒ´L„ÊzÈ[email protected]|GJMA"E ÓI GJ}6}|?AEžMA? <[email protected]|=C<&LKCLUDGH=CF-M ?AFAEV=FAMAEžMAE
Ì ÊA=XÌ
Ô¬E ?A?Ç Fz?Aƒ„{[email protected]‚z?AEªMA?‘}|Ì ÊA{$ƒÐLÓÊzÈ[email protected]|GJMAEª} {~Ì[email protected],DfMAEªIÃ9L L½~=CF‘}|?AGHÅULFD6E
*
n
i+ 21
n
i+ 21
φni+ 1
2
= φ
&< [email protected]
n
n
n
θi+
1 ; Wi , Wi+1 , r
2
M LF
n
i+ 12
n
i+ 21
n
i+ 12
n
M LF
θi+
1φ
i+ 21
2
=
n
i+ 12
n
Win , Wi+1
,r
n
n
n
W
,
W
,
r
,
+ 1 − θi+ 1 φLW,
1
i
i+1
i+
»
(U, V, r) E~DªIHE9?&?Ç;Fz?AƒÐ{[email protected]‚X?)EvMA?9}6Ì ÊA{~ƒ´L-MAEÓÔZL1Ç7 @|GJE$M)@6GJÌ Ê)}փÐ=XMAG8&{VMA=FAFA{
φM LF (U, V, r) =
2
2
1
1
(f (U ) + f (V )) −
(V − U )
2
4r
E~D φ (U, V, r) E~D™I LU<A<[email protected]= ÇGJƒÐLUD6GJ=CFžMA? ?&?ÇFz?AƒÐ{[email protected]‚X?)E }6?AG,ÅULFD™?AFAE M){[email protected],ÅC{~E([email protected]|E$Ì~D6GJ=CF
FAE$IHIJE+E_D?)F <[email protected]Lƒ„Í[email protected] MA? }6Ì ÊA{~ƒ´LÓMAEžÔZL1Ç7 [email protected]|=9MA=CF)FA{ª<&[email protected]
LW,
φLW, (U, V, r) =
1
r f [ϕ̄ (U, V ) + (f (V ) − f (U ))] − f (ϕ̄ (U, V ))
(f (U ) + f (V )) −
,
2
2
= ϕ̄ (U, V ) E$}|Dª?)F;{~DLUD+ƒ„=mÈE$F}6LUDGH} LGH}LFDIÃLÐ@|E$IJLUDGH=CF MAEVÌ$=CFA}|GJ}‡D LFAÌ~E ϕ̄ (U, U ) = U ¿
E$}‡D®Ì ÊA=GJ}6G&D6E$Ih‚z?AEIJLžƒÐ[email protected]Ì~E Ā (U, V ) = A (ϕ̄ (U, V )) ¿E~D IJE'?&?DŽFz?Aƒ„{[email protected]|GJ‚z?AEfMA?´}|Ì ÊA{$ƒÐL
MAEžÔZL1Ç7 [email protected]|=9E$}‡DfMA=CFAFA{ª<&[email protected]
φLW (U, V, r) =
n
θi+
1
02
E_D
E$}‡D Ì ÊA=GJ}6GaMAEfDE$IHIJE}[email protected]|DE‚z?AEÖIJED[email protected]ƒ„EM)EÑIÃLžMAGJ}|}6GH<&LUDGH=CF
n
ηi+
1
2
=
1
r
(f (U ) + f (V )) − Ā (U, V ) (f (V ) − f (U )) .
2
2
=
S(Uin+1 )
1
S(U ) = kU k2
2
−
S(Uin )
+r
n
n
DU Si−
1
2
Uin
E~}|DfIÃ[email protected]ƒ„E+MAEªI {[email protected]¿AE_D
Q=
.
−
φni− 1
2
+
n
DU Si+
1
2
n
ηi+
1
2
}|=CGHD GJF53{[email protected][email protected]‰=C?´{$KLI
n
DU Si+
1
2
= DU S
φni+ 1
2
−
Uin
n
Uin + Ui+1
2
o
º
,
DDI
1D
D 2
1
Analytique
SRNHS
Hybride
0.9
1.9
1.8
0.8
1.7
0.7
1.6
0.6
1.5
0.5
1.4
0.4
1.3
0.3
1.2
0.2
1.1
0.1
1
0
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0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
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0.6
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0.2
0.3
0.4
0.5
0.6
0.7
0.8
K 6LMN:O<P>4B '$(6'$( )QA>R&
SBCED ' $ θn - '0/213%5 "
0.1
0.2
0.3
0.4
0.5
0.6
1
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1.8
0.8
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0.4
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1.3
0.2
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0.1
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"
1
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
2
Representation des courbes d’erreur de SRNHS et Hybride
1
−4
hybride
SRNHS
0.9
−5
0.8
0.7
−6
Log(L1−erreur)
0.6
0.5
−7
0.4
−8
0.3
−9
0.2
0.1
−10
−11
0
K *)MN:O<P>4B '$(6'$( )QA>R&
SBCED ' $ n - '0/213 ( "
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
−10
−9
−8
−7
−6
º P + ,-.0/ - 1

243516798:9;9<=8> [email protected]
Log(dx)
1
D E#"$ & -FG,1 θi+ 1
2
[email protected]
.
' 0.734
−5
−4
1
DDI
1D
©  ¤ Â–Z—™˜-š› œ
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ÖF‘Ì~=CFA}6GHMAÍ[email protected]|EªIJE+}|ÈX}‡DÍ$ƒ„EžÊzÈX<[email protected]|B>=CIHGJ‚z?AE+IHGJFA{[email protected]|EÑÊA=Cƒ„=CKCÍ~FAEӌ^Ë }6?AG,ÅULFD e
1

 ∂W + A ∂W = 0, (x, t) ∈ D × R∗ , D ⊂ R
+
∂t
∂x

W (x, 0) = W0 (x), x ∈ D
0
º Œm L ÅCE$Ì
?)F ÅCE$Ì_D[email protected] Ì$=Cƒ„<a=C}LFD6E$}MAE´‚X?ALFDGHD6{$}Ì~=CFA}|[email protected]|Å{$E~}$¿‰=
Å[email protected]~}vº : M D {~D×L1D R¿ Ω →{_D LΩFDÐ?AF=C?)Å[email protected]|DÐB>[email protected]{ MAE R ¿ A E$}‡D´?AF)E‘ƒÐLUD@|GJÌ~E Ì [email protected]|@6{~E‘M&LUFA}
M (R)
,­=C?)@VÌ$E´<[email protected]=CB)IJÍ$ƒ„E„IJE„}6Ì ÊA{~ƒ´L À Î <[email protected]|{$}6E~FD{-<&[email protected] »®º Ä®E$F)ÉXÊALIJMA=?AF M&LF)} Ä®E$F&’C®E_D
M&LFA}IHE+<[email protected]~ƒÐ[email protected](Ì Ê&L<)GHD@|E+MAEªÌ$E_D6DEªDÊ)Í$}6Ež} {$Ì[email protected],DMAEªIÃLvƒ´LUFAGJÍ[email protected]Ö}|?AGHÅULFDE e
+
m
m



W n
n
αi+
1
1
n
n
n
n
2
A
W
−
W
W
+
W
−
1 =
i+1
i
i+1
i
i+ 2
n
2
2Si+
1
2 

W n+1 = W n − rA W n 1 − W n 1
i
i
i+
i−
=
2
2
n
Si+
max (|λp |) = ρ (A) ,
1 =
L ÅCE$Ì ρ(A) IJEž@L ÈC=F;}6<aE$Ì_D@LUI­MAE A ¿ [λ ] º }|=CFDÖIJE~}fÅ[email protected]|}<)@6=C<[email protected]|E$}ÖM)EVIÃLӃÐLUD@|GJÌ$E
E$}‡Df?AF‘<[email protected]Lƒ„Í[email protected]+MAEªÌ~=CFD@VSIJE
A E_D α
Ô¬Ev}|ÈX}|D6Í$ƒ„E  º Œ^ E$}‡DžÊzÈX<[email protected]|B>=CIHGJ‚z?AEUº ¿³GJI­} E$FA}|?AGHDª‚z?AEÓIÃL ƒÐ[email protected]Ì~E A E$}‡D€M)GÃLKC=F&LIJGH}LB)IJE
M&LFA}(?)FAE+B&L}|E+MAE$}(ÅCE~Ì[email protected]|}f<)@6=C<[email protected]|E$} +F-F)=DE λ ≤ λ ≤ ... ≤ λ }6E~}Å[email protected]|} <[email protected]|=C<[email protected]~}
L}6}|=Ì~GJ{~E$} L?ÇÓÅCE~Ì[email protected]|}®<[email protected]|=C<[email protected]~}
¿zE~D =CF„FA=D6E
IÃL+B&L}|EMAE
[email protected]ƒ„{$Eª<&[email protected]$}(ÅCE~Ì[email protected]|}Ñ<[email protected]|=C<[email protected]~}{rM)E , ...,A º rX=C}GHD R = [r , ...,Br=]{rIÃLv,ƒÐ...,LUD[email protected]|GJÌ$}E+M)Ež<AL}6}6LKCEžRMAE
IÃL-B&L}|EÐÌ$LFA=CF)GJ‚z?AEÓMAE R IÃL BAL}6E B ¿¬GJI } E~FA}6?AG,DV‚z?AE„IÃL-ƒÐLUD@|GJÌ~E A = RΛR ¿¬= Λ E~}|DÖIJLЃÐLUD@|GJÌ~E€MAE~}Å[email protected]|}Ñ<[email protected]|=C<[email protected]~}ÖE~D R E~}|D+IJLЃÐLUD@|GJÌ$EªGHFzÅ[email protected]|}6EMAEVIÃLӃ´[email protected]Ì$EžMAE
<&L}|}LKCE º
Ô {_D L<aEª<[email protected]{~MAGJÌ_DE$?)@f} {~Ì[email protected]|GHD
p=1,...,m
2
p p=1,...,m
n
i+ 21
1
1
2
m
1
m
1
m
m
−1
n
Wi+
1
2
ÖF‘<a=C}6E
V = R−1 W
n
αi+
1
1
n
n
n
=
Wi+1 + Wi − n 2 A Wi+1
− Win .
2
2Si+ 1
¿&I {_D L<aEÖ<[email protected]|{$MAGHÌ[email protected]MAE_ÅGHE$FD
2
n
Vi+
1
2
n
αi+
1
1
n
n
n
=
Vi+1 + Vi − n 2 Λ Vi+1
− Vin
2
2Si+ 1
QR
.
2
m
m
−1
: D
D
E~DfI {~DL<aE+Ì[email protected]|@6E$Ì_D[email protected]M)E~ÅXGJE~FzD
Vin+1
=
Vin
n
n
− rΛ Vi+ 1 − Vi− 1 .
Ô ~{ Ì[email protected],D[email protected]|EªE$}|DfMA=FA̪{$‚z?AGHÅULIHE$FDEžL? }|Ì ÊA{$ƒÐL„M)EªÅ=CIJ?)ƒÐE~}2AFAGJ}*)À* Î }|?AGHÅULFD e
2
2

n
αi+
1

1

n
n
n
n
n
2
V 1 =
Λ
V
−
V
,
V
+
V
−
i+1
i
i+1
i
n
i+ 2
2
2Si+
1
2


V n+1 = V n − rΛ V n 1 − V n 1 ,
i
i
i+
i−
L<A<AIHGJ‚z?A{ªL? }‡È}‡DÍ~ƒÐE e
2
2

 ∂V + Λ ∂V = 0, (x, t) ∈ D × R∗ , D ⊂ R,
+
∂t
∂x

V (x, 0) = V0 (x), x ∈ D.
Á Ev}|ÈX}|D6Í$ƒ„EÓE$}|Dª{~‚z?AGHÅULIHE$FD m ${ ‚z?&LUD6GJ=CFA}ÑGJFAMA{~<>E~FAM&LFD6E$}
‚z?AE€IHE+<[email protected]=BAIJÍ~ƒÐEÖ}|E€@6Lƒ„Í$FAE e ∀p ∈ {1, ..., m} e
m
Å[email protected]|GÃLB)IJE$}~¿>Ì E~}|D [email protected]|E

 ∂vp + λp ∂vp = 0, (x, t) ∈ D × R∗ , D ⊂ R,
+
∂t
∂x

vp (x, 0) = (vp )0 (x), x ∈ D,
E~DfIJE+}|Ì ÊA{$ƒÐL)À Î } {$Ì[email protected]|GHDf<[email protected](Ì$E~}<[email protected]=CB)IJÍ$ƒ„E$}
1D
}|Ì [email protected]|E¿ ∀p

n
αi+
1

1
n
n
n

(vp )i+ 1 =
(vp )i+1 + (vp )i − n 2 λp (vp )ni+1 − (vp )ni ,
2
2
2Si+ 1
2


(vp )n+1 = (vp )n − rλp (vp )n 1 − (vp )n 1 .
i−
i
i
i+
hw2au qckmq) AK )% /1#z\+*J7 \ X7_#J ] /1b,c0/1b)\Ö\_#X01b& 7 \ N
ρ(A)
∀(i, n) ∈ Z × N $ ∀p ∈ {1, ..., m} 7_*Z5$#&7 λ
1) α
≥
|λ |
2
n
i+ 21
2
p
p
6= 0
$
7+\^] 2&% „Ó\$J\ ~c‰*J7('a!cbd]_`'&7;#A„JC0v#±*J/^]6*
2) r ≤ ρ(A)−1 mtXs'>t
ÖF L ∀p
min (vp )ni ≤ min (vp )n+1
≤ max (vp )n+1
≤ max (vp )ni .
i
i
i∈Z
i∈Z
i∈Z
i∈Z
(vp )n+1
= (vp )ni
i
r
λp
−
2
+
"
"
n
(vp )ni+1 + (vp )i −
n
αi+
1
2
n
Si+
1
λp (vp )ni+1 − (vp )i
2
n
αi−
1
r
λp (vp )ni + (vp )ni−1 − n 2 λp (vp )ni − (vp )ni−1
2
Si− 1
>’
2
n
#
#
.
DDI
1D
D ƙF L<)<AIJGH‚X?ALFDIJE+ƒ<;$ƒ„[email protected]LGH}6=CFAF)E$ƒ„E$FD‚z?AEªM&LFA}IJE+Ì$L}}6Ì$LIÃ[email protected]¿A=CF {$Ì[email protected]
n
n
(vp )n+1
= βi−1
(vp )ni−1 + βin (vp )ni + βi+1
(vp )ni+1 ,
i
L ÅCE$Ì
!
n
αi−
1
r
2
= λp 1 + n λp ,
2
Si− 1
2
!
n
n
α
α
1
1
r
i−
i+
2
+ n 2 λp 2 ,
βin = 1 −
n
2 Si−
Si+ 1
1
2
2
!
n
αi+ 1
r
n
2
βi+1
= λp
λp − 1 .
n
2
Si+
1
n
βi−1
ÖF‘<a=C}6E δ = λ º
1) ƙFRL<A<AIHGJ‚z?&LFDªIJEvƒ<;$ƒ„EÓ[email protected]|KC?Aƒ„E$FDª‚z?AEÓM&LFA}ªIJEvÌ$L}ª}6Ì$LIÃ[email protected]¿[email protected] β
≥ 0 E~D
S
¿ ∀(i, n) ∈ Z × N ¿ ∀p ∈ {1, ..., m} º
β ≥ 0 }|=C?A}IÃLvÌ~=CFAMAG,DGJ=F-}6? „}LUFzD6E α
≥
2
h
n
i+ 21
i
p
n
αi+
1
2
n
Si+
1
2
p
n
i−1
2)
n
i+ 21
Ë UL <[email protected]Í~} 1) ¿>=CF MA{$MA?)GHDւz?AE
βin ≥ 0
E$}‡D}LUD6GJ} LGHD6EÑ}6G rλ
p
2
|λp |
1
αn
1
i+ 2
Sn 1
i+ 2
≤
αn
1
i− 2
Sn 1
i− 2
+
≤ αn
2
1
i+ 2
Sn 1
i+ 2
αn
+
‚z?AG¬E$}‡DÅ[email protected]&{ª}6=C?)}fIJLvÌ$=CFAM)GHDGH=CF-}6? Ð}6LFDE
n
i+1
n
i+ 21
i− 1
2
Sn 1
i− 2
|λp |
∀p ∈ {1, . . . , m}
2
;dE_DÖIÃL„Ì$=CF)MAGHD6GJ=CF
¿
∀(i, n) ∈ Z × N ∀p ∈ {1, ..., m}
rρ(A) ≤ 1
¿)Ì~E
º
1 † 9A† E 7:9<CJADC = 2 4 7:9 ‰$C [email protected] ;D5 2 9<EHC E † [email protected]?$=?$C¬ +C
ËÑ E ƒÐLFAGHÍ[email protected]´LF&LIH=CKC?AE I {_D?AMAE<LGHD6E´M&LUFA}IJE„º Ì LU}}6Ì$LIÃ[email protected]¿­=CFp}6E <[email protected]|=C<a=C}6E M)E´[email protected][email protected]
I {~DLUD GJFD[email protected]ƒ„{[email protected]|E (v ) ∀p ∈ {1, . . . , m} ,­=C?)@®Ì$E~IÃL žÌ ÊAL‚z?AEÖGJF)}|D LUFzD =CFÐGJƒ„<>=}6E
}[email protected] v IÃLvÌ$=CF)MAGHD6GJ=CF-}|? „}LFDEž}6?)GHÅULFDE e
(v )
∈ (v ) ⊥(v ) , (v ) >(v )
∀ (i, n) ∈ Z × N E~D ∀p ∈ {1, ..., m}.
0 º /Œ . Ö@
1
n
p i+ 1
2
n
i+ 21
n
p i+ 1
2
n
p i+1
n
p i
(vp )ni+ 1 =
n
p i+1
n
p i
1
1
n
n
1 + [δi+
(vp )ni +
1 − [δi+
(vp )ni+1 ,
1 ]p
1 ]p
2
2
2
2
<>[email protected](‚z?AE€IJLÌ$=CF)MAGHD6GJ=CF  º Œ . ®}6=CG,D@6E~ƒÐ<AIHGJEU¿)GJI>}6? ÓDM L^Å[email protected]
h
i
E~D 1 − hδ i
1+ δ
≥0
>Œ
2
n
i+ 12
p
n
i+ 12
p
≥ 0,
: D
D
GJI³} E~FA}6?)GHDf‚z?AE
n
[δi+
1 ]p ≤ 1
∀p ∈ {1, . . . , m}, ∀(i, n) ∈ Z × N.
» GJFALIJE~ƒÐE~FzD }6=C?A}IJLÓÌ~=CFAMAG,DGH=CF 0 º /Œ . !¿)=CF D@|=C?)ÅE
2

