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PureMathematicsnØêÆ,2020,10(8),771-783
PublishedOnlineAugust2020inHans.http://www.hanspub.org/journal/pm
https://doi.org/10.12677/pm.2020.108090
TheNumericalSchemeofEnergy
ConservationfortheKdVEquation
YuTian,YanfenCui
CollegeofSciences,ShanghaiUniversity,Shanghai
Received:Jul.31
st
,2020;accepted:Aug.18
th
,2020;published:Aug.26
th
,2020
Abstract
WedesignaclassofimprovedschemesatisfyingtwoconservationlawsfortheKdV
equation,whichsatisfiesboththenumericalsolutionandnumericalenergyconserva-
tive.Numerical experimentsshow thatthe schemes have good stability and structure-
preservingpropertyinlongtimenumericalsimulations.
Keywords
Cell-Average,NumericalEnergy,ConservationLaws,StructurePreservation
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DOI:10.12677/pm.2020.108090773nØêÆ
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DOI:10.12677/pm.2020.108090774nØêÆ
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DOI:10.12677/pm.2020.108090775nØêÆ
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DOI:10.12677/pm.2020.108090776nØêÆ
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1
2
,t),
ˆu
h
(x
j±
1
2
,t) = u
h,−
(x
j±
1
2
,t).
(29)
DOI:10.12677/pm.2020.108090777nØêÆ
X…§wý¥
3·‚O‚ª¥†DÚmäk•‚ªØÓ´§©ò‡é1‡Åð•§(25)?1
êŠ[¿±êŠUþÅð"é(25)§3I
j
È©§Œ
Z
I
j
U
t
(x,t)dx+
1
h
n
ˆ
F(x
j+
1
2
,t)−
ˆ
F(x
j−
1
2
,t)
o
= 0.(30)
Ù¥Uþ6¼ê´
ˆ
F
j+
1
2
(t) = 2ˆu(x
j+
1
2
,t)ˆp(x
j+
1
2
,t)−ˆq
2
(x
j+
1
2
,t).
(31)
3©¥§^©ã‚5¼êéêŠ)?1-§=•§(28)êŠ)u
h
∈V
∆x
•
u
h
(x,t) = u
0
j
(t)ϕ
(j)
0
(x)+u
1
j
(t)ϕ
(j)
1
(x),x∈I
j
.
(32)
ùpϕ
(j)
l
(x),l= 0,1,2...´˜|¼êĵ
ϕ
(j)
0
(x) = 1,ϕ
(j)
1
(x) =
2(x−x
j
)
h
,...(33)
ò(32)‘\(28)¥1˜‡•§§Œ±êŠ)ŒlÑ‚ª•
d
dt
u
0
j
(t)+
1
h
[
ˆ
f
j+
1
2
(t)−
ˆ
f
j−
1
2
(t)] = 0.(34)
êŠ6¼ê•
ˆ
f
j±
1
2
(t) =ˆp
h
(x
j±
1
2
,t).(35)
3·‚‚ª¥§ÓžOŽ1‡Åðþ§d(30)Œ
d
dt
U
j
(t)+
1
∆x
[
ˆ
F(x
j+
1
2
,t)−
ˆ
F(x
j−
1
2
,t)] = 0.(36)
Uþ6¼ê•
ˆ
F
j±
1
2
(t) = 2ˆu
h
(x
j±
1
2
,t)ˆp
h
(x
j±
1
2
,t)−(ˆq
h
(x
j±
1
2
,t))
2
.(37)
ØÓ uDÚmäk•‚ª´§ ·‚¤O‚ª•¹üÜ©µêŠ)¥u
0
j
(t)‘ÚêŠU
þU
j
(t)"êŠ)¥u
1
j
(t)Š•˜‡gdÝ"·‚‡¦êŠ)Uþ‚²þÚT‚þêŠUþƒ
§=
1
h
Z
x
j+1/2
x
j−1/2
U(u
h
(x,t))dx=
1
h
Z
x
j+1/2
x
j−1/2
(u
h
(x,t))
2
dx= U
j
(t).(38)
dþªOŽŒ
(u
1
j
(t))
2
= 3(U
j
(t)−(u
0
j
(t))
2
),(39)
DOI:10.12677/pm.2020.108090778nØêÆ
X…§wý¥
duu
1
j
(t)´éu
x
(x
j
,t)Cq§3·‚‚ª¥§mŠÒÎÒ†(u
j+1
(t)−u
j−1
(t))ƒÓ§=
u
1
j
(t) = sgn(u
j+1
(t)−u
j−1
(t))
q
3(U
j
(t)−(u
0
j
(t))
2
).(40)
ddŒ„§ùO‚ªpu
0
j
(t)ÚU
j
(t)Ñ´Åð"
•(½êŠ6¼ê†Uþ6¼ê§éq
h
(x,t)Úp
h
(x,t)Eæ^‚5¼ê?1-§Á&¼
êw,z∈V
∆x
•ϕ
0
(x)Úϕ
1
(x)§Œ±



p
0
j
(t)−
1
h
(ˆq
j+
1
2
(t)−ˆq
j−
1
2
(t) = 0,
p
1
j
(t)−
3
h
(−2q
0
j
(t)+ ˆq
j+
1
2
(t)+ ˆq
j−
1
2
(t)) = 0.
