五阶偏微分方程的时间多区域时空谱方法 Multi-Domain Space-Time Spectral Method in Time for Solving Fifth-Order Partial Differential Equation

doi:10.12677/PM.2021.116117

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PureMathematicsnØêÆ,2021,11(6),1031-1047

PublishedOnlineJune2021inHans.http://www.hanspub.org/journal/pm

https://doi.org/10.12677/pm.2021.116117

Ê ‡©•§žmõ«•ž˜Ì•{

{{{JJJŒŒŒ

þ°ŒÆnÆ§þ°

ÂvFÏµ2021c51F¶¹^FÏµ2021c62F¶uÙFÏµ2021c610F

Á‡

©éÊ ‡©•§JÑžmõ«•ž˜Ì•{.T•{3˜m••þæ^Legendre-

Petrov-Galerkin•{, 3žm••þæ^ õ«•Legendre-tau •{, =: ržm«m©¤õ

‡«•,¿3z‡«•þæ^Legendre-tau•{.Óž,©‰ÑT•{3‚5¯KþØ©

Û,À·Ä¼ê¦XêÝDÕ, éš‚5•§¥š‚5‘æ^3Chebyshev-Gauss-

Lobatto:þŠ?1OŽ.•ÏL˜êŠŽ~yŽ{k5.

'…c

Ê ‡©•§§ž˜Ì•{§Legendre-Petrov-Galerkin•{§Legendre-Tau•{§

Chebyshev-Gauss-LobattoŠ

Multi-DomainSpace-TimeSpectralMethod

inTimeforSolvingFifth-OrderPartial

DiﬀerentialEquation

RongyuYu

CollegeofSciences,ShanghaiUniversity,Shanghai

Received:May1

,2021;accepted:Jun.2

,2021;published:Jun.10

,2021

©ÙÚ^:{JŒ.Ê ‡©•§žmõ«•ž˜Ì•{[J].nØêÆ,2021,11(6):1031-1047.

DOI:10.12677/pm.2021.116117

{JŒ

Abstract

In thispaper,we propose themulti-domain space-time spectral method intime forthe

ﬁfth-orderpartialdiﬀerentialequation.Legendre-Petrov-Galerkinmethodisapplied

inspacedirection,andmulti-domain Legendre-taumethod isappliedintime direction,

thatis: dividingthetimeintervalintomultipledomainsandLegendre-taumethodis

appliedineachdomain. Atthesametime, theerroranalysisofthemethodonlinear

problemsisgiveninthispaper,andwechooseproperbasisfunctionstomakethe

coeﬃcientmatrixsparse, thenonlineartermforthenonlinearequationiscomputed

byinterpolationthroughChebyshev-Gauss-Lobattopoints.Finally,somenumerical

examplesaregiventoverifytheeﬀectivenessofthealgorithm.

Keywords

Fifth-OrderPartialDiﬀerentialEquation,Space-TimeSpectralMethod,

Legendre-Petrov-GalerkinMethod,Legendre-TauMethod,

Chebyshev-Gauss-LobattoInterpolation

2021byauthor(s)andHansPublishersInc.

This work is licensed undertheCreative Commons Attribution InternationalLicense (CCBY4.0).

http://creativecommons.org/licenses/by/4.0/

1.có

3¢S¯K¥, Ê ‡©•§kX2•A^, ù•§Ñy3Ôn!åÆ!1Æ!9D !

Ä!6N$Ä!››±9)ÔXÚ •¡, ~X: 2ÂKorteweg-deVries(KdV) •§[1–8]. 3d

ƒc, IS˜‰ïóŠö®²éÊ ‡©•§?1ŒþïÄ, 3[9] ¥XuÚShuJÑ

Local-Discontious-Galerkin(LDG)•{5¦)ÊÅÄ•§, ¿‰ÑT•{-½5©Û±9

ÏL˜êŠŽ~`²T•{°Ý. 3[10]¥Cheng ÚShu JÑDiscontious-Galerkin(DG)

k••{5¦)žm •6p•§, Óž‰Ñ-½5©ÛÚØO; êŠ(JL², ©¡

õ‘ªgê•k gž, T‚ª°Ýò¬ˆk+1 .3[11] ¥ëùÚo)é˜a|Ü

Ê•§…Ü¯KJÑ˜«wª©‚ª,¿…y²T‚ª3÷v˜½^‡ž´-½.