n
[δi+
1 ]p ≤ 1.
∀p ∈ {1, . . . , m}
2
ºŒ > Ô¬E$}+MAE~?Ç9GJF){$KLIHGHD6{$} α ≥ 3Ì$=CFAM)GHDGH=CF;MAEv}‡D LB)GJIJG,D{žMAEvIÃLÐ<[email protected]=C<a=C}|GHDGH=CF º . º Œ E_D
|λ |
S
GJƒ„<AIJGH‚z?AE$FD ∀(i, n) ∈ Z × N E_D ∀p ∈ {1, ..., m} α = S º ÖFÅC=CG,D
α
≤
MA=CFȀ‚z?AE|λIHE|<[email protected]Lƒ„Í[email protected]MAEfÌ[email protected] α M){$<aE$FAM M)E$} Å[email protected]|} <)@6=C<[email protected]|E$}~¿M =|λÐ|IJEÌ ÊA=CG DŽMAE
IJE+<[email protected]|E$FAM)@6Ež}6=C?A}IJL [email protected]ƒ„EÖM ?AFAEžƒ´[email protected]Ì$EÖMAGJLKC=CFALIJE
n
Si+
1
2
n
i+ 21
n
i+ 21
p
n
i+ 21
n
i+ 12
p
n
i+ 12

n
αi+
1
2
Ö@
V = R−1 W





=





Sn
i+ 1
2
|λ1 |
0
Sn
i+ 1
2
ºº
0
0
ººº
0
...
|λ2 |
0
ºº
...
Sn
0
...
0
1
i+ 2
|λm |
n
i+ 21
p






.





º ƙF‘@|E$ƒ„<AIÃL½$LFDM&LFA}I {_D L<aE+<[email protected]{~MAGJÌ_D[email protected]$¿&=CF =CBDGJE~FD e
n
Wi+
=
1
2
1
1
n
n
−1
n
Wi+1
+ Win − n Rαi+
Wi+1
− Win
1 ΛR
2
2
2Si+ 1
2
=
1
1
n
−1
n
n
Wi+1
+ Win − n Rαi+
A Wi+1
− Win .
1R
2
2
2Si+ 1
ÖF‘<a=C}6E
Q
» GJFALIJE~ƒÐE~FzD IJE+}6Ì Ê){$ƒÐL>)À* Î MAE_ÅXGJE$FD
2
n
i+ 12
n
−1
= Rαi+
.
1R
2

1
1
n
n
n

Wi+1
+ Win − n Qni+ 1 A Wi+1
− Win ,
Wi+
1 =
2
2
2
2Si+ 1
2 
W n+1 = W n − rA W n − W n
.
ƙF‘}|GJƒ„<AIJG8hLFD =CF =CB)D6GJE~FzDIHE+}6Ì ÊA{~ƒ´L)À* Î
i
i
i+ 21
i− 12

1
1
n
n
n

Wi+
− Win ,
Wi+1
+ Win − sgn (A) Wi+1
1 =
2
2
2

n+1
n
n
n
W
,
=W −r F W 1 −F W 1
i
i
i+ 2
>
i− 2
DDI
1D
D L ÅCE$Ì sgn (A) = R sgn (Λ) R E~D sgn (Λ) = diag λ º
|λ |
D tXopi 1r³sZt
,­=C?)@?AF <[email protected]|=CBAIJÍ~ƒÐE+IHGJFA{[email protected]|E F (W ) = AW IHEª}6Ì ÊA{~ƒ´L)À* Î }6Ež@Lƒ„Í$F)E"v?AF‘}6Ì ÊA{~ƒ´L
MA{$Ì~E$FD@|{Lƒ„=CFD(Ì~=CƒÐƒ„E+IJE+}|Ì ÊA{$ƒÐLÓMAEžÀf=XE¿# » »Á E_D*# » À=EÖE~F‘E dE_D e
k
−1
k
n
ΦV F F C Win , Wi+1
1
1
n
n
A Win + Wi+1
− sgn (A) A Wi+1
− Win
2
2
=
1
1
n
n
A Win + Wi+1
− |A| Wi+1
− Win
2
2
n
= ΦRoe Win , Wi+1
.
=
n
ΦSRN HS Win , Wi+1
Ë L?)D@|E<&[email protected]‡D-}6GÖ=F<)@6=1Ž|E_D6DE
ƒ´LUFAGJÍ[email protected]+}6?)GHÅULFDE W − W
n
i+1
2
=
n
n
Win + Wi+1
− sgn (A) Wi+1
− Win
n
= ΦV F F C Win , Wi+1
.
n
Wi+1
− Win
p=m
X
n
=
γ p rp
i
n
Wi+
= Win +
1
Win
1
A
2
=
AM LFA} IJL B&L}6EM)E$} ÅCE~Ì[email protected]|}‘<[email protected]|=C<[email protected]|E$}-MAEIJL
=CF‘[email protected]L
p=1
1
1
n
n
− Win
Wi+1
− Win − sgn (A) Wi+1
2
2
p=m
1X
γp (1 − sgn(λp )) rp
+
2 p=1
= Win +
X
γ p rp
λp <0
n
= WV F Roe 0, Win , Wi+1
,
n
0, Win , Wi+1
9
p = 1, . . . , m
= W
~E }|DZIJL }6=CIH?)DGH=CFÖMA?ª<[email protected]|=CBAIJÍ~ƒÐE­MAE ÀfGHE$ƒÐLFAF³¿ γ = l · W − W
E~D l >< [email protected]
IH$E }ÖÅCE~Ì[email protected]|}V<[email protected]|=C<[email protected]~} -KLU?AÌ ÊAE„MAE A ÅC{[email protected] l · r = δ
M = ‘<[email protected](Ì~EªÌ L}fIJGHFA{ [email protected]
V F Roe
t
p
p
t
p
t
p
n
i+1
q
pq
n
n
n
n
n
ΦSRN HS Win , Wi+1
= AWi+
= ΦV F Roe Win , Wi+1
.
1 = AWV F Roe 0, Wi , Wi+1
2
1 Q † 5 462 4 $?$4DG¬† ?&9<C
ÖF‘Ì~=CFA}6GHMAÍ[email protected]|EªIJE+}|ÈX}‡DÍ$ƒ„EžÊzÈX<[email protected]|B>=CIHGJ‚z?AE+FA=F‘IHGJFA{[email protected]|EÑÊA=Cƒ„=CKCÍ~FAEvŒ^Ëj}6?AG,ÅULFD e
1


 ∂W + ∂F (W ) = 0, (x, t) ∈ D × R∗ , D ⊂ R
+
∂t
∂x

W (x, 0) = W (x), x ∈ D
0
>.
0
º Œm n
i
: D
D
L ÅCE$Ì W : D × R → Ω ?)F ÅCE$Ì_D[email protected] m Ì$=Cƒ„<a=C}LFD6E$}MAE´‚X?ALFD GHD6{$}Ì~=CFA}|[email protected]|Å{$E~}$¿‰=
Å[email protected]~}®M {~DLUD ¿XÌ$=ƒÐƒ„EÑIJLVƒ´LU}6}6EU¿)IJL€‚z?&LFDG,D{ÖM)EփÐ=?)ÅCE~ƒÐE~FzD E~D(I {[email protected]|KCGJEM&LFA} IHEÑÌ$L}
MAE$}­<[email protected]=BAIJÍ~ƒÐE~}­MAE IÃLƒÐ{~Ì LF)GJ‚z?AE MAE$}?&?AGJM)E$º }$¿1E~DyIÃLÊ&L?)[email protected]~DyIÃL‚z?&LFDG,D{ MAE®º ƒ„=C?)ÅE$ƒ„E$FD
M&LFA}IJE™Ì$L}MA?€<)@6=CBAIHÍ$ƒ„E™MAE)LGHFzD (#‰E~F&LFD Ω {~D LUFzD­?AFž=C?)Å[email protected]|D­[email protected]{‰MAE R ÔZL 3=CFAÌ_DGH=CF º
F MA{ &FAGHEÖMAE Ω M&LF)} R @6E$<)@6{$}|E$FDEžIÃL 3=CFAÌ~D6GJ=CF?&?Ç-‚z?AE+I =F }6?A<)<>=C}|EªL}6}|E [email protected]|{$KC?AIHGJÍ[email protected]
Ô¬Ež}|Ì ÊA{$ƒÐL )À Î <>=C?)@(Ì$Eª<[email protected]=BAIJÍ~ƒÐE+} {~Ì[email protected],Df}6=C?A}IJ9L [email protected]ƒ„E+}6?AG,ÅULFDE e
+
m
m



W n
n
αi+
1
1
n
n
n
n
2
W
+
W
F
W
−
F
(W
)
,
−
1 =
i+1
i
i+1
i
i+ 2
n
2
2Si+
1
2 

W n+1 = W n − r F W n 1 − F W n 1
,
i
i
i−
i+
L ÅCE$Ì
2
n
Si+
max
1 =
2
max
λni ,p , λni+1 ,p
= [λ ]
}6=CFD+IHE$}ÑÅ[email protected]|}Ñ <[email protected]|=C<[email protected]~}+MAEIJL´ƒ´L1D@6GHÌ$E Ž6LÌ$=BAGJE~ FAFAEMAE F ¿ λ @|E$}6<aE$Ì DG,ÅCE$ƒ„E$FD λ ÓÌ$LIJÌ~?AIJ{~E$}RI {_D LUD W [email protected]~}6<aE$Ì~D6GHÅE$ƒ„E$FDE I {~D L1D W vE~D α ?AF
<&[email protected]ƒÐÍ_D@|EžM)E€Ì~=CFD@VSIJE º ÖF }6?A<)<>=C}|EV‚z? GJI¬E!Ç)GH}|D6E€?AF {~DLUDGJFD[email protected]ƒ„{[email protected]|E V̄ W , W DE~I¬‚X?)E
F W
− F (W ) = A V̄ W , W
W −W .
ÔZLӃÐ[email protected]Ì~E+MAEž Àf=XE À[email protected])Œ Z‚z?&LFAM E$IHIJEÖE!Ç)GH}|D6EªÅ{[email protected]|G &EžE_Ç)LÌ_DE$ƒ„E$FDfÌ$E_D6D6E€Ì~=CFAMAG,DGJ=F º
ËÖLFA}fÌ$EªÌ$L}I {~D LU<>E+<[email protected]|{$MAGHÌ[email protected]MAE_ÅGHE$FD e
2
p=1,...,m
p p=1,...,m
n
i+1 ,p
n
i
n
i+1
n
i ,p
n
i
n
i+1
n
Wi+
1
2
n
i
n
i
n
i+1
n
i+1
n
i+ 21
n
i+1
n
i
n
αi+
1
1
n
n
n
n
Wi+1 + Wi − n 2 A V̄ Win , Wi+1
=
Wi+1
− Win .
2
2Si+ 1
Á =Cƒ„ƒÐE IHE }|ÈX}‡DÍ$ƒ„E 0 º Œm aE$}|DZÊÈX<[email protected]=CIHGJ‚z?AE¿$GJI} E~FA}6?)GHDZ‚z?AE‰IÃL ƒÐLUD@|GJÌ~E A V̄ W , W E$}‡DvMAGÃLUKC=CF&LIHGJ}6LBAIJE º ÖF FA=D6E λ¯ ≤ λ¯ ≤ ... ≤ λ¯ IJE~}VÅ[email protected]}€<[email protected]=C<)@6E$}L}6}|=Ì~GJ{$E~}vL?Ç
ÅCE~Ì~D[email protected]º } <[email protected]=<[email protected]~} {r̄ , ..., r̄ } ¿CE_D =FÐFA=UDE B̄ = {r̄ , ..., r̄ } IÃLÖBAL}6EMAE~} ÅE$Ì[email protected]|}®<[email protected]|=C<[email protected]|E$}
MAE R =CG,D R̄ = [r̄ , ..., r̄ ] IÃL‘ƒ´[email protected]Ì$E„MAE-<&L}|}LKCE´MAE IÃL‘B&L}6E´Ì LFA=FAGJ ‚z?AE M)E R IÃL B&L}6E [email protected]ƒ„{$E„<&[email protected]€IJE~}€ÅCE~Ì[email protected]|}„<)@6=C<[email protected]|E$}M)E
¿­GHI } E~FA}6?AG,Dv‚z?AE IJL
ƒ´L1D@6GHÌ$E B̄A V̄ W , W = R̄ Λ̄ R̄ ¿ A= V̄Λ̄ W ,E$W}‡D-IÃL ƒ´[email protected]Ì$E MAE$} Å[email protected]|}
<[email protected]=<[email protected]~}+E~D R̄ E~}|DªIJL´ƒ´L1D@6GHÌ$EžGJFÅ[email protected]}|E„M)EIÃL´ƒÐ[email protected]Ì~EVMAEV<&L}6}6LKCE º G =CF Ì$=CFA}|GJMAÍ[email protected]
Ì$=Cƒ„<[email protected]|D6E$ƒ„E$FD º MA?´}|ÈX}|D6Í$ƒ„E+IJ=XÌ LIHE$ƒ„E$FD®E~FD@6EÑIJE$} ƒ´LUGJIJIHE$} x E_D x ¿XI {~DL<>E<[email protected]{~MAGJÌ_D[email protected]
MA? }|ÈX}|D6Í$ƒ„E  Œ^ ®MA=CFAF)E e
2
n
i
1
1
m
2
m
1
m
1
n
i
m
m
m
n
i+1
i+ 12
n
i
−1
i+ 12
i+ 12
−1
i+ 12
n
i+1
i+ 21
i
ÖF‘<a=C}6E
n
R̄−1
Wi+
1
i+ 12
2
i+1
n
αi+
1
1 −1
n
n
n
n
= R̄i+ 1 Wi+1 + Wi − n 2 Λ̄i+ 1 R̄−1
W
−
W
.
1
i+1
i
i+ 2
2
2
2
2Si+ 1
n
−1
−1
n
n
V̄i+
Wi+
Win
1 = R̄
1 , V̄i = R̄
i+ 1
i+ 1
2
2
n
V̄i+
1
2
2
n
i+1
2
E_D
2
n
n
V̄i+1
= R̄−1
Wi+1
i+ 1
=CF L?)@L
2
n
αi+
1
1
n
n
n
=
V̄i+1 + V̄i − n 2 Λ̄i+ 1 V̄i+1
− V̄in .
2
2
2Si+ 1
>Q>
2
DDI
1D
D ™Æ F D6E$F&LUFzD Ì$=Cƒ„<)DEÑMAE+I LF&LI,ÈX}6EÑM)? }|Ì ÊA{$ƒÐL )À* Î M&LUFA} IJEÑÌ L} IHGJFA{[email protected]|EfÅE$Ì_D[email protected]|GJE$I¿)M&LUFA}
IJE+Ì$L}= λ 6= 0 ∀p ∈ {1, . . . , m} =CF <aE$?)Df{~Ì[email protected]@6E+MAEªƒÐLFAGHÍ[email protected]ªLUF&LIJ=KC?AE e
p
n
Q̄ V̄ Win , Wi+1
L ÅCE$Ì

n
αi+
1
2
−1
n
= R̄i+ 1 αi+
,
1 R̄
i+ 1
2
2
Sn
i+ 1
2
|λ¯1 |





=





ºº
0
0
ºº
...
0
0
ººº
0
...
0
Sn 1
i+ 2
|λ¯2 |
...
Sn
0
ËÖLFA}fÌ$EªÌ$L}IJE+}6Ì Ê){$ƒÐL>)À* Î MAE_ÅXGJE$FD
1
i+ 2
|λ¯m |
2






.