(41)



q
0
j
(t)−
1
h
(ˆu
j+
1
2
(t)−ˆu
j−
1
2
(t) = 0,
q
1
j
(t)−
3
h
(−2u
0
j
(t)+ ˆu
j+
1
2
(t)+ ˆq
j−
1
2
(t)) = 0.
(42)
Ù¥ˆu
j±
1
2
(t)Úˆq
j±
1
2
(t)•(29)¥¤L«/ª"
2.4.2.ꊂªžmlÑ
·‚òŒlÑ•§(34)Ú(36)3žm••þlѧæ^Crank−Nickson‚ª§
u
0,n+1
j
= u
0,n
j
−λ(
ˆ
f
n+
1
2
j+
1
2
−
ˆ
f
n+
1
2
j−
1
2
),
U
n+1
j
= U
n
j
−λ(
ˆ
F
n+
1
2
j+
1
2
−
ˆ
F
n+
1
2
j−
1
2
).
(43)
êŠ6¼ê
ˆ
f
n+
1
2
j±
1
2
ÚêŠUþ¼ê
ˆ
F
n+
1
2
j±
1
2
©OXe
ˆ
f
n+
1
2
j±
1
2
= (p
h
)
+,n+
1
2
j±
1
2
,(44)
ˆ
F
n+
1
2
j±
1
2
= 2(u
h
)
−,n+
1
2
j±
1
2
(p
h
)
+,n+
1
2
j±
1
2
−((q
h
)
h,n+
1
2
j±
1
2
)
2
,(45)
±9
(p
h
)
+,n+
1
2
j+
1
2
=
1
2
((p
h
)
+,n
j+
1
2
+(p
h
)
+,n+1
j+
1
2
),(46)
(q
h
)
+,n+
1
2
j+
1
2
=
1
2
((q
h
)
+,n
j+
1
2
+(q
h
)
+,n+1
j+
1
2
),(47)
(u
h
)
−,n+
1
2
j+
1
2
=
1
2
((u
h
)
−,n
j+
1
2
+(u
h
)
−,n+1
j+
1
2
).(48)
DOI:10.12677/pm.2020.108090779nØêÆ
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3.ꊎ~
Ž~3.1 •§(1)üf)
u(x,t) = Asech
2
(κx−ωt−x
0
),(49)
Ù¥
A= 12κ
2
,ω= 4κ
3
,(50)
Figure1.Numericalresultsatdifferenttimes
ã1.ØÓž•êŠ(J
DOI:10.12677/pm.2020.108090780nØêÆ
X…§wý¥
ùpkÚx
0
´~ê§Š•k= 0.3Úx
0
= 0§u(x,0)•ÐŠ"‚«m•(-15,15)§æ^±Ï
>.^‡l(50)Œ±wÑÅ„Ý´0.36§±Ï•T=30/0.36=83.3§3Ž~¥§ò‚ ªO
ŽêŠ)²Lê±Ï†Ð©ž•êŠ)'"
3·‚ êŠ[¥§‚Ú•'•λ=τ/h= 0.5"·‚^100‡‚!:(=h= 0.3)?1O
Ž§ã1¥©O‰Ñt= 83.3(1‡±Ï)§t= 416.7(5‡±Ï)±9t= 833.3(10‡±Ï)êŠ(
J"l㥌wÑêŠ)3•žmêŠ)UéÐ[)("
Ž~3.2 •§(1)Vf)
u(x,t) = 12
κ
2
1
e
θ
1
+κ
2
2
e
θ
2
+2(κ
2
−κ
1
)
2
e
θ
1
+θ
2
+a
2
(κ
2
2
e
θ
1
+κ
2
1
e
θ
2
)e
θ
1
+θ
2
(1+e
θ
1
+e
θ
2
+a
2
e
θ
1
+θ
2
)
2
,(51)
Figure2.Numericalresultsatdifferenttimes
ã2.ØÓž•êŠ(J
DOI:10.12677/pm.2020.108090781nØêÆ
X…§wý¥
Ù¥
κ
1
= 0.4,κ
2
= 0.6,a
2
= (
κ
1
−κ
2
κ
1
+κ
2
)
2
=
1
25
,(52)
θ
1
= κ
1
x−κ
3
1
t+x
1
,θ
2
= κ
2
x−κ
3
2
t+x
2
,x
1
= 4.0,x
2
= 15.(53)
3·‚êŠ[¥§‚«m•(-40,40)§æ^±Ï>.^‡"u(x,0)•Ð©Š§‚Ú
••h=0.4(=^200‡:OŽ)§‚Ú•'•λ=τ/h=0.5"ã2©O‰Ñt=100ž•§
t= 400Út= 1000ž•êŠ)"l㥌±wü‡ÅZœ¹"
4.(Ø
©éKdV•§E˜‡÷vü‡ÅðÆ©‚ª§æ^ ©Žf{òKdV•§©¤
Åð•§Ü©†Ñ•§Ü©§éÅð•§Ü©ÚnÑ•§Ü©ÓžO÷vü‡ÅðÆ
ꊂª"êŠÁL²ùa‚ªkéÐ-½5§AO3õ‡Åœ¹e§ËÄ5²wkU