òXeÊ ‡©•§Š•.•§?1ïÄ:

DOI:10.12677/pm.2021.1161171032nØêÆ

{JŒ











∂

U+∂

U= f(x,t),x∈I

,t∈I

;

U(±1,t) = ∂

U(1,t) = 0,t∈I

;

U(x,−1) = U

(x),x∈I

(1.1)

Ù¥I

= (−1,1),I

= (−1,1),U= U(x,t).

éu¦)žm•6 ‡©•§, •§°()3žmÚ˜m••þv 1wž, XJ3ž

m••þæ^©‚ª, ¬N´—Ì°Ý¿”. Óž, ž˜õ‘ªg ê˜½ž, æ^žm ü«•

ž˜Ì•{?1¦)¤ØŒ. Ïd©éÊ ‡©•§JÑžmõ«•ž˜Ì•

{,±B~Ø.

©‰Ñ•§(1.1)žmõ«• ž˜Ì‚ª. ˜m••þæ^Legendre-Petrov-Galerkin •

{, žm••þæ^õ«•Legendre-tau •{, =:ržm«m©¤õ‡«•, ¿3z‡«• þæ

^Legendre-tau •{.

©e5SN•):1)0©Ž{‚ª9nØ©Û¥I‡^˜ÎÒPÒÚÚn;

2) ‰ÑÊ ‡©•§'užm õ« •ž˜Ì•{lÑ‚ª, ¿£ãŽ{¢–; 3) éÊ

‚5 ‡©•§lÑ‚ª?1Ø©Û;4) ÏL˜êŠŽ~y©Ž{k5.

2.ý•£9PÒ

!Ì‡0©Ž{‚ª9nØ©Û¥I‡^˜ÎÒPÒÚÚn.

2.1.ÎÒPÒ

-Ω=I

⊗I

. σ•šKê, PH

) •˜„½ÂSobolev ˜m, …‰ê†Œ‰ê©O

P•k·k

,σ

Ú|·|

,σ

.•Äàg>.^‡, és≥1, ½Â[12]:

) = {v∈H

) |v(±1) = v

(±) = ···= v

(s−1)

(±1) = 0}.

˜m••lÑ:

éu?¿êN,P

)L««mI

þ¤kgêØ‡LNgõ‘ª¤˜m.

½Â:

2,3

) = {v∈H

)∩H

) |∂

(1) = 0}, H

2,2

) = H

)∩H

du˜m••"é¡5,·‚ÀÁ&¼ê˜mØÓuu¼ê˜m:

= P

)∩H

2,3

),V

∗

= P

)∩H

2,2

L

(x)•gêØ‡LjLegendreõ‘ª, é0 ≤j≤N−5,½Â[12]:

DOI:10.12677/pm.2021.1161171033nØêÆ

{JŒ

(x) = L

(x)+a

(j)

j+1

(x)+a

(j)

j+2

(x)+a

(j)

j+3

(x)+a

(j)

j+4

(x)+a

(j)

j+5

(x),

Ù¥

(j)

= −

2j+3

2j+7

(j)

= −

2(2j+5)

2j+7

(j)

2(2j+3)

2j+9

(j)

2j+3

2j+7

(j)

= −

(2j+3)(2j+5)

(2j+7)(2j+9)

3ùp·‚-γ

(x)•V

þÄ¼ê,KéN≥5, k

= span{γ

(x),γ

(x),···,γ

N−5

(x)}.

L

(x)•gêØ‡LnLegendre õ‘ª,é0 ≤n≤N−5, ½Â:

(x) = c

n+1

(x)−L

n+2

(x))−c

n+3

n+2

(x)−L

n+4

(x)).