1
1
n
n
n
n

− F (Win ) ,
Wi+1
+ Win − n Q̄ V̄ Win , Wi+1
F Wi+1
Wi+
1 =
2
2
2Si+ 1
2 
W n+1 = W n − r F W n 1 − F W n 1
.
i
i
i+
i−
ƙF‘}|GJƒ„<AIJG8hLFD =CF‘[email protected]
2
2

1
1
n
n
n
n

Wi+
Wi+1
+ Win − sgn A V̄ Win , Wi+1
Wi+1
− Win ,
1 =
2
2
2

W n+1 = W n − r F W n 1 − F W n 1
.
i
i
i+
i−
» GJFALIJE~ƒÐE~FzD IJE+}6Ì Ê){$ƒÐL>)À* Î ~E }|Df?AFAE+Å[email protected]|GÃLFDE+M)?‘}|Ì ÊA{$ƒÐL>À Î ‚z?AG¬} {~Ì[email protected]|GHD e
2
2

−1
1
1
n
n
n

Wi+
Wi+1
+ Win − A V̄ Win , Wi+1
1 =
2
2
2

W n+1 = W n − r F W n 1 − F W n 1
,
L ÅCE$Ì
i
i
A V̄
n
Win , Wi+1
i+ 2
−1
n
F (Wi+1
) − F (Win ) ,
i− 2
= R̄i+ 1
Λ̄−1
i+ 21
R̄−1
i+ 12
E~D
Λ̄−1
i+ 21
= diag
1
.
|λp |
D tXopi 1r³sZt %
ËÖLFA}ÐIJE BA?)D„MAE‘‚z?&LIJG8&[email protected]Ì~E‘}|Ì ÊA{$ƒÐLpFA=?A}´LUIJIJ=FA}vIJE Ì$=ƒÐ<&[email protected][email protected]„L ÅCE~Ì MAE~}Ð}|Ì ÊA{$ƒÐL}ÐMAE
ÅC=CIH?Aƒ„E$}AFAGJ}„E„Ì LUÌ$E$}-<[email protected]„IÃL9}6GHƒ?AIJLUDGH=CF FX?)»ƒÐ»{~Á@6GJ‚z?AE‘MAE$Á }´}|ÈX}|D6Í$ƒ„E$} ÊÈX<>[email protected]=CIJGH‚z?AE$}~¿
}L Å[email protected];}|Ì ÊA{$ƒÐL ?&?Ç Ì[email protected]LÌ_D{[email protected]}‡DGH‚X?)E #
ÏžÒªÔ RQ=)¿(Ä =C? [email protected]¿ M)=CFD IJED?&?Ç
Fz?AƒÐ{[email protected]‚X?)Ež} {~Ì[email protected]|GHDf}6=C?)}fIJ9L [email protected]ƒ„E+ƒÐLUD@|GJÌ$GHE$IHIJEf}6?AG,ÅLUFzD6E e
g V F F C (V, W ) =
2
1
1
(F (V ) + F (W )) − sgn(A(Ū )) (F (W ) − F (V ))
2
2
>
: D
D
&B L}|{€}|[email protected]IHE$}Ì[email protected]$}fÌ [email protected]Ì~D6{[email protected]|GJ}|D6GJ‚z?AE~}
3{[email protected]|E$FDGHE$IJIHE$}(}|?AGHÅULFDE~} e
‚z?AGZ}6=CFDMAE$}f}6=IJ?)D6GJ=CFA}(M)E$}Ñ{~‚z?&LUDGH=CFA}MAG8B
Xk (t)
dXk
= λk (V (Xk , t)).
dt
¬Ô E‘}|Ì ÊA{$ƒÐL # » »Á L9?AFAE [email protected]LF)MAE‘@|E$}|}6E$ƒVBAIÃLF)Ì$E‘L^ÅE$̑IJE }6Ì ÊA{~ƒ´L MAE Àf=XE M&LFA}„IJE Ì$L}
MAE$}€}|ÈX}|D6Í$ƒ„E$}ÊÈX<[email protected]=CIHGJ‚z?AE$}žÊA=Cƒ„=CKCÍ~FAE$}~¿ZMA=CFDVIHE?&?ÇRFz?AƒÐ{[email protected]‚X?)EÐ<aE$?)D€} {$Ì[email protected]@6E„MAE´IJL
ƒ´LUFAGJÍ[email protected]+}6?)GHÅULFDE e
g Roe (V, W ) =
1
1
(F (V ) + F (W )) − sgn ARoe (F (W ) − F (V ))
2
2
E$F Ì Ê&LFAKE LFDIÃLƒ„=mÈE$FAF)E€MAEžÀ=E+<[email protected](?AFAEžƒÐ=^ÈE$FAFAEž[email protected]|GHD6ÊAƒÐ{_DGH‚X?)E Ū = 1 (V + W ) º
2
ÖF FA=D6E€{~KLIHE$ƒ„E$FDIJEª}6Ì Ê){$ƒÐL# » À=XE ¹ » ÏERQR¿>Ï Î )’ 0¿hMA=FzDIJE?&?XÇ FX?)ƒÐ{[email protected]‚z?AEžE$}‡D
MA=CFAF){ž<[email protected]
g
(V, W ) = F (W
(0; V, W )) ,
= W
(0; V, W ) MA{~}6GHKCFAE+IÃLž}6=CIH?)DGH=CF MA?-<)@6=CBAIHÍ$ƒ„EÑMAEÖÀfGHE$ƒÐLFAF´IHGJFA{[email protected]}6{}6?AG,ÅLUFzD e
V F Roe
V F Roe
V F Roe

∂W
∂W


+ A(Ū )
= 0,


 ∂t
 ∂x
V, x < 0,



W
(x,
0)
=


W, x > 0,
E$F }6?A<)<>=C}6LFDтz?AEªIJLÓ}|=CIJ?DGJ=F-MA?‘<)@6=CBAIHÍ$ƒ„EÖFA=CF IJGHFA{ [email protected]E$}‡DÌ~=CFAFz?AE"„I GHFA}|DLFD t º
Ô¬Ež}|Ì ÊA{$ƒÐL´Lv{_D{ª{$Ì[email protected]<[email protected](IHE$}<[email protected]=BAIJÍ~ƒÐE~}(ÊzÈX<[email protected]|B>=CIHGJ΂z?AE~}fº Ì~=CFA}|[email protected]|ÅULUD6G3}~¿&E~Df<&[email protected]IÃLv}|?AGHD6E
Lv{~D{ž<[email protected]|=CIJ=CF)KC{+LÎ ?Ç-<[email protected]=CB)IJÍ$ƒ„E$}(F)=CF‘Ì~=CFA}|[email protected]|»ÅULUD6G3} Ä(Ï ’C’
Ô¬E+}6Ì ÊA{~ƒ´L )*À @|E~ D@|=C?)ÅE+IJE+}6Ì Ê){$ƒÐL # À=EÖ<[email protected]®IJE$}(}‡ÈX}|DÍ~ƒÐE~}ÊzÈX<[email protected]|B>=CIHGJ‚z?AE~}(FA=CF
ÊA=Cƒ„=CKCÍ~FAE$}E$F !E >E~DfI {_D L<aE+<[email protected]{~MAGJÌ_D[email protected]} {$Ì[email protected]|GHD
n
m
n
Wi+
1
2
=
Win
1X
+
γ¯p 1 − sgn(λ̄p ) r̄p
2 p=1
= Win +
X
γ¯pr̄p
λ̄p <0
n
= WV F Roe 0; Win , Wi+1
,
M&LFA}¬IHE™Ì L}¬=+I {~D L1D¬ƒÐ=^ÈE$F V̄ W , W = 1 W + W , E~D γ¯ = ¯l · W − W ¿
<[email protected](<AIH?A}MAEªMA{~DLGJIH}®Å[email protected](<&[email protected]@6L<AÊAE 0 º . º > 2 º
,­=C?)@„?)FAE-IJ=CG MAE-Ì$=FA}[email protected]|ÅULUDGH=CF }6Ì$LIÃ[email protected]Ð}[email protected]Ó?AF ƒÐLGJIHIÃLKCEÓ@|{[email protected]$¿IJE-}6Ì Ê){$ƒÐL À Î Ì$ =Cƒ„ƒ„E IHE$}„}6Ì ÊA{~ƒ´LU<} # » »Á E~<D # » À=XE }|=CFD´[email protected]Í$}„<[email protected]|=Ì ÊAE~} º ƙFE!>E~D$¿ }|G=CF Ì~=CFA}6GHMAÍ[email protected]|E
I {$‚z?&LUD6GJ=CF MAEªÌ~=CFA}[email protected]|ÅULUD6GJ=CF }6Ì$LIÃ[email protected] e
n
i
n
i+1
n
i
∂v ∂f (v)
+
=0
∂t
∂x
>=
n
i+1
p
t
p
n
i+1
n
i
DDI
1D
D = v ∈ R E~D f ?AFAE3=CFAÌ~D6GJ=CFž@6{~KC?AIHGJÍ[email protected]|E M)E R M&LFA} R ¿1IJE~}­E!Ç<[email protected]~}6}6GH=CFA}™MAE~} ?&?XÇFz?Aƒ„{[email protected]|GJ‚z?AE$}
}6=CFDfMA=CF)FA{$E~}<&[email protected] e

}6G f (û) > 0

f (v)


}6G f (û) < 0
g
(v, w) = f (w)




}|G f (û) = 0
f (ū)

}|G f (û) > 0
f (v)





}|G f (û) < 0
g (v, w) = f (w)




}|G f (û) = 0
 1 (f (v) + f (w))
2

}|G f (ū) > 0
f (v)




}6G f (ū) < 0
g
(v, w) = f (w)




}|G f (ū) = 0
(f (v) + f (w))
0
SRN HS
0
0
0
0
Roe
0
0
V FFC
0
1
2
=


 f (w) − f (v)
0
w−v
f (û) =

f 0 (v)
}6G
0
}|G
w 6= v
w=v
E~D
1
ū = (v + w).
2
D tXopi 1r³sZt A
Î
g
(v, w) = g
(v, w) M&LFA}ÑIJEVÌ$L}Ö=C?I {_D LUDуÐ=^ÈE$F û MA?}|Ì ÊA{$ƒÐL )À* E~}|D
º
Î
<[email protected]}­{~KLI ū Ô¬E }|Ì ÊA{$ƒÐL)À* VLƒ„=CFD@|{ ?)FAE®@|=CBA?)D6E$}6}|E E~D­?AFAE®[email protected]LF)MAE E „Ì Lº­Ì~¹GHD{ <>[email protected]
IÃL }6GHƒ?AIÃL1DGJ=F;Fz?AƒÐ{[email protected]‚X?)EÐMAE„Ì[email protected]|DLGJFA}ª<[email protected]|=C»yBA»­IHÍ$Á ƒ„E$}ªÊÈ» <[email protected]|B>=IJGJ‚z?AE~}€Ê)=CƒÐ=KCÍ$FAE~} LGH}ªMAE
ƒ´LUFAGJÍ[email protected]™LFALIJ=CK?AE™L?Ç+}6Ì ÊA{~ƒ´L}ZMAE À=EU¿ # E~D# Àf=XE ‚z?AG}|E BAL}6E~FzD­E$}|}6E~FzD6GJE~IJIJE~ƒÐE~FD
}[email protected] LU<A<[email protected]= ÇGJƒÐLUD6GJ=CF MA? <[email protected]|=CBAIJÍ~ƒÐE‘MAEÀGJE~ƒ´LF)F IJ=XÌ LUI<&[email protected] ?AF<A @|=CBAIJÍ~ƒÐE‘IJGHFA{ [email protected]¿ Ì~E
}6Ì ÊA{~ƒ´L‘<>E~?)DVM)[email protected]€MAE~}V}|=CIJ?)D6GJ=CF)}+FA=CFp<)ÊzÈX}|GJ‚z?AE$}VE~D€ƒÐE~ FAE$"@ I LU<A<& [email protected]|GHDGH=CFMAE„Ì ÊA=XÌ$}
}|DLUDGH=CFAF&[email protected]~}$¿[email protected]|} ‚z?AE+IÃL€}6=CIH?)DGH=CF <AÊÈX}6GH‚X?)Ež‚z?AGaÅC{[email protected] AE+I GJFA{~KLIJG,D{M E$FD@|=C<AGHE+E$}|D?)FAE
=CFAMAE´MAE MA{_DE$FD6Eº ¿ Ì~E$IÃL‘FA=C?A}=BAIJGHKC<E [email protected]|E´?AF)E Ì[email protected]|@6E~Ì~DGH=CF [email protected]=C<)GJ‚z?AE-L?pFAGHÅE L? MAE~}
<>=GJFD}(}|=CFAGJ‚z?AE~}
SRN HS
V F Roe
1 Š 2IA6? =d† [email protected]?$2 4 CK4 7:92 D? ;DC
ËÖLFA}ªI {_D L<aE<[email protected]|{$MAGHÌ~D[email protected]ªMA?9}6Ì ÊA{~ƒ´L )À* Î >¿³=CF L-E$}|}L ÈC{„MAED@|=C?)Å[email protected]ª?AFAEÓBa=CFAFAEÓ L<
<[email protected]= ÇGJƒÐLUD6GJ=CF MAE IJL }6=CIH?)DGH=CFRL<A<[email protected]|=Ì ÊA{~E-MA? <[email protected]|C= BAIJÍ~ƒÐE„MAE-ÀGJE~ƒ´LUFAFpIHGJFA{[email protected]}6{„<&[email protected] E!Ç7
<[email protected]~}6}|GJ=CF e
1
n
Wi+
1 =
2
1
1
n
n
n
Wi+1
+ Win − sgn A V̄ Win , Wi+1
Wi+1
− Win .
2
2
>P
: D
D
Ö F<)@6=1Ž|E_D6DE W − W M&LFA}vIJL B&L}|E M)E$}ÅCE~Ì[email protected]|}Ð<[email protected]|=C<[email protected]|E$}
LÌ$=BAGJE~FAFAE A Ì$LIJÌ~?AIJ{~EªLvI {~D L1DƒÐ=^ÈCE~F³¿&=CF‘[email protected] e
n
i+1
Ö@
L ÅCE$Ì
n
i
n
Wi+1
− Win =
n
A V̄ Win , Wi+1
n
sgn Λ̄ V̄ Win , Wi+1
sgn Ā V̄
Ö@
p=1
n
= R̄ sgn Λ̄ V̄ Win , Wi+1
t¯
lq r̄p

λ̄1
 |λ̄1 |


 0

n
sgn Λ̄ V̄ Win , Wi+1
=




0
λ̄k = 0
n
Win , Wi+1
= δpq
¿A=CF <a=C}6E
n
Wi+1
sgn(λ̄k ) = 0
−
Win
ºº
0
...
λ̄2
|λ̄2 |
...
0
ººº
...
0
¿AGJI³} E~FA}6?)GHDf‚z?AE
R̄−1 ,