õ"
ë•©z
[1]“f.‚5DÑ•§÷võ‡ÅðÆ©‚ª[D]:[a¬Æ Ø©].þ°:þ°ŒÆ,2005:No.11903-
99118086.
[2]Li,H.,Wang,Z.andMao,D.(2008)NumericallyNeitherDissipativeNorCompressiveSchemeforLinear
AdvectionEquationandItsApplicationtotheEulerSystem.JournalofScientificComputing,36,285-331.
https://doi.org/10.1007/s10915-008-9192-x
[3]où_,dx.ü‡Åð.•§ÑÑ‚ª¥ÑѼêE[J].OŽÔn,2004,21(3):319-331.
[4]Cui,Y.andMao,D.(2012)ErrorSelf-CancelingofaDifferenceSchemeMaintainingTwoConservation
LawsforLinearAdvectionEquation.MathematicsofComputation,81,715-741.
https://doi.org/10.1090/S0025-5718-2011-02523-8
[5]Cui, Y.and Mao, D.(2007)Numerical Method SatisfyingtheFirst TwoConservationLawsforthe Korteweg-
deVriesEquation.JournalofComputationalPhysics,227,376-399.
https://doi.org/10.1016/j.jcp.2007.07.031
[6]Holden, H., Hvistendahl, K. andRisebro, N. (1999) Operator SplittingMethods forGeneralized Korteweg-de
VriesEquations.JournalofComputationalPhysics,153,203-222.
https://doi.org/10.1006/jcph.1999.6273
[7]Strang,G.(1968)OntheConstruction andComparisonofDifferenceSchemes. SIAMJournalonNumerical
Analysis,5,506-517.
https://doi.org/10.1137/0705041
[8]LeVeque,R.J.(2002)FiniteVolumeMethodsforHyperbolicProblems.CambridgeUniversityPress,Cam-
bridge.
https://doi.org/10.1017/CBO9780511791253
[9]Harten,A.(1989)ENOSchemeswithSubcellResolution.JournalofComputationalPhysics,83,148-184.
https://doi.org/10.1016/0021-9991(89)90226-X
[10]Harten,A.,Harten,A.,Engquist,B.,Osher,S.andChakravarthy,S.R.(1987)UniformlyHighOrder
AccurateEssentiallyNon-OscillatorySchemes,III.JournalofComputationalPhysics,71,231-303.
https://doi.org/10.1016/0021-9991(87)90031-3
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X…§wý¥
[11]Yan, J. and Shu, C. (2002) A Local Discontinuous Galerkin Method for KdV Type Equation. SIAMJournal
onNumericalAnalysis,40,769-791.
https://doi.org/10.1137/S0036142901390378
DOI:10.12677/pm.2020.108090783nØêÆ

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