Ù¥c

2n+1

,3ùp·‚-ψ

(x)•V

∗

N−1

þÄ¼ê,KéN≥5, k

∗

N−1

= span{ψ

(x),ψ

(x),···,ψ

N−5

(x)}.

½ÂÝKŽfP

α,β

˜ω

α,β

) →P

),PP

:= P

0,0

,…÷v

α,β

u−u,v)

,˜ω

α,β

= 0, ∀v∈P

•;•3nØ©Û¥ÑyÊê, ·‚ I‡Ú\XeÝKŽf. ½ÂP

∗

: H

2,3

→V

÷v

(∂

∗

u−u),∂

= 0,∀v∈V

∗

N−1

.(2.1)

Œ•P

∗

u•3…•˜, ¦

∗

u: =

∂

−3

N−3

∂

u,(2.2)

Ù¥∂

−1

v(x) = −

v(y)dy,∂

−m

v(x) = (∂

−1

)

d½Â•,

u(1) = ∂

u(1) = 0,…k

u(−1) = −(x+1)∂

−2

N−3

∂

u(x) |

−1

(x+1)∂

u(x)dx

= (x+1)∂

u(x) |

−1

−

−1

∂

u(x)dx= 0.

∂

u(−1) = −(x+1)∂

−1

N−3

∂

u(x) |

−1

(x+1)∂

u(x)dx

= (x+1)∂

u(x) |

−1

−

−1

∂

u(x)dx= 0.

DOI:10.12677/pm.2021.1161171034nØêÆ

{JŒ



u∈V

,…∀v∈V

∗

N−1

(∂

(

u−u),∂

= (P

N−3

∂

u−∂

u,∂

= 0,

Óž,ŽfP

∗

•÷v

(∂

∗

u(x))(−1) = ∂

−1

N−3

∂

u(−1) = ∂

u(−1).(2.3)

e5`²žm••lÑ:

éu‰½êK,·‚ò«mI

=(−1,1)©)¤K‡f«m:I

=(b

k−1

],h

−b

k−1

,k= 1,2,···,K,

−1 = b

<···<b

= 1.

-Ω = I

⊗I

,ω

α,β

= (b

−t)

(t−b

k−1

)

,˜ω

α,β

= (1−x)

(1+x)

•Ω

þ¼ê. P

(·,·)

Ω

,ω

,˜ω

Úk·k

Ω

,ω

,˜ω

©O•L

,˜ω

(Ω

)SÈÚ‰ê. ϕ•šKê, PH

)•˜„½Â

Sobolev ˜m,…‰ê†Œ‰ê©OP•k·k

,ϕ

Ú|·|

,ϕ

.Ú\©¡Sobolev ˜m:

) = {v: v

≡v|

∈H

),1 ≤k≤K},

ÙþŒ‰ê½Â•:

|v|

)

= (

k=1

ϕ,I

)

éu?¿êM

, ·‚À

) ˜‡f˜mX

Š•žm••þÁ&¼ê˜m, §

þ¤kgêØ‡LM

©¡õ‘ª|¤. À˜mY

−1

Š•žm••þu¼ê˜m,

= Y

∩

),Y

= {v: v|

∈P

),1 ≤k≤K},

Ù¥P

)L««mI

þ¤kgêØ‡LM

gõ‘ª¤˜m.

ÓžÚ\•N:

v(t) =¯v(

t),t=

t+b

k−1

),t∈I

t∈I

dþªŒ•,∂

v= (

)

∂

v,s= 1,2,···.

L

(

t)•gêØ‡LiLegendre õ‘ª,c

2i+1

,é1 ≤k≤K, ½Â:











(t) = c

(

t)+L

(

t));

(t) = c

i−1

(

t)−L

i+1

(

t)),1 ≤i≤M

−1;

˜ϕ

(t) = L

(

t),0 ≤i≤M

−1.