ºº
0 




,

0 


λ̄
m
|λ̄m |
= R̄ sgn Λ̄ V̄
n
Win , Wi+1
= R̄ sgn Λ̄ V̄
n
Win , Wi+1
=? ¯l ~E }|DIJE+‚ GHÍ$ƒ„EÑÅCE~Ì[email protected]Ì$=CIH=CFAFAE+MAEªIJLƒ´[email protected]Ì$E
q
n
sgn Ā V̄ Win , Wi+1
MAEª<AIJ?)}f=F‘L e
p=1
R̄−1
p=1
1
1
n
n
Wi+1
+ Win = Win +
Wi+1
− Win
2
2
m
= Win +
Ë L ?)D@|Eª<&[email protected]|D
1X
γ¯pr̄p .
2 p=1
1
1
n
n
n
Wi+1
+ Win = Wi+1
−
Wi+1
− Win
2
2
m
=
n
Wi+1
>@
1X
−
γ¯pr̄p .
2 p=1
R̄
−1
m
X
¿&M =
m
X
n
Wi+1
− Win =
γ¯p sgn(λ̄p )r̄p ,
MAE-IJL
γ¯pr̄p .
E$}|DfMA=FAFA{$Eª<&[email protected]
M&LFA}IHE+Ì L}=
m
X
B̄ = {r̄1 , ..., r̄m }
m
X
γ¯pr̄p
p=1
γ¯p R̄−1 r̄p .
DDI
1D
D Ë = m
n
Wi+
= Win +
1
2
1X
γ¯p 1 − sgn(λ̄p ) r̄p
2 p=1
m
=
ÖF‘}|?A<A<a=C}6Ež‚z? GHI³E_ÇGH}|DE
n
Wi+1
1X
γ¯p 1 + sgn λ̄p r̄p .
−
2 p=1
j ∈ {1, . . . , m}
DE~I¬‚z?AE
λjl < 0 < λjr
= λ E$}‡DÑIJLŽ GJÍ~ƒÐEÅLUIJE$?)@(<[email protected]=C<)@6EžMAEªIÃLŽ6LÌ$=BAGJE~FAFAEª Ì LIHÌ$?AIH{$EªLÓI {~DLUD W E~D λ E$}‡DÑIJL
Ž GHÍ$΃„EÖÅLUIJE$?)Î @f<)@6=C<[email protected]|E€M)E€IJL֎6ÎÑLÌ~Î =CBAGHE$FAFAEžÌÎ LUIJÌ$?)IJ{$EžLÓÎ I {~DLUD W ΏΠ¿aMALFA}fÌ$EVÌ$L}f} GJFA}|<[email protected]
MAE [email protected]|D6E$F³¿ ÈXƒÐLF´E_D ÔZL1Ç Ô4P =aE~D [email protected]|D6E$FÐE~D ÈXƒ´LF @ .>=F´@6E~ƒÐ<)IÃLÌ~E λ <&[email protected]
λ̂ E_D λ <[email protected] λ̂ L ÅCE$Ì
jl
n
i+1
n
i
jr
jl
jr
jl
jr
λ̂jl = λjl
λjr − λ̄j
λjr − λjl
λ̂jr = λjr
λ̄j − λjl
.
λjr − λjl
E~D
ƙFA}6?AG,DEª=CF I GJF1Ž|E~Ì~D6EÖM&LFA}fI E_Ç<[email protected]|E$}|}6GJ=F‘MAE
n
Wi+
1
2
E~D
=
Win
2
MAEªIÃL9L½~=CF }6?AG,ÅLUFzD6E e
m
1 1 X
γ¯p 1 − sgn(λ̄p ) r̄p + γ¯j 1 − sgn(λ̂jl ) r̄j
+
2 p=1,p6=j
2
n
n
Wi+
1 = Wi+1 −
2
n
Wi+
1
m
1 X
1 γ¯p 1 + sgn(λ̄p ) r̄p − γ¯j 1 + sgn(λ̂jr ) r̄j .
2 p=1,p6=j
2
0
º Œ= 0
º ŒP ÖF LM)MAGHD6GJ=CFAF)E 0 º Œ= ®E_D 0 º ŒP =F L?)@L
n
Wi+
1
2
m
1 X
1
1 n
n
=
Wi + Wi+1 −
γ¯p sgn(λ̄p )r̄p − γ¯j sgn(λ̂jl ) + sgn(λ̂jr ) r̄j .
2
2 p=1,p6=j
4
=C?)}LUIJIJ=FA}®DE~}|[email protected]+}6Ì Ê){$ƒÐL>)À* Î > ¿AD=?)DM LB>[email protected] }[email protected]MAE~}<)@6=CBAIHÍ$ƒ„E$}(ÊA=ƒÐ=CKÍ$FAE~} º
>R
: D
D
1 [email protected]; 7)† [email protected] 4 ; EHG9? ;DCK5
65­x)7Xoptt B i&q k -žt i k
Ô¬ E$}9{$‚z?&LUD6GJ=CF)};MAEC)LGHFzD (#‰E~F&LFDƒ„=CFA=XMAGJƒ„E$FA}|GJ=CF)FAE$IHIJE$} ƒ„=M){$IJGH}LFD I {$Ì$=?AIJE~ƒÐE~FzD;MAE
I E L? M&LUFA}?AF‘Ì$LF&LI¬} {~Ì[email protected]|GHÅCE~FD e
1

∂h
∂


 ∂t + ∂x (hu) = 0, x ∈ [−10, 10]
h2
∂
∂

2

 (hu) +
hu + g
=0
∂t
∂x
2
= h $E }‡DfIÃLvÊ&L?)[email protected]MAEªI E$L?³¿ uº E$}|DfIJLVÅG,DE~}6}6EžM {$Ì~=C?AIJE~ƒÐE~FDMAEªI E L? E~D
I LÌ$Ì~{$IH{[email protected]L1DGJ=F-MAEªIÃL<aE$}[email protected]
ÔZLӃÐ[email protected]Ì~E LÌ$=BAGJE~FAFAE+E$}‡DÑM)=CFAFA{~Ež<[email protected]

0
A(W ) = 
= c = pgh º
Ô¬EžM)E$?Ç Å[email protected]|}f<)@6=C<[email protected]|E$}fMAE A }|=CFD
<&[email protected]

R(W ) = 

1

c2 − u2 2u
u−c
E~D
u+c
1
¿&IÃLvƒÐLUD@|GJÌ$E+M)Ež<AL}6}6LKCEžE$}|DMA=CF)FA{$E

1

u−c u+c
E~DfI GJFÅ[email protected]}|EªMAEªIJE+ƒÐLUD@|GJÌ$EÑMAEª<&L}6}6LKCEžE$}‡D e
R−1 (W ) =

u+c
g = 9.81ms−2
1 
−u + c
2c
−1
1

.
ZÔ L€ƒÐLUD@|GJÌ$E}|GJKCFAEMA{ &FAGHEÑM&LF)} IHE}|Ì ÊA{$ƒÐL )À* Î >¿XE$}|D®E$FÐ<[email protected]Ì$GH<>EMA{ &FAGJ√Ef<&[email protected] IJE~} √{_D LUD6}
ƒÐ=^ÈE$FA}žMAEÓÀ=E À[email protected]Œ ™‚z?AG™}6=FzDžMA=FAFA{$E~}€<[email protected] h̃ = h + h E_D ũ = u h√+ u h º
Á E$<aE$FAMALFD´M&LUFA}vIJE~}DE$}‡D}ÓFz?Aƒ„{[email protected]‚z?AE$}v=CFL;}|GJƒ„<AIJE~ƒÐE~Fz2D<[email protected]}IJE~}v{~D L1D}vƒ„=mhÈE$hFA} h̄ =
u + u ¿d= h + h E_D
ū =
h E~D u }6=FzDª@|E$}|<>E~Ì~DG,ÅCE~ƒÐE~FzDªIJL´Ê&L?DE$?)@+E~D+IJLÐÅXGHDE~}6}|EvMAE
I E L2?Ð}6G x < 0 ¿ h E~D 2u }|=CFD @|E$}6<aE$Ì_DG,ÅCE$ƒ„E$FD®IÃLªÊ&L?)[email protected] E~D IÃL+ÅG,DE~}6}6EMAEÑI E$L?„}6G x > 0 º
,­=C?)@fÅ[email protected](IJEª}6Ì ÊA{~ƒ´>
L )*À Î E~DIÃLӃÐ=XMA8G &Ì LUD6GJ=CF´E$FD@|=C<AGJ‚z?AE€FA=C?A2} LGH}6=CFA}MAE~}fD6E$}‡D}
Fz?AƒÐ{[email protected]‚X?)E$}V}|[email protected])E$}VÌ$L}V= pGJIH}ªE_ÇGJ}‡Dº E$FDvM)E$}V<a=CGHFzD6}€}|=CFAGH‚X?)E$}VE_DVF)=C?A}VGHFD@6=XMA?AGH}6=CF)}€IJL
Ì[email protected]|@6E~Ì~DGH=CFE$FD@|=C<AGJ‚z?AE Ì$E FAGHÅE L? ,[email protected] Ì$E$IJL=F:<[email protected]|E$FAM Ì$=Cƒ„ƒÐEÌ~=CFAMAG,DGH=CF GJFAG,DGÃLUIJE
h = 20m ¿ h = 1m ¿ u = 0 E_D u = 0 t = 0.3s L ÅCE~Ì ?AFRƒ´LGHIJIJLKCEMAE 100 <a=CGJFD6}
&[email protected]~}
 º @)¿  º R ™E~D®?AF´Ì$L} D6E$}‡D(L ÅCE~%Ì 3=CFAMÐ}6E$Ì &[email protected]~}  º Œ^’)¿  º ŒCŒ ’
l
l
r
l
r
l r
l
l
r
r
l
r
l
r
l
r
l
r
r
DDI
1D
D º ƙFA}6?)GHDE®FA=C?A}‰LIHIJ=CF)}[email protected]~<[email protected][email protected]|E(‚X?)E$IJ‚z?AE~}‰E_ÇE$ƒ„<AIJE~} DE~}|D{~}‰<&[email protected]$& º Ä ?³[email protected]³¿ & º ϞLUIJIJ=?AE~D
E~D )º ¹ ºÎ {[email protected] [email protected] Ä(Ï Î [email protected]&LFA} ?AF€}6Ì ÊA{~ƒ´Lf}6GJƒ„<AIHE™<>=C?)@­IHE$}{$‚z?&LUD6GJ=CFA}MAº E+)LGHFzD (#‰E~F&LFD
<>[email protected] Å[email protected]I E„Ì LÌ~GHD6{+MA?‘}|Ì ÊA{$ƒÐLÓ<>=C?)@(IÃ[email protected] {~}6=CIH?)D GH=CF MAEªÌ$E~}f<)@6=CBAIHÍ$ƒ„E$}
#^'a#X7 X7f" !|@
X7 \_# /1,b ´/1#Xc*c*J%Öe GJIz} LKCG,DyM ?AFv<)@6=CBAIHÍ$ƒ„E M)E ÀGJE$ƒÐLF€MAE®Ì~=CFAMAG,DGJ=F
GJFAG,DGJLIJE~} e
(h, Q)(x < 0, t = 0) = (1, 0),
(h, Q)(x > 0, t = 0) = (0.3, 0)
FAEӌ! =CFAMAEªMAEªMA{_DE~FzD6E€}|?ABA}|=CFAGJ‚z?AEžE~D?)F;- Ì ÊA=X̞}|[email protected]ƒ„E$FD º
#^'a#X7 X7V" _|@
X7Ö\_#XE /1,b Ó\^7 ]ve Ô¬E~}fÌ~=CFAMAG,DGH=CFA} GHFAGHD6GÃLIHE$}®}|=CFDfƒ´LGHFDE$FALFD e
(h, Q)(x < 0, t = 0) = (1, 0),
(h, Q) (x > 0, t = 0) = (0, 0)
FAEӌ! =CFAMAEÑMAEªMA{~D6E$FDEª}6?)BA}6=CF)GJ‚z?AEªE~D?AF‘- Ì ÊA=XÌ+M LUƒÐ<AIHGHD6?AMAE[email protected]Í~}2LUGJBAIHE$}®}|[email protected]ƒ„E$FD º
ÖF Ì$=FA}|DLUDE€BAGJE~F ‚z?AEVIHE$}[email protected]{~}6?AI,D LUD6}ÑFz?Aƒ„{[email protected]‚z?AE$}f}[email protected]<AIJ?)}[email protected]}fÌ LU}DE~}|DÑ=CB)DE~Fz?A}Ñ<&[email protected]IHE
}6Ì ÊA{~ƒ´L)À* Î 3=C?6# » À=E Ì$=CGHFAÌ$GHMAE$F DL ÅCE~̀Ì~E$?Ç MA? }|Ì ÊA{$ƒÐLÐMAEžÀ=E º
ÖFRÌ~=CFA}|DLUDEÐL?A}|}6G‰‚z?AEÓIJEÓÌ LUIJÌ$?)I MAE„IÃL LÌ~=CBAGJE~FAFAE„E$F ?)DGHIJGH}LFD+IJL ƒ„=^ÈCE~FAFAEÐ[email protected],DÊAƒ„{ DGH‚X?)E€LU? IHGJE~?-MAEªIÃLvƒÐ=^ÈE$FAFAEªMAE€À=E+FAEªÌ Ê&LUFAKCEª<&L}IJE~}(@6{~}6?AI,D LUD6}fFz?Aƒ„{[email protected]|GJ‚z?AE~} º
Œ
: D
D
20
14
sans correction
avec correction
sans correction
avec correction
18
12
16
10
14
12
8
10
6
8
6
4
4
2
2
0
−10
−8
−6
−4
−2
0
2
4
6
A
B ' ? A9 PB
G /213 'I/213 P5 1*1 > ( ) ' "
8
0
−10
10
−8
−6
−4
5
−2
0
2
4
6
8
(6' "$"$ B
G /2143 '0/213 U5 11 > ( ) ' "
10
12
sans correction
avec correction
sans correction
avec correction
4.5
10
4
3.5
8
3
2.5
6
2
4
1.5
1
2
0.5
0
−10
−8
−6
−4
−2
0
2
4
6
K M ' P P
G /213 'I/213 P5 1*1 > ( ) ' "
8
10
0
−10
−8
−6
−4
−2
0
2
4
6
8
<P (@' N PB
G /213 0' /2143 5 11 > ( ) ' "
'
8
10
DDI
1D
D 65­x)7Xopt sx3tL
Ô¬E$} {$‚z?&LUD6GJ=CF) }-MAEIÃL M)ÈXF&LUƒÐGH‚X?)EMAE$}-KL<>=C?)@-?AF &? ?AGJM)E<&[email protected] LUGHD } {~Ì[email protected]|GHÅCE~FD‘E~F:?)FAE
MAGJƒ„E$F)}6GJ=F-M E~}6<&LUÌ$EV}|=C?A}IÃ[email protected]ƒ„E


 ∂W + ∂F (W ) = 0,
∂t
∂x

W (x, 0) = W (x).
0
x ∈ [0, 10],
­=C?)@[email protected]ƒ„[email protected](IHE+}|ÈX}|D6Í$ƒ„E¿h=CF‘LmŽ|=C?)D6E+I {$‚z?&LUD6GJ=CF M {~DLUDfMAE$}KL+<[email protected] LG,D}
,
1 2
p = (γ − 1) E − ρu .
2
Ô¬E$}(Å[email protected]~}(Ì$=CF)}[email protected]‡ÅULUDG,ÅCE$}fE~DIJL 3=CFAÌ~D6GJ=CF-M)?6?&?XÇ <AÊÈ}|GJ‚z?AEž}6=CFDfMA=CF)FA{$}f<&[email protected] e

ρ


E~D




W =  ρu  ,


E
ρu

2

F (W ) =  ρu + p

(E + p) u





= ρ @|E$<[email protected]|{$}|E$FDEÖIJL€MAE~FA}6G,D{MA? KLU¿ u E$}|D®IÃLªÅG,DE~}6}6EU¿ p E$}|D®IÃLž<[email protected]~}6}6GH=CF E_D E E$}‡D I {$F)[email protected]
D=DLIJE º γ E$}|D®[email protected]L<)<>[email protected]‡D®M)E$} Ì Ê&LUIJE$?)@6º }®}|<>{~Ì$G8&‚z?AE$} º I>E~}|D®}6?A<A<a=C}|{ÖÌ$=CF)}|D LUFzD E~D {~KLI 1.4
Ì$L}M ?)F‘KLž<[email protected] LG,D(MAGJLUD=Cƒ„GJ‚z?AE ÖF M)!{ &FAG,Dž{$KLIHE$ƒ„E$FD+I E~FzD6Ê&LIH<AGJEVD6=D LUIJE H <&[email protected] H = 1 (E + p) E_DžIÃL Ì${~IJ{[email protected]{vMA?9}6=CF
ρ
r
γp º
c=
ÔZL LUÌ$=CρBAGHE$FAF)EžM)E F E$}|DMA=CF)FA{$Eª<&[email protected]

ÔZLӃÐ[email protected]Ì~E
0
1
0

2
2

(3 − γ)u
γ−1
∇F (W ) =  (γ − 1)H − u − c

(γ − 2)uH − uc2 H − (γ − 1)u2 γu
∇F (W )
j
λ2 = u,
E~Df}6E~}fÅE$Ì_DE$?)@6}Ñ<)@6=C<[email protected]|E$}fL}|}6=XÌ$GJ{~}}6=CFDfMA=CFAF){$}f<&[email protected]
1


R1 (W ) =  u − c

H − uc


.

E~}|DMAGÃLUKC=CF&LIHGJ}6LBAIJEU¿z}6E$}Å[email protected]}(<)@6=C<[email protected]|E$} λ }6=CFDfMA=CFAF){$E$}f<&[email protected] e
λ1 = u − c,





,


1



 u 
R2 (W ) = 

 1 
2
u
2
.
λ3 = u + c
E_D

1


R3 (W ) =  u + c

H + uc



.

: D
D
¾Ñ<[email protected]|Í$}+L Å[email protected]E!>E$Ì~D6?A{IJL„Ì[email protected]@|E$Ì~D6GJ=CF E$FD@|=C<AGJ‚z?AE€GJFD@|=M)?AGHD6E€M&LUFA}ÑIHE€<&[email protected][email protected]<AÊAE0 º . º > _¿
Ì$E}|Ì ÊA{$ƒÐL <>[email protected]ƒ„E~D-IJLp}6GHƒ?AIÃL1DGJ=F Fz?Aƒ„{[email protected]‚z?AE;MAE‚z?AE~IJ‚z?AE$} Ì$L}-Ì[email protected]|GHDGH‚z?AE$} E~F <[email protected]~F&LFD
t = 0.005 E_DIJLvÌ$=CFAM)GHDGH=CF-MAEª}|DLBAGHIJGHD6{Ñ{~KLIJE 0.9 e
ƙF 3=CFAÌ_DGJ=F MAE~}ÑÌ$=FAMAGHD6GJ=CF)}GHFAGHD6GÃLIHE$} IHE€<[email protected]|=CBAIHÍ$ƒ„E€M ƙ[email protected]ÑLUMAƒÐE_D[email protected]=CGH}Ì ÊALƒÐ<)}ÑÌ [email protected]LÌ D{[email protected]}‡DGH‚X?)E$}$¿ IJE‘<[email protected]~ƒÐ[email protected]´E_D-IJE D@|=CGJ}|GJÍ$ƒ„E Ì Ê&Lƒ„<A} Ì [email protected]LÌ_D{[email protected]|GJ}‡DGJ‚z?AE~} }6=FzD´Å@6LGJƒ„E$FD´FA=CF
IJGHFA{ [email protected]~} · R 6= 0 ¿A<[email protected] j = 1, 3 E_D<>E~?)ÅCE~FzD ;[email protected]+}|=CGHD(M)E$}=CFAMAE~}MAEª@[email protected]|!{ LÌ_DGJ=F³¿
}6=CG,DVM)E$}V=∇λ
FAMAE$}€MAE„Ì ÊA=ÌU¿­<[email protected]€Ì~=CFD@|E´IJEÓMAE~?ÇGJÍ~ƒÐE„Ì Ê&Lƒ„<pÌ[email protected]LÌ_D{[email protected]}‡DGH‚X?)EÐE~}|DIHGJFA{[email protected]|E ƒÐE~FDvMA{$K{$FA{[email protected]{ ∇λ .R =º 0€E_DÌ[email protected]@|E$}6<a=CFAM ?)FAE MAGH}6Ì$=FzD6GJFz?AG,D{´M)E Ì$=CFDLÌ~Dv‚z?AG®}6E
<[email protected]=<&LKCEžL ÅCE~̀IJLVÅG,DE~}6}6E u
=D6=CFA} ‚z?AE9»<>[email protected] Ì$E9<[email protected]=BAIJÍ~ƒÐE;IHE$} º @6{~}6?AI,D LUD6}-Fz?AƒÐ{[email protected]‚X?)E$} MA? }6Ì ÊA{~ƒ´L )À Î E~D IHE
}6Ì ÊA{~ƒ´LÓMAE # Àf=XE+}6=CFDfIJE~}ƒ ;~ƒÐE~}
j
j
2
2
Ci&uvk^tXu~k^n
b,X7Ö\_#^'&7_ \^/1b&5$#&7#" |% ~X]_0/1b
Ô¬E+<[email protected]~ƒÐ[email protected] Ì Ê&LUƒÐ< Ì [email protected]LÌ_D{[email protected]|GJ}‡DGJ‚z?AE+E~}|D?AF)Ež=FAMAE+MAEª@[email protected]{!LUÌ~DGH=CF ‚z?AG¬Ì~=CFDGJE~FD?AF <>=CGHFD
}6=CF)GJ‚z?AE¿dÌ$E<[email protected]|=CBAIHº Í$ƒ„º?> EVº E$}‡DÖD@6LGHD6{ I LGHMAEžMAEIÃLÐÌ[email protected]|@6E$Ì_DGH=CF;[email protected]=C<AGH‚z?AEvGJFD@|=M)?AGHD6E€M&LUFA}
IJE+<&[email protected][email protected]<AÊAE  . {~D L1DKCL?AÌ ÊAE x ≤ 5 {~DLUD[email protected]=GHD x > 5
º
ρ =1
ρ = 0.01
l
r
ul = 0
pl = 105
ur = 0
pr = 103
ρl = 1
ul = 300
pl = 105
ρr = 1
ur = −300
pr = 105
=D6=CFA}‚z?AE´IJE„}6Ì ÊA{~ƒ´L # » »Á º <[email protected]žÌ$E´Ì L}~¿y} E$}‡D„L ÅC{[email protected]|{ FA=CF E~FD@6=<AGJ‚z?AEЃ;~ƒÐE´L<[email protected]Í~}
Ì[email protected]|@6E~Ì~DGH=CF E$FD@|=C<AGH‚X?)E E$KC’
,b X;7 X7v] 2&/^;] X/1#)"$*J7Ö\_#)"_\^/1b&5$#h7
ËÖLFA}fÌ$EªÌ$L}GJIdÈ LvMAE~?Ç Ì ÊA=XÌ$}‚z?AEž}6Eª<[email protected]=<&LKCE~FzDfMAEªIJLKL?AÌ ÊAEªÅ[email protected]}fIÃ[email protected]|=CGHD6E º
{~DLUDKL?AÌ Ê)E x ≤ 5 {_D [email protected]|=CGHD x > 5
>
DDI
1D
D 4
1
10
SRNHS
Rusanov
x 10
SRNHS
Rusanov
0.9
9
0.8
8
0.7
7
0.6
6
0.5
5
0.4
4
0.3
3
0.2
2
0.1
1
0
0
1
2
3
4 ( G /241 3
> () ' "
8 ?)4" (@' < '0/2143 1*15 5 111
4
5
6
7
8
9
0
1
2
3
4
5
6
7
8
M L
8 >P"" ( ) 4 ( G /21 3 '0/21 3 1145 5P1*11
> ( ) ' "
5
2.8
0
10
9
10
300
x 10
2.6
200
2.4
100
2.2
2
0
1.8
−100
1.6
1.4
−200
1.2
1
0
1
2
3
4
5
6
7
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i+ 21
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1
2
=
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uni
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i+ 1
2
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i+ 2
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f (uni )
+
−
−
= uni − r f (uni+ 1 ) − f (uni− 1 ) + ∆tsni .
αn
+
1
i+ 2
2S n
1
i+ 2
∆xŝn (xi , xi+1 , uni , uni+1 ),
‹$1Ui"51&T0'G(*'—„83|#%T0;'T0!8+8 A 51=Y"!;'/T;"51 @"A &
4L
!L9'/:!b0&! .5 A .e0L:"U.'/ α = S V
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i
2
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2
n
i
n
i+ 21
n
i+ 12
0
n
i+ 21
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2
n
i+ 12
1
1 n
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ŝn (xi , xi+1 , uni , uni+1 )
ui+1 + uni − sgn f 0 ani+ 1
uni+1 − uni + 0 n
2
2
2
2|f (ai+ 1 )|
2
4L
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2
2K
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f 0 ani+ 1 uni+1 − uni = f (uni+1 ) − f (uni ).
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2
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i
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n
n
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0
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0
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n
i
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xi+1
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i
n
n
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1
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0
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uni+1 − uni

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2
2
2

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+ ˛˛ „∆x «˛˛ ŝn xi , xi+1 , uni , uni+1
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xi , xi+1 , uni , uni+1
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Z
xi+1
xi
Z
tn +θ
u(x, t)
tn
dz
(x)
dx
1 uni+1 + uni
(zi+1 − zi ) ,
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2
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1 n
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2
1
f uni+ 1 − f uni− 1 =
uni− 1 + uni+ 1 uni+ 1 − uni− 1 .
2
2
2
2
2
2
2
:
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f
0
ani+ 1
2
=

1
1 n

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(uni+1 + zi+1 ) − (uni + zi )
2
2
2
2
r

n
n
n
n
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+
u
u
−
u
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u
1
i
i
i− 21
i+ 21
i− 21
2 i+ 2
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2
1 n
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2
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1
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2
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2
2
2
2
2
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ui+ 1 + uni− 1
uni+ 1 − uni− 1
2
2
2
2
2∆x
1 n
n
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1
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2
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i
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2
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i
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n
n
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ui+1 + ui (zi+1 − zi ) .
2
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2.6
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2.4
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2
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1
1
n
n
n
n
2
2
F
W
−
F
(W
)
+
∆xQni+ 1
W
+
W
−
1 =
i+1
i
i+1
i
n
n
i+ 2
2
2
2Si+
2Si+
1
1
2
2 

W n+1 = W n − r F W n 1 − F W n 1
+ ∆tQni ,
i
i
i−
i+
%E ZY ’"K
4L
"U–'‡9'/!&‡"&‰0~'/Z52.3/
S
[λ ]
= max |λ |, |λ
|
M/:_0 F P λ : λ 3"U=`'!'4T33Q::.—9":52:SJg>'G(‚4:. W : W P\
!15 A .3T0"U. "'/P Q :3'G(*3`+&5bq./70'N_V.T"1"51 @"A &
α
0')(*4\Y_4:0/\.:!0'/ 9"'/!&52_0^:"U.'/ [t , t + θ[×[x , x [ m*hi(™ -pY
. 3!Y"1RS!ƒ(*'–\+83.1!“4:.^/U.:5140//3
Pr';RS!B'N}52.3/D0
V̄ W , W
<=- 9"4:—ˆ^'N>:'N/"
2
n
i+ 21
p=1,...,m
n
ip
n
ip
n
i+ 21
n
i+1 p
n
i+1 p
2
1L
p p=1,...,m
n
i
n
i+ 21
n
n
i
n
i
n
i+1
i+1
n
i+1
n
n
n
F Wi+1
− F (Win ) = Ā V̄ Win , Wi+1
Wi+1
− Win .
{$(*4\Y^4:0/\.!&=0&:9-/:SZ'"
n
Wi+
1
2
n
n
αi+
αi+
1
1
1
n
n
n
n
n
n
2
=
Wi+1 − Wi + n 2 ∆xQni+ 1 .
Wi+1 + Wi − n Ā V̄ Wi , Wi+1
2
2
2Si+ 1
2Si+ 1
•~"5152_" B!&Q"34^R-!&f'_3„-3 A 51FE%ZY K~3=U„-Q:MY"'/RS!P'/"3T"cQ:!&=43/3
0^'N>52 A J!—9S
Ā V̄ W , W
2
n
i
2
n
i+1
n
Ā V̄ Win , Wi+1
= R̄i+ 1 Λ̄i+ 1 R̄−1
i+ 1
P R̄ \ R̄ "U–33Q::.—9":52:S–'NZ52/:;0‰9'!3x&"3:P'/Z52.3/
Λ̄
0 . @ :~'N]5b=0J. @ /U93_`'!'4ZgD'G(‚4:. V̄ W , W P&0ƒ(‚4LF!&
4:.Q^340:!RS!v(‚4:—IK
4L
i+ 21
i+ 21
−1
i+ 21
2
2
2
n
i
n
Wi+
1
2
n
i+1
n
n
αi+
αi+
1
1
1
n
n
−1
n
n
2
=
Wi+1 + Wi − n R̄i+ 1 Λ̄i+ 1 R̄i+ 1 Wi+1 − Wi + n 2 ∆xQni+ 1
2
2
2
2
2
2Si+ 1
2Si+ 1
2
’"n
2
E*9
n
R̄−1
Wi+
1
i+ 21
2
#%% %
n
n
αi+ 21
αi+ 21
1 −1
−1
n
n
n
n
= R̄i+ 1 Wi+1 + Wi − n Λ̄i+ 1 R̄i+ 1 Wi+1 − Wi + n ∆xR̄−1
Qn 1 .
i+ 12 i+ 2
2
2
2
2
2Si+ 1
2Si+ 1
2
2
'
%E ZY K
]'/B^02V„-3 A 52:D52 @A :]”c! "!89"^')(*,+&&3"”0!“5 A .3B0
"U"' α = S |Λ̄ | Y
. e351'N:_0fE%ZY 'K\P')(*4\Y_4:0/\.:!0:9-/:U
⇐⇒
n
i+ 21
n
i+ 21
n
Wi+
=
1
2
i+ 21
−1
1
1
n
n
n
Wi+1
+ Win − R̄i+ 1 |Λ̄i+ 1 |−1 Λ̄i+ 1 R̄−1
W
−
W
1
i+1
i
i+ 2
2
2
2
2
2
1
n
∆xR̄i+ 1 |Λ̄i+ 1 |−1 R̄−1
1 Qi+ 1 ,
i+
2
2
2
2
2
"5151
sgn Ā V̄ W , W
= R̄ |Λ̄ | Λ̄
:
|Ā V̄ W , W
| = R̄ |Λ̄ | R̄
'/T'/_345226&<ZHZI 6}Q!T'/_3"M' A 511E)ZY "K~(‚4:—
+
n
i
n
i+1
n
i
n
i+1
i+ 21
−1
i+ 21
i+ 21
−1
i+ 12
−1
−1
i+ 21 R̄i+ 1
2
−1
,
i+ 12

1
1

n
n
n
n
n
n
n

W
W
+
W
−
sgn
Ā
V̄
W
,
W
W
−
W
1 =

i+1
i
i
i+1
i+1
i
i+

2
2
2


n
+ 21 |Ā V̄ Win , Wi+1
|−1 ∆xQni+ 1

2




n+1
n
n
n
W i
= Wi − r F Wi+ 1 − F Wi− 1
+ ∆tQni .
E%ZY Œ"K
 k ‘ 
='N7&./RS!_" Q:!&=:03]')(*4\51C„":”m*˜^{‡•|n"’€p
2
#"
2
1
n
n
V̄ Win , Wi+1
Wi+1
+ Win .
=
2
.,),8 '(:),+ - & := 8 0;<' @+ &2 .,4
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8 (:+ + 52 ' ),4>)32*2*0 & .,)254
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p
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0J6&/UV?XW;UC9Z!1#d0234 @ !'/:–04Y&0U~0&Z'/J9NM'/T3./'/ x :0"A &4
z(x) P h(x, t) 3343UU~'N !8.!&0&Z')(*!B: ζ(x, t) = h(x, t) + z(x) '/^3!3#%
'/MP u(x, t) :3x'NT98—.:0'G(‚`!]: Q(x, t) = h(x, t)u(x, t) V‰'0&4M—`PC'G(‚4RS!q./]0&
6&SV?XW;:UiC9."Y @ T/34 @ !'/ A ~(‚430&:Z"!;'/_#d51T!—9S K

∂h
∂(hu)


(x, t) = 0
 (x, t) +
∂t
∂x

∂
1
∂z
∂(hu)


(x, t) +
(hu2 + gh2 )(x, t) = −gh(x, t) (x).
∂t
∂x
2
∂x
™
'
E%ZY n"K
%E
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2 *
*
h(x,t)
2
*9*
u(x,t)
fond de la rivière
z(x)
Profil du lit de la rivière
86 "— W (x, t) = h(x, t) ! !&F'/!&/"“3.."B0aE%ZY*nUK^(*:3V?›gq?X0/3 W V
3kT34 @ !' A i943ˆŽhu(x,
U t) (x, t) = 0 : u(x, t) = 0 P0–T–'/J0!-+& A 51|4RS!q./
0! 3„-3 A 51FE%ZY n"K~3=4:R-!&9'/:U.]g
∂W
∂t
1 ∂(gh2 )
∂z
(x, t) = −gh (x)
2 ∂x
∂x
⇐⇒
h(x, t) + z(x) = c.
Z‡ b #"%$&)(+*0DS*~-=8.a+B/Tb;Q#B/@/" ƒ*LDt; C−[email protected]@v":B/$eBL-/"
DS*L-=8.a+B/Tb;U;/PxPDF"9&)(+BOU(E2*L-=57DF(1$9":57 -/$:;#$":57+;#"%@H*[email protected]*/[email protected](E#*
*/$ u = 0 ∀(i, n) ∈ Z × N,
h + z = c,
jOkml_n6%9oprq57( -sDt;`8.57#"%$9":57uE*gDt;<#"S-=8/@HB/$9"S-);#$":57u#(`$e*/@vTU*w-=57([email protected]*Kx;#ZD?^yB/$N;/P+*[email protected]/@H*.8/$J*/([email protected]~-=57(x-0D?^ (12*iE*.-0E*/(#z[email protected]*.--/(E"%7;#+$J*.-i€
‡
g i) s = −
h
+h
(z − z )
4∆x
‡
g
ii) s = −
h + 2h + h
(z − z )
DS*L-=8.a+B/Tb;Q8∆x
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jOkE‘p’“
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VZ4:WRS!(x,
9'UZg
n
i
n
i
n
i
n
i
i
n
i+ 12
n
i− 12
i+1
n
i+1
n
i
n
i−1
i−1
i+1
i−1
∂F
(W (x, t)) = Q (x, t) ,
∂x
!
!
h(x, t)
0
F (W (x, t)) =
1
0
gh2 (x, t)
2
`9" W (x, t) =
P
{vT9'/:!3":"U=0"&4=
λ¯1 = −c̄
\ '8
λ¯2 = c̄
: Q (x, t) =
0
∂z
−gh(x, t) ∂x
(x)
!
.
E*9
#%% %
:='N752/:_0^3. @ R :='N752/: 0^3. @ ]U9"3
!
1 1
\ R = 1 c̄
R=
−1
−c̄ c̄
2c̄
`9" c̄ = g h + h Y
šx"!&=:^`='G(‚4:.2QJ4:0/\.!&=:(*4:3
n
i
2
n
i+1
n

Wi+
1 =
:='G(‚4:.Y_"3\.:!
"
`9"