DOI:10.12677/pm.2021.1161171035nØêÆ

{JŒ

é0 ≤i≤M

−1,©O-ϕ

(t),˜ϕ

(t)•X

−1

þÄ¼ê,KéM

≥1,1 ≤k≤

K, k











= span{ϕ

(t),ϕ

(t),···,ϕ

−1

(t)},

−1

= span{˜ϕ

(t),˜ϕ

(t),···,˜ϕ

−1

(t)}.

½ÂÝKŽfP

) →P

)

÷v

u−u,v)

= 0,∀v∈P

)

½ÂÝKŽfP

) →P

)

÷v

u= u(b

k−1

−1

∂

u(s)ds.

½ÂÝKŽfP

:dP

1,M

) →P

))¤, Pu(t) |

= u

(t),…÷v

u(t) |

≡P

1,M

(t) := P

¯u

(

t).(2.4)

½ÂChebyshev-GaussŠŽf:

:C(

) →P

N,M

:C(

Ω

) →P

×P

÷v

u(x

) = u(x

),Π

N,M

u(x

) = u(x

Ù¥x

•I

þChebyshev-Gauss-Lobatto(CGL):, (x

)•Ω

þCGL :.

= V

⊗X

= V

∗

N−1

⊗Y

−1

2.2.nØ©Û¥I‡^Ún

Ún1.(„[13]:Lemma3.3)eu∈H

),…ϕ≥1,é1 ≤k≤K,Kk

1,M

u−uk

≤Ch

−ϕ

k∂

,ω

ϕ−1,ϕ−1

.(2.5)

Ún2.(„[14]:Lemma3.1)eα!β>−1,…u∈H

),Kk

DOI:10.12677/pm.2021.1161171036nØêÆ

{JŒ

k∂

α,β

u−u) k

,˜ω

α+s,β+s

≤CN

s−r

k∂

α,β

u−u) k

,˜ω

α+r,β+r

≤CN

s−r

k∂

,˜ω

α+r,β+r

,0 ≤s≤r.

(2.6)

Ún3.eu∈H

2,3

)∩H

),…σ≥3,Kk

k∂

∗

u−u) k

,˜ω

l−3,l−3

≤CN

l−σ

k∂

,˜ω

σ−3,σ−3

,0 ≤l≤3.(2.7)

y:(1) l= 3 ž,PI•ðŽf,d(2.2)ªÚ(2.6)ªŒ±á=

k∂

∗

u−u) k

=k(P

N−3

−I)∂

≤CN

3−σ

k∂

σ−3

N−3

−I)∂

,˜ω

σ−3,σ−3

= CN

3−σ

k∂

∗

−I)uk

,˜ω

σ−3,σ−3

≤CN

3−σ

k∂

,˜ω

σ−3,σ−3

(2)l= 2 ž,-g= ∂

∗

u−u),d(2.3)ª•, g∈H

).ÏLeHardy’s Øª[14],

kg˜ω

−1,−1

∈L

−1

(x)(1−x

)

β−2

dx≤sup{

1−β

(1−β)

}

−1

(∂

(x)(1−x

)

dx,

Ù¥v∈H

),β<1,d(2.6)ª

k∂

∗

u−u) k

˜ω

−1,−1

= (∂

∗

u−u),∂

∂

−1

[g˜ω

−1,−1

])

=|((P

N−3

−I)∂

u,∂

−1

[g˜ω

−1,−1

]) |

=|((P

N−3

−I)∂

u,(I−P

N−3

)∂

−1

[g˜ω

−1,−1

]) |

≤k(P

N−3

−I)∂

ukk(I−P

N−3

)∂

−1

[g˜ω

−1,−1

] k

≤CN

3−σ

k∂

σ−3

N−3

−I)∂

˜ω

σ−3,σ−3

−1

kg˜ω

−1,−1

˜ω

1,1

= CN

2−σ

k∂

∗

−I)uk

˜ω

σ−3,σ−3

kgk

˜ω

−1,−1

≤CN

2−σ

k∂

˜ω

σ−3,σ−3

kgk

˜ω

−1,−1

Ïd,l= 2 ž,(2.7)ª•¤á.