2
hn+1
i
qin+1
#
=
"
hni
qin
− 2c̄
#
−
hni + hni+1
2
hni − hni+1 −

R−1
−1
c̄
zi+1 −zi 2
0
1
3"U=0"4:=
!
,


E)1YC™"K

rg 
 + ∆tQni
2
2
n
n
2
(hi+ 1 ) − (hi− 1 )
2
Qni =
"
0
#
2
.
~• "5152–'/|"'!&." :33.."/P'/"3 W = W \ h −h = − (z
/'ƒ:(*:!&=RS!f'/70!-+& A 51^"51Q.U._0&bE%ZY—C™UK~V=4RS!9'U.]g
sni
n+1
i
n
i
n
i+1
n
i−1
i+1
− zi−1 )
g n
hi+ 1 + hni− 1
hni+ 1 − hni− 1
2
2
2
2
2∆x
g n
hi+ 1 + hni− 1 hni+1 − hni−1
=
2
2
4∆x
g
n
n
h 1 + hi− 1 (zi+1 − zi−1 )
= −
2
4∆x i+ 2
sni =
0ƒ(‚4Le'/7:52 A 3_0/34\../"Y
P/'r(‚3!=RS!
. 
1
h
=
h +h
n
i+ 21
2
n
i
n
i+1
1
1 n
hi+ 1 + hni− 1 =
hni+1 + 2hni + hni−1 ,
2
2
2
4
0ƒ(‚4Le'/70!-+& A 51^0/34\../"Y
]
9' 7 418Ž ' &(?2541= =?8 0;<'@=
$Zˆ 0 RS!'/—ˆ>'}4:5b 6&<=HJIJ6 '|VbS43Ub0& :"51:>.O#d"5D!'N/"
`9"f!e!&.3^4:5bBQ:3#d"35bS0&4:9":'/"&Q4]Q!'N757!'/."FS!514RS!^0=q?
M' A 52:i&"b52 @A :)Y-‹$b\+8'".U'G(‚651`:4=0&!b&452D04:U.34 51"U0Z „-Y
[email protected]; ' 2*' )#=?+ - & =:8 0;<'
'4-
P
%E
* %% *
2 *
*
2
*9*
<=-PƒW8?›{v:P_YtYY—PvY"!^'N}57!'/."LS!514RS!B0]M' 52:]U„-YMY"'/RS!^q?
52 @"A :FEdM' A 52:]0&D514R-!&]0]Š!/0&K^h~:57j0&kbA \fW lkRS!:kLm‚h| |W_nŒP
hiW_noPUWZ•|n"npY"U~&"Y"4=!Z0/334:../"B04::S4:_0Z'/]3/"2"51 @"A &=\
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!^:!'/^4\Y^0^'N7#% K
Win+1 = Win −
∆t Roe
Roe
Fi+ 1 − Fi−
+ ∆tQni
1
2
2
∆x
V'N #d"\./>0!1Š!8+BS!514RS!=g^')(*3.U t !B—9"`!20')(*SV#% i + 1
2
0"&4]
4L
Roe
Fi+
1
n
2
Roe
Fi+
1 =
1
1
n
Fi+1
+ Fin − Ai+ 1
2
2
2
n
Wi+1
− Win
V|'/f52.3/0J<=-J\ A = R |Λ |R P Λ
'N}52.3/D0_9'/:!^3"3P R '/}5b70B. @ B: R 'G(‚/U9":B0B'/
5bq.=0J P8RS!r04:Q:0}0Z0!8+b4:..9"" : Y8{vZ.—.51Ui0!
.:51]3"!3^@ Q"34^(‚43=0^'ND#% "c!&9U.! K W W
2
4L
Ai+ 1 = Ri+ 1 Λi+ 1 R−1
i+ 1
2
2
2
2
i+ 21
Qni =
`9"
:
n
i
`9"
2
2
Y
&43U.>!&
∆t ∆t V az
n
n
Φ (Win , Wi+1
, Qni+ 1 ) − ΦV az (Wi−1
, Win , Qni− 1 ) +
Qni+ 1 + Qni− 1
2
2
2
2
∆x
2
1 n
F + Fin − Ai+ 1
2
2 i+1
{v]345226&<ZHZI 6}(‚4:=3"!'N7#d"352 3!9U
2
`9"
2
i
h
1
)
· Qni+ 1
QR
=
I
−
sgn(A
i+ 2
i+ 21
2
1
n
n
n
Qi+ 1 = Q xi+ 1 , Wi + Wi+1
2 2
2
1
i+ 2
n
ΦV az (Win , Wi+1
, Qni− 1 ) =
Win+1 = Win −
i+ 12
i
h
QLi− 1 = I + sgn(Ai− 1 ) · Qni− 1
I
Win+1 = Win −
−1
i+ 12
1 L
,
Qi− 1 + QR
1
i+ 2
2
2
V_'N15bq.^0ƒ(‚/0:U.4D\
3`+&5bq./F0!c.352^3"!:]gB')(*U.V#%
‹$3. @ `U=\]4:3.!&Pe!.
4L
i+ 21
i+ 12
−1
i+ 21
n
i+1
i+ 21
n
Wi+1
− Win − ∆x sgn(Ai+ 1 )Qni+ 1
2
2
∆t SRN HS n
n
n
Φ
Wi , Wi+1
, Qni+ 1 − ΦSRN HS Wi−1
, Win , Qni− 1 +∆t˜Qni
2
2
∆x
n
n
ΦSRN HS Win , Wi+1
, Qni+ 1 = F Wi+
1

'
2
2
E*9
#%% %
`9"
:
n
Wi+
1 =
2
1
1
1
n
n
Win + Wi+1
− sgn Ani+ 1 Wi+1
− Win − |Ai+ 1 |−1 ∆xQni+ 1
2
2
2
2
2
2
˜Qni
1
=
∆x∆t
Z
xi+ 1
2
Z
tn+1
Q(x, t)dxdt
VZ!&D`+8/52/"F0&]'/7V./_&"e&"52 @"A _0'G(‚4:.Q_:"3\.!&)Y
1'NL34:—.!3c0! 3452 68<ZHZI 6 :1'/}3452“0& W lkRS!:ke\bhi5Dj0kP;"
:5b3RS!bRS!ƒ(‚/'$„ e!&b0—[r4:&b0D'/ 52 A 310&2.^'/B.:5123"!3b\7'/
#d"\./ 0&! Š!-+cS!524:R-!&P'/:T0!-+ 4:5bB^"U=0"&]4RS!9'U.:)Y
!(:41- =:),+ - & =?8 0;<' &- + 2254B01.,47 0
}'L`}0ƒ(*!& V„-3 52OU„-YMY"'/RS!L'/4/3"51 Pi'/e4:5b 6&<=HJIJ6 3
.5 A &B!a4:5b}'/A 3/RS!>04::S4251"U`Pr::$"&ˆ@"51A 7M/:aR-!&1:>4:5bcV
0ƒ(‚"03a! .:52c\c 3[Y=šx"!}0&c`c52 @A :}—.R-!&}34:UU
0bQ"U.1"RS!2L3452”e"U9": @ 293F'N“"'!&." :U./RS!Pi0
/'=V}4:3.O0ƒ(‚/U.380&!/3!"3\." U"RS!OQ!b'/'/B"M&' A 51#Y
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0'ƒƒ(‚VZ/0RS!4_0:]0_9""!&'/"T51'/:|\.]0S/S!—.4 =!&
"0e0 04:U.P~52/>'|„ a! &"M' A 51cR-!&=Y33.e!1'Na'/!&/" S!514RS!
! —9"`! 0L'N“0/3"U/S!40! #d0 Ed9"/1ˆ @ !3 TZY -o : TZY -’UKvY 'J
4:4f::./]0_#%_'G(‚\+-3/" 0! 4:5b26&<=HJI 6cgB'G(‚"03_0!-+ fQ"!&
043^\.>0& 51!'—.4[Yƒ &_.37`Pƒ' '/51.:!3"U^/U.380!&.J!O—9"`!O0
'G(‚4:.Q>&40&/:!^Y"! '_9/M'J&S„-3/RS!FE%"a. V '/79:!]0:^93NM&'/
U„-/RS!:J0^:"52Y"U. (v , . . . , v )K|0^'ND#%:"e3!9U K
t
2
tn
m
1
n
n
n
(vp )+
i = (vp )i + φ(θp ) (vp )i+1 − (vp )i
2
:
`9"
0'_`4L
1
xi− 1
1
n
n
n
(vp )−
i = (vp )i − φ(θp ) (vp )i+1 − (vp )i
2
θp =
λni+ 1 > 0
(vp )ni − (vp )ni−1
(vp )ni+1 − (vp )ni
\
2


0


φ(θp ) = θp




1
3
' o
(vp )ni+1 6= (vp )ni
θp ≤ 0
0 ≤ θp ≤ 1
θp ≥ 1.
%E
* %% *
2 *
*
2
*9*
‹$!—.}" 4:37'D93NM&'/f"&V9.—9"B: #d":/"”0793NM'fU„83/RS!:1\
0:f'G(‚4:Q_340:!0\98UCK
n
Wi+
=
1
2
+
4L
Ai+ 1
1
1
−
−
Wi+1
+ Wi+ − sgn Ai+ 1 Wi+1
− Wi+
2
2
2
∆x
Ai+ 1
2
2
−1
−
Qni+ 1 (xi , xi+1 , Wi+1
, Wi+ )
2
3='/J3-M/:_`'/!&'/4^>')(*4\0]<8_: #d:/"c0
2
01=:& . (*'(:= >
&;<025),%!&41=
−
Vi+1
: V Y
+
i
jOk g 9n x‘p=Ml k  0 2‘
f2`:Px"“.:3.2'NcQ:3#d"35bB0! 3452O6&<=HJIJ6“3!D! 4:"!'/:52:U7gc9€?
/."a.0#Y‡•J(‚V7!“.:3f"Y"342D{vXW;RS!m*{vXW n"ŒpGP:!—.B.:3.42f68:,+
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4.5
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3.5
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4.5
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4
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8
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6
3.5
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 ∂W (x, t) ∂F (W (x, t))
+
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∂t
∂x
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αl ρl u2l
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,
√
√
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√
√
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ūl =
,
√
√
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ρ̄v =
p̄ γ1
c
;
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* q
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!t$'"
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≤ max un+1
≤ max uni .
i
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Ci ∈T
Ci ∈T
Ci ∈T
Ci ∈T
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= uni −
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j∈Ni
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a⊥b = min (a, b)
Ci ∈T
Ci ∈T
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sup {|f10 (u)| , |f20 (u)|} = M ∈ R,
u∈X
X
(C+3uAO|Klm"7-;$'>N>@(–C+Mn-63u
X = {u ∈ R+ /|u| ≤ ku0 kL∞ (R2 ) }.
š "ˆC+MUiU7$.C.(•AO(?UEGfM C•gBM7(•GV( UWD'C•AO(•3d(+>#U7C
u
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Ai
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∀i , n ∈ N
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∀(i, n) ∈ N2
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||un ||L∞ (R2 ) ≤ ||u0 ||L∞ (R2 ) .
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j∈Ni
X
|eij |g (uni , uni , ηij ) =
j∈Ni
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X
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v2
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2Ai j∈N
C1
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uni ⊥unj , uni >unj
Nsv2 <(,"
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νijn
nij ∈ uni ⊥unj , uni >unj
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ij
j
i
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1−
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uni
2Ai j∈N
i
+
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7
τijn
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~ n · ηij un .
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ij
j
2Ai j∈N
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n
n
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2Ai
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X
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τijn .
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τin ≤
j∈Ni
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j∈Ni
((
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Ai
n
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j∈Ni
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S |∂Ci |.
Ai i
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N
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i=2
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7
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αij
1 n
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n
n

u
u
(f (unj ) − f (uni ))vijn · ηij
=
+
u
−

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i
j
n

2
2Sij



n X
 n+1
∆t
|eij |g uni , unj , Sijn , ηij
ui = uni −
Ai j∈N

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i
Z


1
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0
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W (x, y, 0)dxdy,
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i
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1 2
! "
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1
=
|eij |
vijn
n
v(x, tn )dσ = vji
eij
3
3
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g
?- 1 n
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2
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n
n
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1
n
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2
Sij
n
n
αij
1
n
0
n
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1 + n f aij vij · ηij unj − uni .
2
Sij
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1
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2
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g uni , unj , Sijn , ηij
B ,
X
j∈Ni
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(uni )
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i
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Z
eij
v(x, tn ) · ηij dσ
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Ci
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=
uni
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n
∆tn X
n
1−
|eij | δijn − 1 f 0 θij
vij · ηij
2Ai j∈N
i
,
n
∆tn X
n
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n
1
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n
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2
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θij
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n
n
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0
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a
ij vij .ηij ,
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δijn =
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n
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n
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vij · ηij unj .
2Ai j∈N
i
τijn =
n
∆tn
n
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vij · ηij ,
2Ai
!
uni
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3 (
=
1−
X
τijn ≥ 0
!
τijn
!
uni +
j∈Ni
n
n
f 0 θij
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X
τijn unj .
X
et
X
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j∈Ni
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' ∀(i, n) ∈ N2 , ∀j ∈ Ni , ∃ γijn ≥ 1
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f 0 anij vijn · ηij
δijn − sgn f 0 (u)vijn · ηij ≥ 0
⇐⇒
δijn ≥ 1
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δijn ≥ 1,
Sijn
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n
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2Ai
≤
n
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sup |f 0 (θij
γin = sup γijn .
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uj + uni −
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j
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ij
j
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f 0 unij vijn · ηij
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(1 + γsij ) uni + (1 − γsij ) unj
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n
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j∈Ni
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j∈Ni
P # # ∂H
(sij + γ) f 0 unij vijn · ηij
X
<
|eij | (γ + sij ) f 0 (unij )vijn · ηij ≤ (γ + 1) ψin |∂Ci |.
P O( M , ,)( D#
)( D# /" / 2 3 O( ∆t ≤ 2A
j∈Ni
n
uni , u{j∈Ni } ≥ 0.
∂uni
1 - 1
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1
i
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ij
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i
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= uni −
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i
unij =
∆tn X
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Ai j∈N
i
g uni , unj , Sijn , ηij
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n
αij
1 n
ui + unj − n F (unj ) − F (uni ) · ηij .
2
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!
1 2
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,
B
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V
f10 (anij )
f20 (bnij )
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F (unj ) − F (uni ) · ηij = V
j
i
ij
!
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anij
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n
αij
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~ n · ηij un − un
ui + unj − n V
j
i
ij
2
2Sij
n
αij
1
n
n
~ · ηij un − un .
= ui +
1− nV
ij
j
i
2
Sij
unij =
,
’
δijn =
n
αij
~ n · ηij
V
Sijn ij
3 &
"
? (R D! > " ? -J
1
1 − δijn unj − uni .
2 F (un ) η θn n ∈ un⊥un , un>un ij ij
ij
ij
i
ij
i
ij
n
n
n
n
g ui , uj , Sij , ηij = F uij · ηij
1
n ~n
n
n
n
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,
=
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2
!
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f20 (nij )
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X
|eij |g uni , uni , Sijn , ηij =
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j∈Ni
j∈Ni
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X
j∈Ni
O(
|eij | · ηij
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un+1
i
=
uni
n
∆tn X
~ ij · ηij unj − uni
−
|eij | 1 − δijn U
2Ai j∈N
i
=
n
∆tn X
~ · ηij
1−
|eij | δijn − 1 U
ij
2Ai j∈N
i
+
n
∆tn X
~ · ηij un .
|eij | δijn − 1 U
ij
j
2Ai j∈N
i
)%
!
uni
uni
,
’
`
B n
∆tn
~
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ij
2Ai
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~ ijn · ηij
~ ijn · ηij ,
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=
U
2Ai
τijn =
,
un+1
=
i
( # >
1−
X
j∈Ni
!
τijn ≥ 0
δijn
≥1
⇐⇒
j∈Ni
n
|τijn |
- < +J
-! δijn ≥ 1,
' ≥1
n
αij
γijn
=
Sijn
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V
ij
∆tn |∂Ci |
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2Ai
∀ (i, n)
3 ,
X
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N2
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γin = sup γin .
j∈Ni
( H D H/ " 3 ? ( " /-! ∃
≤
1
2Ai
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≤
γijn
τijn ≤ 1.
δijn − 1 ≥ 0
∀(i, n, j) ∈ N × Ni
j∈Ni
( ∆t
+/ j∈Ni
2
~ n · ηij = φn ,
sup U
ij
i
τijn unj .
j∈Ni
X
⇐⇒
X
- X
uni +
1 ™hHh25 D <G/I:K LM- 1 > "
τijn ≥ 0
τijn
!
H
:$ C 3 D<< - # 1 ( " i
un+1
= H uni , un{j∈Ni } ,
i
∆tn X
H uni , u{j∈Ni } = uni −
|eij |g u, uj , Sijn , ηij .
Ai j∈N
i
q
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( # F/'1 1 Hh 2 3 D PG/IKL ? ( " H
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!
/
n
ij
n
ij
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*
n
i
n
ij
n
{j∈N
i } ! .
n
ij
n
ij
ij
γijn
1 n
n
n
~
=
uj + u i −
sgn Vij · ηij unj − uni .
2
2
$ # 7 3
N2 × N i
#
"
γijn = γ ∀(i, n, j) ∈
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2
ij
n
ij
ij
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V
ij
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0
n
ij
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2
2
1
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2
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∆tn X
ui , u{j∈Ni } = −
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∂uj
2Ai j∈N
/: ij
ij
unij =
◦
γ≥1
-
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(+3
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<
+ ∀(i, n) ∈ N2
i
3 O( - ∂H n
ui , u{j∈Ni } ≥ 0
sij − γ ≤ 0 ⇐⇒ sij ≤ γ
∂uj + 9
γ≥1
∂H n
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u
,
u
|eij | (sij + γ) F 0 unij · ηij
=
1
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i
n
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i
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u
,
u
≥0
{j∈N
}
i
i
∂uni
∆tn X
(sij + γ) F 0 unij · ηij ≤ 1
2Ai j∈N
⇐⇒
i
B ,
’
X
j∈Ni
,
’
(sij + γ) F 0 unij · ηij
≤ (1 + γ) sup F 0 unij · ηij |∂Ci |.
(
ψin = sup F 0 unij · ηij ,
j∈Ni
X
j∈Ni
j∈Ni
|eij | (1 + γ) F 0 (unij ) · ηij
j∈Ni
( / : X
≤
∆tn ≤
(1 + γ) F 0 unij · ηij ≤ (1 + γ) ψin |∂Ci |.
2Ai
1
∀n ∈ N,
|∂Ci | (1 + γ) ψin
# ∂H n
ui , u{j∈Ni } ≥ 0.
n
∂ui
J/ < &# 9 (R / = # & B / ;/ # B (R "<$! u " 9 D&! ; D
3 >< u
$ $ /-! "
$
ij
/ unij =
A" 6-! ;9 # % #
n
ij
unij ∈ uni ⊥unj , uni >unj .
B
$"
n
’
δijn ≥ 1
1
1
1 + δijn uni +
1 − δijn unj ,
2
2
1 + δijn ≥ 0
Sijn
n
αij
≤
~ n · ηij
V
ij
n
S
ij
n
αij =
~ n · ηij
V
ij
+
3 "( δ
"$, - $,/7 / 1 − δijn ≥ 0
3 D G/I:K LBG '
n
ij
≤1
" 1 ! ( 3
B
]
D QG/I:K L = # - "(R :$ = -! & B< /"/ 2
F 0 · ηij