(3) l=1 ž, ÓnŒ-g=∂

∗

u−u), g∈H

). 2g$^þãHardy’s Øª, k

g˜ω

−2,−2

∈L

),d(2.6)ª

DOI:10.12677/pm.2021.1161171037nØêÆ

{JŒ

k∂

∗

u−u) k

˜ω

−2,−2

= (∂

∗

u−u),∂

∂

−2

[g˜ω

−2,−2

])

=|((P

N−3

−I)∂

∂

−2

[g˜ω

−2,−2

]) |

=|((P

N−3

−I)∂

u,(I−P

N−3

)∂

−2

[g˜ω

−2,−2

]) |

≤k(P

N−3

−I)∂

ukk(I−P

N−3

)∂

−2

[g˜ω

−2,−2

] k

≤CN

3−σ

k∂

σ−3

N−3

−I)∂

˜ω

σ−3,σ−3

−2

kg˜ω

−2,−2

˜ω

2,2

= CN

1−σ

k∂

∗

−I)uk

˜ω

σ−3,σ−3

kgk

˜ω

−2,−2

≤CN

1−σ

k∂

˜ω

σ−3,σ−3

kgk

˜ω

−2,−2

Ïd,l= 1 ž,(2.7)ª•¤á.

(4)l=0ž,ÓnŒ-g=P

∗

u−u,g∈H

).2g$^þãHardy’sØª,k

g˜ω

−3,−3

∈L

),d(2.6)ª

∗

u−uk

˜ω

−3,−3

= (P

∗

u−u,∂

∂

−3

[g˜ω

−1,−1

])

=|((P

N−3

−I)∂

u,(I−P

N−3

)∂

−3

[g˜ω

−3,−3

]) |

≤CN

3−σ

k∂

σ−3

N−3

−5thbfI)∂

˜ω

σ−3,σ−3

−3

kg˜ω

−3,−3

˜ω

3,3

= CN

−σ

k∂

∗

−I)uk

˜ω

σ−3,σ−3

kgk

˜ω

−3,−3

≤CN

−σ

k∂

˜ω

σ−3,σ−3

kgk

˜ω

−3,−3

Ïd,l= 0 ž,(2.7)ªE¤á.d,l•šêž,Œd˜mŠ¼(Ø. nþ,é(2.7)ªy

²dÜ¤.

3.Ê ‡©•§lÑ‚ª

!Ì‡QãÊ ‡©•§'užmõ«•ž˜Ì•{lÑ‚ª.˜m••æ^

Legendre-Petrov-Galerkin•{, žm••æ^õ«•Legendre-tau•{. À·Ä¼ê¦

XêÝDÕ,Jp¦)„Ý.

é1 ≤k≤K, e¡•Ä•§(1.1)3Ω

þž˜Ì‚ª:•3U

= U

(x,t) ∈W

,¦











(∂

,v)

Ω

+(∂

,∂

Ω

= (f,v)

Ω

, ∀v∈

(x,−1) = P

∗

(x).

(3.1)

W

= U

−P

∗

,K(3.1)ªŒ¤: •3W

∈W

,¦











(∂

,v)

Ω

+(∂

,∂

Ω

= (f−∂

∗

(x),v)

Ω

, ∀v∈

(x,−1) = 0.

(3.2)

DOI:10.12677/pm.2021.1161171038nØêÆ

{JŒ

-W

−1

i=0

N−5

j=0

˜w

,v=˜ϕ

,m= 0,1,···,M

−1,n= 0,1,···,N−5.(3.3)

òW

,v“\(3.2)ª, PW

= (˜w

)

×(N−4)

= F

,(3.4)

Ù¥

= (a

)

×M

= (ϕ

,˜ϕ

)

= (b

)

×M

= (ϕ

,˜ϕ

)

= (f

)

(N−4)×(N−4)

= (γ

,ψ

)

= (f

)

(N−4)×(N−4)

= (γ

000

,ψ

)

= (f

)

×(N−4)

= (f−∂

∗

(x),˜ϕ

)

Ω

••BOŽ, ©•Äžm•• þõ« •œ¹õ‘ªgêþ†ü«•œ¹õ‘ªgêƒ

Ó,…«måy©, =:M

= M,h

,k= 1,···,K.

qÏ•











k−1

(t)˜ϕ

(t)dt=

−1

(

t)˜ϕ

(

t)d

k−1

(t)˜ϕ

(t)dt=

−1

(

t)˜ϕ

(

t)d

(3.5)

k











= A,A = (a

)

M×M

= (ϕ

,˜ϕ

)

B,B = (b

)

M×M

= (ϕ

,˜ϕ

)

= (f

)

M×(N−4)

= (f−∂

∗

(x),˜ϕ

)

Ω

(3.6)

ò(3.6)ª“\(3.4)ª,Kk

.(3.7)

e¡•Äk= 1(1˜‡«m)¦)‚ª, Kk

.(3.8)

|^ÜþÈÎÒ⊗,(3.8)ªŒU¤

⊗A+

⊗B)w

,(3.9)

DOI:10.12677/pm.2021.1161171039nØêÆ

{JŒ

Ù¥w

©O•W

•þ/ª.

d(3.9)ªŒ±•§(1.1)31˜‡«mþêŠ):U

(x,t)=W

(x,t) + P

∗

(x),X

U

(x,−1)Š•1‡«mÐŠ, “\(3.7) ªŒ±•§(1.1)31‡«mþêŠ), ···,

Xd?1e·‚Œ±•§(1.1)3z‡«mþêŠ).

4.Ø©Û

!Ì‡élÑ‚ª(3.1)?1Ø©Û.

-U!U

©O••§(1.1)!•§(3.1)),du°()Ué•§(1.1)f/ªE¤á,é

1 ≤k≤K, keØ•§¤á:

(∂

(U−U

),v)

Ω

+(∂

(U−U

),∂

Ω

= 0, ∀v∈

.(4.1)

Pˆu

= P

∗

1,M

U,˜u

= U

−ˆu

,´•˜u

(x,−1) = 0.(4.2)

ò(4.2)ª“\(4.1)ª,

(∂

˜u

,v)

Ω

+(∂

˜u

,∂

Ω

= (∂

(U−ˆu

),v)

Ω

+(∂

(U−ˆu

),∂

Ω

.(4.3)

du(2.1)ª!(2.4)ª,þªmàŒ±{z¤

(∂

(U−ˆu

),v)

Ω

+(∂

(U−ˆu

),∂

Ω

= (∂

(U−P

∗

1,M

U),v)

Ω

+(∂

(U−P

∗

1,M

U),∂

Ω

= (∂

(U−P

∗

U),v)

Ω

+(∂

(U−P

1,M

U),∂

Ω

(4.4)

-v= ω

0,−1

˜ω

0,−1

˜u

,“\(4.3)ª†à, …d(3.5)ª•,k

(∂

˜u

,v)

Ω

(∂

˜u

,˜u

)

Ω,ω

0,−1

,˜ω

0,−1

k˜u

Ω,ω

0,−2

,˜ω

0,−1

k˜u

(x,1) k

,˜ω

0,−1

.(4.5)

(∂

˜u

,∂

Ω

(3∂

v+(1+x)∂

v,∂

Ω,ω

0,1

k∂

Ω,ω

0,1

k∂

v(1,t) k

,ω

0,1

(4.6)

òv= ω

0,−1

˜ω

0,−1

˜u

“\(4.3)ªmà,…d(3.5)ªÚd–]Øª

|(∂

(U−P

∗

U),v)

Ω

|≤

k∂

(U−P

∗

U) k

Ω,ω

0,−1

,˜ω

0,−1

k˜u

Ω,ω

0,−1

,˜ω

0,−1

.(4.7)

|(∂

(U−P

1,M

U),∂

Ω

|≤

k∂

(U−P

1,M

U) k

Ω,ω

0,−1

,˜ω

0,−1

k˜u

Ω,ω

0,−1

,˜ω

0,−1

.(4.8)