1 n
1

n
n
n
n

F
(u
)
−
F
(u
)
· ηij
u
=
u
+
u
−

j
i
ij
i
j

2

~ n · ηij

V
2
ij



n X
∆t
un+1
= uni −
|eij |g(uni, unj , Sijn , ηij )
i

A
i j∈N


i
Z


1

0

u(x, y, 0)dxdy,

 ui = A
i Ci
1 2

1
1 n
n
n
n

~

ui + uj − sgn Vij · ηij unj − uni
uij =


2
2


P
n+1
∆tn
n
ui = u i − A i
|eij |g uni , unj , Sijn , ηij

j∈Ni

R


1
0

ui = Ai Ci u(x, y, 0)dxdy.
1 2
O
’ H /H "# 1 q2 - / $# 3 / <&# L /&# L / " D< <- 1 - < 3
<#$
3
!
"

 ∂u + ∂u + 2 ∂u = 0, (x, y) ∈ D, t ∈ R+
∂t
∂x
∂y

u(x, y, 0) = u0 (x, y) = cos2 2π (x − 0.5)2 + (y − 0.25)2 ,
1 #q 2
- D = [0, 2] × [0, 1] /A # " -! : y 3 "< @)( 8 $ #
.- " ># 1D3
( # " = < &! $ x # - 3 B)( # &#$;-#! / $ & / & ? " j
B
&
?
#
O
(
:
&
#
Hmax
Hmax = max max |e |
? D 7" ?= 8&#? D<<G/IKLBG : 6-!
h
i
j∈Ni
ij
maillage 150x60
condition initiale,iso−concentration
0.25
1
0.9
0.2
0.8
0.7
0.15
Y
0.6
0.1
0.5
0.4
0.3
0.05
0.2
0.1
$ % )( / % n
/) n & "
0
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
0.45
0
0.5
0
0.2
0.4
0.6
0.8
1
X
1.2
1.4
1.6
1.8
Hh .*W"()#5#n
2
solution numrique−iso−concentration
1
solution exacte−iso−concentration
1
0.9
0.9
0.8
0.8
0.7
0.7
0.6
Y
Y
0.6
0.5
0.5
0.4
0.4
0.3
0.3
0.2
0.2
0.1
0.1
0
0
0.2
0.4
0.6
0.8
1
X
1.2
1.4
1.6
1.8
2
/) #"H / %& $/
% O ! ∆t = 0.00047
0
#" $ /) %)" % ! 0
0.2
0.4
0.6
1
0.8
1
X
1.2
1.4
1.6
1.8
pente=1.17
−4.5
0.9
0.8
−5
0.7
Log(L1−erreur)
u(t)
0.6
0.5
0.4
−5.5
−6
0.3
0.2
−6.5
0.1
0
0
0.2
0.4
0.6
0.8
1
x
1.2
1.4
1.6
1.8
&' ( )+*-,*/.103246587:9<;>[email protected]
@4?HFI2JKJL4NMPO%QRICST467
2
−7
−3.6
−3.4
−3.2
−3
−2.8
−2.6
Log(Hmax)
−2.4
−2.2
−2
−1.8
&' ( )U*V,WX.0324NQRYZ7=\[ 7<QFQR7%46Q]BD^G
DF2L_%IO<7`L5Z7%JaHF7
' 1.17
bNc-b
2
%
. "
#/
!$
. !
/
B
)
$
"
, "$
!
0
.
Q&# ; "< D#/ < / D< I .- " /A D< G/I:K L #; 0 J+ ( & ’ Q !& ?)( &> +
- 0 $# -!

 ∂W + div(F (W )) = 0 ∀x ∈ R , ∀ t ∈ R ,
1 2
∂t
2

W (x, 0) = W0 (x) ∀x ∈ R2 ,
"!
+
- F (W ) = (F (W ), F (W )) F F * + = R - " R , m ≥
$ C W : R × R → R D<;B- # @ 8 11 2 ( " :$ < " -! 1 1
2
2
"!
+
g Win , Wjn , ηij
m
$#
m
∆tn X
|eij |g Win , Wjn , ηij
Ai j∈N
Win , Wjn , ηij
g
i
1
=
|eij |
Z
F (W ) · ηij dσ.
D<7
eij
J+0- /
" %#
!"#%$&('
;= Q
:$ = Q J+ 7" /&#
# .+J " 2
Win+1 = Win −
1
m
J+ 7" D I - &#
1 2
O( < D&# # 7 1
1
g Win , Wjn , ηij =
F (Win ) · ηij + F (Wjn ) · ηij − Sijn Wjn − Win ,
2
2
- n $0- <I
/SijC &#
.- ! 7&#
i
Sijn
max
1≤p≤m
[λni ]p
, max λnj p
1≤p≤m
,
-!# </ <H• ∇F (W ) · η
:>& ? < A )( !
Q- > ’ Q-#" B= @ $ = 0
J+ 7" D " -!
6- <
= max
n − λj p
p
∇F (W n) · η ∆tn = tn+1 − tn jC ij
i
[λni ]p
&#
hrh
n
i
i
ij
:/ < Ž Sx‡Ÿ$'-63
Mn"a(^1J$'"a;+3S-/$'"
m
2 MO"a(E1J$'"a;+3Q-/$'"NAO(I;+GPD'CC.(
1 !‘(+3
AiD)"WC ∞ 2 AO$'"7"a|( -šu"LFC+M:UiU7R$.C.(t→AO(ZRUEGfMiCIgBM7(YGV( UWD'CIAO(t3d(+>#U7C C n !GV( WU70|+*+-6>@c+3S*v(
∆t
|∂Ci |
(+3Go€ D)L
-6*v( (R AO) ( GPDk;(+G6GfMOGV(
&)|+*,- lu(+"[email protected]*v(+GPD)3Q-/$'"
Ai
Ci
Ai
,
|∂Ci |Sin
∆tn ≤
$
(C+3GPD4&.-632(CC.(\AO(tM CBD)"a$'&
Sin = sup Sijn
šu" D ! D)GV$'j∈N
*vC iGPDk;$'"EA)-63S-/$'"_AO(<C+3/DieB-6Gf-632|
'[email protected]"a$'*+>@(
L∞
-
∀i, n ∈ N,
L∞
||W n ||L∞ (R2 ) ≤ ||W0 ||L∞ (R2 )
U7$'MO* Mn"w&)(;+3d(+MO*
(C+3TAO$'"7"a|(uUWD)*
X = (X1 , . . . , Xm ) ∈ Rm
||X||L∞ = sup |Xr |.
1≤r≤m
<
X |e
1 52 = j∈Ni
(Win ) · ηij = 0,
ij |F
-!
= Win −
;"< D< 7- n i
n
j ∆tn X
|eij | F (Wjn ) − F (Win ) · ηij − Sijn Wjn − Win .
2Ai j∈N
i
Ae W n, W n; η η &# i j ij ij
!
#
=
Win
<
/$ &# #/ 7& 7;= < &#
@ J+ -!# 4 λ /
D< I .- "(R :# 8$ < " -! Win+1
i
*/ ∆tn X
|eij | g Win , Wjn , ηij − F (Win ) · ηij
Ai j∈N
(i) Ae (Win , Win ; ηij ) = A (Win ; ηij )
(ii) Ae Win , Wjn ; ηij (Wj − Win ) = F (Wjn ) − F (Win ) · ηij
(iii) Ae Win , Wjn ; ηij
Rn p = 1, . . . , m ij p
p = 1, . . . , m
-
#
’ < "# B I & +/
W W /-! 7$ -#"
8 /-! RI Win+1 = Win −
# n
ij p
h
i
∆tn X
n
n
n
n
n
n
n
e
−
|eij | A Wi , Wj ; ηij Wj − Wi − Sij Wj − Wi
2Ai j∈N
i
=
+
"
#
n X
∆t
Im −
|eij | −Ae Win , Wjn ; ηij + Sijn I m Win
2Ai j∈N
i
∆tn X
|eij | −Ae Win , Wjn ; ηij + Sijn I m Wjn ,
2Ai j∈N
i
hZ
1 ™h2
Im
: B > M (R) 7 * ' W 3 W W & '@& <
, m
n
i
n j
p=m
p=m
X
αin+1 p Rnij p ,
Win+1 =
Win =
O( , D 1 ™h2 : /-!# :9
"
X p=m
n+1
αi
p=1
p
−
X
p=1
p=1
n+1
i
[αin ]p Rnij p ,
Rnij p
p=m
Wjn =
p=1
!
#
∆tn X
|eij | − λnij p + Sijn
[αin ]p Rnij p
1−
2Ai j∈N
i
#
"
X ∆tn X
|eij | − λnij p + Sijn αjn p Rnij p = 0.
−
2A
i
p=1
j∈Ni
n Rij p
p = 1, . . . , m
"
X p=m
/" αin+1 p
p=1
−
3 # #
#
!
∆tn X
n
n
n
|eij | − λij p + Sij
[αi ]p Rnij p
1−
2Ai j∈N
i
X ∆tn X
|eij | − λnij p + Sijn αjn p Rnij p = 0
2Ai j∈N
p=1
p=m
−
i
!
∆tn X
1−
|eij | − λnij p + Sijn
[αin ]p
2Ai j∈N
n+1 −
αi
p
i
−
∆tn X
|eij | − λnij p + Sijn αjn p = 0,
2Ai j∈N
i
p = 1, . . . , m
n+1
⇐⇒ αi
p
=
!
∆tn X
|eij | − λnij p + Sijn
1−
[αin ]p
2Ai j∈N
i
+
’ ,
∆tn X
|eij | − λnij p + Sijn αjn p ,
2Ai j∈N
i
p = 1, . . . , m
n
∆tn
|eij | − λnij p + Sijn
γij p =
2Ai
h8"
p = 1, . . . , m
3
X αjn p Rnij p ,
p=m
1≤p≤m
,<# γ ≥ 0 p = 1, . . . , m
/ " /0$ D# L 3 8 X γ ≤ 1 p = 1, . . . , m.
’ H
’
#
n
ij p
n
ij p
∞
7 /- ;$ j∈Ni
X
γijn
j∈Ni
(
Sin = max Sijn
X ∆tn
|eij | − λnij p + Sijn
2Ai
j∈N
=
p
i
≤
∆tn n X
S
|eij |
Ai i j∈N
≤
∆tn n
S |∂Ci |,
Ai i
i
,
j∈Ni
∆tn ≤
Ai
n
Si |∂Ci |
O( , / 0 D# $
2 4
<5
"
"
- A
B
γijn
j∈Ni
M
p
≤ 1,
p = 1, . . . , m
2D
89/ " ’
L∞
X
∀ (i, n) ∈ N2 ,
J+ " > "# /-!

 ∂W + A ∂W + B ∂W = 0
∂t
∂x
∂y

W (x, y, 0) = W0 (x, y) ,
1 ™h 2
8& M (R) " / " 3 ( / L #
m
J (ηij ) = A · (ηij )x + B · (ηij )y
$ &# ! R 3 ! m !- # # [λ ] J+
- # [R ] D <EF- # HGJIKL ( " /,
1 ™h 20 < "
ij p
ij p
# 1≤p≤m
h
1≤p≤m
3
6-!

n
αij
1

n
n
n

W
+
W
−
W
=
(A · (ηij )x + B · (ηij )y ) Wjn − Win

i
j
ij

n

2
2Sij


n X

∆t
|eij |g Win , Wjn , Sijn , ηij
Win+1 = Win −
Ai j∈N


i
Z



1

0

W (x, y, 0)dxdy,
Wi =
A i Ci
- g Win , Wjn , Sijn , ηij = J (ηij ) · Wijn
1 ™hrh#2
:$ <# D :$ -J I .- " #>&# ?O( > D # # >>$ " &#
ηij
n ij S
Ci
Sijn = max
[λij ]p
1≤p≤m
AO$'"7"a|(t$'" C+M.UiU7$.C.(
z Ÿ$'-63
Mn"a(1J$'"a;+3S-/$'"yAiD)"WC ∞ 2
W0
L R × R+
gBM^(?GV(tUWD)*„D)>@c+3S*v( OA (†;$'"73S*'GV( n &)|+*,- lu( [email protected]*v(+GPD)3Q-/$'"
:/
< Ž
αij
n
min αij
≥ max
j∈Ni
'‘(
UWD'CuAO(T32(+>#U7C
∆tn
!aGV(‘U7|+*+-6>@c+3S*v(
∆tn ≤
j∈Ni
|∂Ci |
(+3Go€ D)-6*v(
|∂Ci | max
j∈Ni
(
n
Sijn
[λij ]p
Ai
)
.
A)M†&)$'GfMO>@(uAO(–;$'"73S* 'GV(
2Ai
[λij ]p 1 +
δijn
Ci
&)|+*+- lu(+"73
o
n
αij
D)GV$'*vC\GV( C.;57|+>4DAO(\&)$'GfMn>@(Clm"7-VC?Ÿ ),+ -0D+UiUEGf-SgBM7|IU7$'Mn*
= n [λij ]p ,
GV(ŒC+bC+3dc+>@($S),+ ij-(04&)|+*,- lu( GPDk;$'"EA)-63S-/$'"wAO(ŒC+3/DieB-6Gf-63d| ∞ C+Mn-6&'D)"73d(
L
D)&)(;
δijn
||W n||L∞ (R2 ) ≤ ||W0 ||L∞ (R2 ) .
<
P 7 W , W , W
- # : J (η ) /-!
’
n
i
n
j
n
ij ij
p=m
Win
=
X
p=1
Win+1
& & = 77&# p=m
[ξin ]p
[Rij ]p ,
Wjn
=
X
p=m
[ξjn ]p
[Rij ]p ,
Wijn
p=1
=
X
h
[ξijn ]p [Rij ]p
p=1
p=m
Win+1 =
X
p=1
[ξin+1 ]p [Rij ]p
(R # >/"/ ? D/< G/I:K L 1 ™h6" 25
,
δijn
n
αij
1 n
[ξi ]p + [ξjn ]p − n
2
2Sij
[ξijn ]p =
’
n
αij
= n λij
Sij
3 1 HhZC25 -J
3 p = 1, . . . , m
[λij ]p [ξjn ]p − [ξin ]p .
1 ™hZ2
1
1
1 + δijn [ξin ]p +
1 − δijn [ξjn ]p .
2
2
[ξijn ]p =
3 # # 5 0> (R D! /8 D G/I:K L 1 Hh8" 25 ?$ & >= 7>#? @- / / "(R
X
j∈Ni
[ξin ]p
∆tn X
|eij | [λij ]p [ξijn ]p − [ξin ]p
Ai j∈N
[ξin+1 ]p = [ξin ]p −
=
|eij |J (ηij ) Win = 0
i
n
n
1
1
∆tn X
n
n
n
|eij | [λij ]p
1 + δij [ξi ]p +
1 − δij [ξj ]p − [ξi ]p
−
Ai j∈N
2
2
i
=
X
1−
γijn
j∈Ni
!
X
[ξin ]p +
γijn [ξjn ]p ,
j∈Ni
- γ = ∆t |e | [λ ] −1 + δ ( # 5:2A
1 Hh™h#2* "A$ / "? γ ≥ 0 X γ ≤ 1 n
n
ij
ij
i
ij p
n
ij
n
ij
# L∞
n
ij
j∈Ni
γijn
⇐⇒ [λij ]p −1 + δijn ≥ 0
sij = sgn [λij ]p
≥0
- H# &
’ H
O(
−sij + δijn ≥ 0
⇐⇒
:# @ 7 8$ 0 ?# "
γijn ≤
X
n
o
∆tn
,
|∂Ci | max [λij ]p sij + δijn
j∈Ni
2Ai
?$ 0 D! j∈Ni
∆t ≤
|∂Ci | max
n
2Ai
[λij ]p sij + δijn
D< GJIKL -#" = $ 0 D! j∈Ni
hrq
L∞
o
5
-#
j∈Ni
X
3 γijn ≤ 1
δijn ≥ 1,
2 <5
"8/9 D/<PQ- 6-!
g
4
G/IKL 7 1 2 "(R" H < "
#
!