ò(4.4)ª!(4.5)ª!(4.6ª)!(4.7)ª!(4.8)ª“\(4.3)ª,k

DOI:10.12677/pm.2021.1161171040nØêÆ

{JŒ

k˜u

Ω,ω

0,−2

,˜ω

0,−1

k˜u

(x,1) k

,˜ω

0,−1

k∂

Ω,ω

0,1

k∂

v(1,t) k

,ω

0,1

≤k∂

(U−P

∗

U) k

Ω,ω

0,−1

,˜ω

0,−1

k˜u

Ω,ω

0,−1

,˜ω

0,−1

k∂

(U−P

1,M

U) k

Ω,ω

0,−1

,˜ω

0,−1

k˜u

Ω,ω

0,−1

,˜ω

0,−1

.(4.9)

k˜u

Ω,ω

0,−2

,˜ω

0,−1

≤4 k∂

(U−P

∗

U) k

Ω,ω

0,−1

,˜ω

0,−1

k∂

(U−P

1,M

U) k

Ω,ω

0,−1

,˜ω

0,−1

.(4.10)

qdÚn1!Ún3Ú(3.5)ª•,(4.10)ªŒz¤

k˜u

Ω,ω

0,−2

,˜ω

0,−1

≤CN

−σ

k∂

∂

Ω,ω

0,−1

,˜ω

σ−3,σ−3

−ϕ

k∂

∂

Ω,ω

ϕ−1,ϕ−1

,˜ω

0,−1

.(4.11)

Ïd,dL

½n1.U!U

))∩L

)∩

2,3

)),Ké1 ≤k≤K,•3~êC,¦

kU−U

Ω,ω

0,−2

,˜ω

0,−1

≤CN

−σ

kUk

σ,ϕ

(Ω)

−ϕ

kUk

(Ω)

(4.12)

Ù¥

kUk

σ,ϕ

(Ω)

= k∂

Ω,ω

0,−2

,˜ω

σ−3,σ−3

+ k∂

∂

Ω,ω

0,−1

,˜ω

σ−3,σ−3

−ϕ

k∂

∂

Ω,ω

ϕ−1,ϕ−1

,˜ω

σ−3,σ−3

kUk

(Ω)

=k∂

Ω,ω

ϕ−1,ϕ−1

,˜ω

0,−1

+ k∂

∂

Ω,ω

ϕ−1,ϕ−1

,˜ω

0,−1

kU−U

Ω,ω

0,−2

,˜ω

0,−1

=kU−(ˆu

+ ˜u

) k

Ω,ω

0,−2

,˜ω

0,−1

≤kU−ˆu

Ω,ω

0,−2

,˜ω

0,−1

+ k˜u

Ω,ω

0,−2

,˜ω

0,−1

=kU−P

∗

1,M

Ω,ω

0,−2

,˜ω

0,−1

+ k˜u

Ω,ω

0,−2

,˜ω

0,−1

.(4.13)

DOI:10.12677/pm.2021.1161171041nØêÆ

{JŒ

kU−P

∗

1,M

Ω,ω

0,−2

,˜ω

0,−1

=k(I−P

∗

)U−(I−P

∗

)(I−P

1,M

)U+(I−P

1,M

)Uk

Ω,ω

0,−2

,˜ω

0,−1

≤k(I−P

∗

)Uk

Ω,ω

0,−2

,˜ω

0,−1

+ k(I−P

∗

)(I−P

1,M

)Uk

Ω,ω

0,−2

,˜ω

0,−1

+ k(I−P

1,M

)Uk

Ω,ω

0,−2

,˜ω

0,−1

.(4.14)

dÚn1!Ún3Ú(3.5)ª•,(4.14)ªŒn¤

kU−P

∗

1,M

Ω,ω

0,−2

,˜ω

0,−1

≤CN

−σ

k∂

Ω,ω

0,−2

,˜ω

σ−3,σ−3

+CN

−σ

−ϕ

k∂

∂

Ω,ω

ϕ−1,ϕ−1

,˜ω

σ−3,σ−3

−ϕ

k∂

Ω,ω

ϕ−1,ϕ−1

,˜ω

0,−1

.(4.15)

(Ü(4.11)ªÚ(4.15)ª,(4.13)ªŒn¤

kU−U

Ω,ω

0,−2

,˜ω

0,−1

≤CN

−σ

kUk

σ,ϕ

(Ω)

−ϕ

kUk

(Ω)

nþ,½n1(Ø¤á.