n
αij
1

n
n
n
n
n

Wi + Wj − n F (Wj ) − F (Wi ) · ηij
Wij =



2
Sij


n X

∆t
Win+1 = Win −
|eij |g Win , Wjn , Sijn , ηij
Ai j∈N


i

Z


1

0
W i =
W (x, y, 0)dxdy,

A i Ci
$ = Q J+ 7" D<<G/IKL
Win , Wjn , Sijn , ηij
g
Wijn
=F
/ &#
· ηij
1 ™h8"C2
Sijn
= max
1≤p≤m
h
n
λf
ij
i p
h i
7
<
#
#
D
7 !- # ; # 70$ < η
λf
I RI / & B D I .- ; B D W W 3 6-!
$; η ;= Q J+ 7" D <G/I:K L D " -! - n
ij
ij
!
#
p n i
ij
:/
αij
n
min αij
≥ max
j∈Ni
'‘(
UWD'CuAO(T32(+>#U7C
∆t
D)&)(; n δij
p
=
j∈Ni
n !aGV(‘U7|+*+-6>@c+3S*v(
∆tn ≤
Sin
|∂Ci |








Sijn
h i .


en 

 λ
ij
(+3Go€ D)-6*v(
p
Ai
A)M†&)$'GfMO>@(uAO(–;$'"73S* 'GV(
2Ai
,
n
δij p
|∂Ci |
1 + sup
Ci
&)|+*+- lu(+"73
j∈Ni
n h
i (+3
αij
D)GV$'*vC†$'" D8GPDk;$'"EA)-63S-/$'"_AO(<C+3/DieB-6Gf-632| ∞
en
λ
Sin = max Sijn ,
L
ij
n
j∈Ni
Sij
p
||W n ||L∞ ≤ ||W0 ||L∞ .
<
’
n
j
< Ž Ž Ÿ$'-/(+"73
MO"a(1J$'"a;+3S-/$'" OA ( m &)(+*vC 2 !tAO(N;+GPD'CC.( 1
R
nM "a(m1J$'"a;+3S-/$'"_AiD)"WC ∞F =2 (FAO1$', "7F"a2 )|( u$'" C+M.UiU7$.C.( Bg M^( RGV(tUWD)*„D)>@
c+3S*(?AO(N;$'"73Q* C
'GV(
W0
L R
n &)|+*,- lu( GPD8*v(+GPD)3S-/$'"
(+3
H
Wijn
1
=
2
Win
+
Wjn
n
αij
n
n
− n F (Wj ) − F (Wi ) · ηij .
Sij
h
!
’ , @$ < B I RI
/ & ? D<
W W 3 6-! η 3 n n
j
i
!
Win
"!
Wjn
+
p
n
ij
p=m
X
=
[ξin ]p
p=1
,
h
en
R
ij
i
p
B
Wjn
,
p
n i
X h i
en ,
=
ξjn p R
ij
n
j
p=m
Wijn
p
p=1
X
=
n
Rij p ,
p
ξijn
1 ™h 2
p=1
X
h ni
e
ξin+1 p R
.
ij
p=m
Win+1
=
p
p=1
3 , 1 ™h 2
1
1
1 + δijn p [ξin ]p +
1 − δijn p [ξjn ]p .
2
2
3 ( / : |eij |F (Win ) · ηij = 0
j∈Ni
1≤p≤m
n h
i
n
αij
en
δij p = n λ
Sij ij p
X
p=1,...,m
p=m
ξijn =
’
@/$ &# #/
A J+ -#! / * $ & W
W
’
p=1,...,m
Win
n
αij
n
n
n
n
− n Ae Wi , Wj ; ηij Wj − Wi
.
Sij
Ae Win , Wjn ; ηij
h i en
R
ij
p
n
ij
? 1 2 @ 3 # #
R # m - / h i e
λ
3 7 F D
- # hRe i 3 I .- #
ij
1
=
2
Wijn
Win+1 = Win −
∆tn X
|eij | F (Wijn ) − F (Win ) · ηij .
Ai j∈N
; </ "" 3 !( 9 / $ < "# I
8 -! :$ / η Ă W , W ; η
W
W
K ! :# # = # T ?& 8 +
Ae W , W ; η Ă W , W ; η 3 8& ::- 3 / O’
hλ˘ i -#! / # 7 Ă W , W ; η / 3
@ B D&! p -!# # hλ̆ i heλ i @ s B " ?$ ' ,)( D# ? 8- : ’
n
ij
n
i
n
ij
n
i ij
n
j
i
p
p=1,...,m
n+1
ξi
p
=
n ij
n
i
n
ij
ij
&#
ij
n
i
ij
n
ij
n
ij
p
n
ij
n
i
p !
h i ∆tn X
|eij | λ˘nij
1−
−1 + δijn p
[ξin ]p
2Ai j∈N
p
i
h
ij
n
ij
h i n n
∆tn X
n
˘
−1 + δij p ξj p .
+
|eij | λij
p
2Ai j∈N
, γ = ∆t |e | λ˘ i −s + δ p = 1, . . . , m 2A
/ "8 0 # L 3 : "
X p = 1, . . . , m.
≤1
γ
≥0
γ
i
’
h
n
n
ij p
ij
i
n
ij
p
∞
n
ij p
n
ij p
n
ij
n
ij p
j∈Ni
H# &
’ H
γijn
p
≥0
⇐⇒
δijn
≥1
p
0-! "7 "8$ 0 ?$# n
αij
≥ ⇐⇒
X
j∈Ni
X
i
n
∆tn n n
≤
S sij + sup ( δij p ) |∂Ci |.
2Ai i
j∈Ni
3 D L #
S
/ & % ( &# = # /% *)( &# "#$!
/ " Q :" 9 /- " , D G/I:K LBG :$ = n
≤
n
i
∆t
h i ∆tn X
−1 + δijn p
|eij | λ˘nij
2Ai j∈N
p
γijn =
j∈Ni
"( : γijn ≤ 1
Sijn
.
λnij p
2Ai
n
1 + sup δij p |∂Ci |
∞
j∈Ni

1
1
n
n
n
n
n
e

e
W
=
W
+
W
−
sgn
A
W
,
W
;
η
Wjn − Win

ij
ij
i
j
i
j
2
2



 n+1
n P
n
n
n
|e
|g
W
,
W
,
S
,
η
Wi
= Win − ∆t
ij
ij
i
j
ij
Ai
j∈Ni


R

1

0

Wi = Ai Ci W (x, y, 0)dxdy.
1 ™hrq#2
/ 1 " D HG/I:K LBG 7 - <
D %I L B 2 D 7 O( F </$"B I ;&#> ! /9;)( D ? "<$# Ae
Win , Wjn , ηij
Win + Wjn
= Ae
, ηij .
2
%
!#"
)
)
" $
+ 0
"!
K ?# " D 8 / ( "8 " ? 7G/# & - ;= B "4& > "#; > B A :" & ? /G # & : > (R :9 = /
!= 3 7 ?9; - 1 & :; -" "# 2 "
J
&%$"
$
$
$


 ∂W + ∂F (W ) + ∂G(W ) = 0
∂t
∂x
∂y

W (x, y, 0) = W (x, y),
0
- 

h


,
hu
W =



hv

hu



1
2
2 
hu
+
gh
F (W ) = 
2

,
huv

G(W ) = 


hv
huv
hv 2 + 21 gh2



>?- @ -!## " -# - 3 h @ & 5 ( 3 u v
4 - O- O(R *& ' ' #
?- ! 3 g : ( " 0 8 J+; "(R" A (W ; η) = A (W )η + A (W )η - A (W )
? < 8 ?= 5 J+0/ 3 "
A (W )
/ :&#
W
2
1
x
2
y
1

0
1
0

2

A1 (W ) =  −u + gh 2u 0

−uv
v u








A2 (W ) = 

< # /-! :$ η=
ηx
ηy
0
v
3 #
1


u 
.

−v 2 + gh 0 2v
−uv
A(W, η) = A1 (W )ηx + A2 (W )ηy

0
ηx
ηy

2
2

uηy
=  (c − u )ηx − uvηy 2uηx + vηy

−uvηx + (c2 − v 2 )ηy
vηx
uηx + 2vηy
0



,

= ~ · η = u · ηx + v · ηy V
~ · η,
λ2 (W, η) = V
Q H 8 $! /-!
/ 0- ? )( #> C- (h, u, v)(x < 6, t = 0) = (5, 0, 0),
H
λ3 (W, η) = V~ · η + c,

$ #



0
1
r1 (W, η) =  u − cηx  , r2 (W, η) =  −ηy 
ηx
v − cηy
B "
8- # ~ · η − c,
λ1 (W, η) = V
$ "" > ,& ?B-
" <-!# < #
J+ : #
p
gh
c=


1
r3 (W, η) =  u + cηx  .
v + cηy
D/< :O( (h, u, v)(x > 6, t = 0) = (1, 0, 0).
1.
Hauteur de l’eau−Schéma SRNHS
Mesh NS 300x30
5
1
0.9
4.5
5
0.8
4
4.5
0.7
4
3.5
h(x,y,t)
3.5
0.6
3
0.5
3
2.5
0.4
2.5
0.3
2
0.2
1.5
0.1
0
2
0
1
0
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
1.5
0.5
y
2
!
$&%' 4
6
x
8
*),+- .0/21
10
:
3D (
∆t = 0.0002
3456784/
5
)
1
1
12
!" #
-/
.
+
erreur 2d srpc0.95
1
4.5
0.5
4
3.5
0
3
−0.5
2.5
2
−1
1.5
1
E
0
2
!E )E
F
4
9;:=<?>,@BACEDGFHI$JK
R 4I$JTS
6
8
10
?L ?MBNPONI Q
RVU,W$X Y Z[H\LP]$^N R
12
−1.5
−3
F
−2.5
E _E E
!E E E F
−2
9;:=<?>[email protected]&A`_'aDbFHI$JK
H'h4i7j k']
_ ll
4\ R
−1.5
−1
−0.5
cLed fJJ 4I$JgN Q
' 0.95
0
/ >O( /"8/ "(R
ρ


 ∂W + ∂F (W ) + ∂G(W ) = 0
∂t
∂x
∂y

W (x, y, 0) = W (x, y).
0
-

?$ = 6-#




 ρu 


W =
,
 ρv 


ρu



 ρu2 + P 


F (W ) = 



ρuv



ρv


ρuv

G(W ) = 
 ρv 2 + P








$ - < 3 V ;<- D& < 0$Q-J 0 "! ρ
7 u
?)( " & D# &# BB- 3 P ?
Q- & ;v 3 e C &! = # $#
" : ( " # &#
(e + P )u
e
&#
&#
(e + P )v
u2 + v 2
P = (γ − 1) e − ρ
2
,
- # :D# & C 3 #
$< " / T
: R&=# CAv −(W,Cpn) = ∇F (W )η + ∇G(W )η -
γ
x




∇F (W ) = 



0
1
[(γ
2
−uv



∇G(W ) = 



- c
1
[(γ
2
0
−uv
v
− 3)v 2 + (γ − 1)u2 ]
(1 − γ)u
2
c
−v[ γ−1
+ (1 − γ2 )(u2 + v 2 )] (1 − γ)uv
3
p
γP
= γRT .
ρ
Z"
0
1
0

c2
γ−1


0 


(3 − γ)v (γ − 1) 


− (γ − 23 )v 2 + 21 u2
γv
u
& ? C 3 ( +// H C#9
s
c=
0

(3 − γ)u (−γ + 1)v (γ − 1) 


v
u
0 


3
1 2
2
− (γ − 2 )u + 2 v (1 − γ)uv
γu
0
:$ - " #
c2
γ−1
2
c
+ (1 − γ2 )(u2 + v 2 )]
−u[ γ−1

y
1
− 3)u2 + (γ − 1)v 2 ]
/" 3
&#
/ : < > 8- λ1 (W, η) = V · η − c,




R (W ; η) = 



&!
1
0

1
c
ηy
u
c
+ ηx
v
c
− ηx
v
−ηx
v
c
+ ηx
|V |2
2
u · ηy − v · η x
c
γ−1
λ3 (W, η) = V · η + c
u
−v·η+
− ηx
2
V · η + (γ − 1) |V2c|
|V |2
2c
+v·η+
c
γ−1
2
−V · η + (γ − 1) |V2c|







−(ηx + (γ − 1) uc ) −(ηy + (γ − 1) vc )
(2(γ − 1) cu2 )
2(γ − 1) cv2
2ηy
−2ηx
(ηx − (γ − 1) uc )
ηy − (γ − 1) vc
Q H 8 $!
u
c


|V |2
)
2(1
−
(γ
−
1)

1
2c2
R−1 (W ; η) = 
2
 2(ηx · v − ηy · u)


B "
1
c
|V |2
2c
:-!# 8 /
λ2 (W, η) = V
, -" : &#
(ρ, p, u) (x < 6, t = 0) = 12, 106 , 0 ,
γ−1
c


−2 γ−1
c2 



0

γ−1
c
(ρ, p, u) (x > 6, t = 0) = 1.2, 105 , 0

Mesh NS 150x15
12
1
’r11’
’r11’
0.9
10
0.8
0.7
8
0.6
6
0.5
0.4
4
0.3
0.2
2
0.1
0
0
0.1
0
0.2
#
0.3
0.4
'
1
0.5
0.6
0.7
!
0.8
0.9
#
7
V),+$- ++".
0
1
2
4
6
8
10
12
8 #
(
440
1.1e+06
’t11’
’t11’
’p11’
’p11’
1e+06
420
900000
400
800000
380
700000
360
600000
500000
340
400000
320
300000
300
200000
280
100000
0
78
#
2
b
4
$
V),+$- ++".
6
8
10
12
$ #
0
456 (
2
"
#
4
6
$
(
*),+$- ++.
8
10
#
12
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X
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,
X
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j∈N (i)
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2
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X
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Y+W
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2
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r
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r
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r
r r
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h ¸iY+X h j«TjoR h k T('iQ5jqXj°T d XWYl|-H+W†WV+¸oY™ d +W d W h j ` h ¸iY+T£ d H[Xj°T d X k TVj•YlW}?„
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Hauterde l’eau−Schéma SRNHS
1
0.9
4.5
0.8
4
5
0.7
3.5
4.5
4
0.6
3
3.5
0.5
3
0.4
2.5
h(x,y,t)
2
2.5
2
1.5
0.3
0.8
0.5
0
−10
0.4
0.5
−5
0.1
x
0.2 y
0
5
0
10
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
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0.9
1
0.6
0.2
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1.5
1
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1
Hauteur de l’eau−SRNHS
5
Iso−Concentration de la hauteur
Analytique
SRNHS (sans MUSCL)
1
4.5
0.9
4
0.8
3.5
0.7
h(x,y0,t)
3
0.6
2.5
0.5
2
0.4
1.5
1
0.3
0.5
0.2
0
−10
0.1
−8
−6
−4
−2
0
x
2
4
6
f
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8
10
0
−10
−8
−6
−4
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0
2
4
6
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8
10
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2
Vitesse de l’eau−Schéma SRNHS
Champ de la vitesse de l’eau
4
8
10
3.5
8
3
6
6
2.5
4
2
4
2
u(x,y,t)
1.5
2
0
−2
1
0
−4
1
0.5
0.8
−2
0.6
0
10
0.4
y
−0.5
5
0
0.2
−5
x
0 −10
−1
−10
−8
−6
−4
−2
0
2
4
6
NPORQTSº VU X4c)b jC`^ ƒb A„ z f ^ v+v ^
8
NPORQTSaº 9VU X4Z[>\+]7^.`^.ƒ b A„ z f ^ v+v ^
10
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{}€ q z f- \+b f zZy v
'
∆t = 0.0001
Vitesse de l’eau−SRNHS(sans MUSCL)
10
Analytique
SRNHS (sans MUSCL)
Iso−Concentration de la vitesse
1
8
0.9
0.8
6
0.7
u(x,y0,t)
4
0.6
2
0.5
0.4
0
0.3
−2
0.2
−4
−10
0.1
−8
−6
−4
−2
0
x
2
4
6
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8
10
0
−10
−8
−6
−4
−2
0
2
4
6
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8
10
1
Schéma SRNHS−Débit de l’eau
0.9
10
5
0.8
5
0.7
0
0
0.6
−5
0.5
−5
q(x,y,t)
0.4
−10
0.3
−15
0.2
−20
1
−10
−15
10
0.8
0
0.4
y
0
5
0.6
0.1
0.2
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0 −10
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
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1
Débit de l’eau − SRNHS (sans MUSCL)
10
Analytique
SRNHS (sans MUSCL)
Iso−Concentration
1
0.9
5
0.8
0
q(x,y0,t)
0.7
0.6
−5
0.5
−10
0.4
0.3
−15
0.2
−20
−10
0.1
−8
−6
−4
−2
0
x
2
4
6
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8
10
0
−10
−8
−6
−4
−2
0
2
4
6
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2
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Mesh NS 300x30
5
1
4.5
0.9
5
0.8
4
4.5
0.7
3.5
4
0.6
3.5
0.5
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3
3
2.5
2.5
0.4
2
2
0.3
1
1.5
0.8
0.2
0.1
0.4
2
4
0.2
6
8
10
12
x
0
1.5
0.6
1
0
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0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
y
0
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5
Iso−Concentration
SRNHS (avec MUSCL)
Analytique
1
4.5
0.9
4
0.8
3.5
0.7
0.6
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