5.êŠ(J

ŠŽ~mà‘Úš‚5‘æ^CGL Š?n,¿¦^¯„Fp“C†(FFT)?1OŽ.

~1.•ÄXeÊ•§:











∂

U+∂

U= f(x,t),x∈I

,t∈I

;

U(±1,t) = ∂

U(1,t) = 0,t∈I

;

U(x,−1) = U

(x),x∈I

(5.1)

Ù¥I

= (−1,1),I

= (−1,1),U= U(x,t).

U(x,t) = (x−1)tcos

(

πx

),x∈(−1,1),t∈(−1,1]

••§(5.1)°(),K•§ÐŠU

(x),mà‘f©O•:

(x) = −(x−1)cos

(

πx

DOI:10.12677/pm.2021.1161171042nØêÆ

{JŒ

x−1

t−

t)cos(πx)+

x−1

t(1−π

)sin(πx).

dêŠŽ~¢(Jdã1-ã3‰Ñ.

ã¥L

∞

-error½ÂXe:

kU−U

∞

= max

i,j

|U(x

)−U

) |,

kU−U

∞

= max

i,k

|U(x

)−U

) |,

Ù¥(x

)!(x

) ©O•Ω ÚΩ

þCGL :,i=0,1,···,M−1,k=0,1,···,M

−1,j=

0,1,···,N−5. ML«žm••ü«•õ‘ªgê, M

L«žm••f«•õ‘ªgê, N

L«˜m••õ‘ªgê.

Figure1.Example1Convergenceofsingledomainintime

ã1.~1žmü«•Âñ5

Figure2.Example1Convergenceoffourdomainsintime

ã2.~1žmo«•Âñ5

~2.•ÄXeÊ•§:

DOI:10.12677/pm.2021.1161171043nØêÆ

{JŒ

Figure3.Example1Convergenceofsixdomainsintime

ã3.~1žm8«•Âñ5











∂

U+∂

U+UU

= f(x,t),x∈I

,t∈I

;

U(±1,t) = ∂

U(1,t) = 0,t∈I

;

U(x,−1) = U

(x),x∈I

(5.2)

Ù¥I

= (−1,1),I

= (−1,1),U= U(x,t).

´••§(5.2)žmõ«•ž˜Ì‚ª'…3u?nš‚5‘UU

,džŒæ^š‚5‘3

CGL:þŠéÙ?1OŽ,T•{(ÜChebyshev˜{¯„C†OŽ`:†Legendre

•{ûÐ-½5.•§(5.2)Ù¦‘?n•ª†•§(1.1)ƒÓ.

3é•§(5.2)žmõ«•ž˜Ì‚ª?1êŠOŽž,·‚æ^S“{?1¦),=:S“

gê•l,S“Ø•ε,•gé•§(5.2)žmõ«•ž˜Ì‚ª?1S“,cügS“Ø

•Œ5uεž,ªŽS“.

Figure4.Example2Convergenceofsingledomainintime

ã4.~2žmü«•Âñ5

DOI:10.12677/pm.2021.1161171044nØêÆ

{JŒ

U(x,t) = (x−1)sin

(πx)e

−t

,x∈(−1,1),t∈(−1,1]

••§(5.2)°(),K•§ÐŠU

(x),mà‘f©O•:

(x) = (x−1)sin

(πx)e,

f=2π

(x−20π

−1)e

−t

cos(2πx)+(x−1)e

−t

sin

(πx)(e

−t

sin

(πx)+π(x−1)e

−t

sin(2πx)−1)

+2π(8π

x−8π+1)e

−t

sin(2πx).

S“#NØ•ε= 10

−12

,S“gê•l= 40.dêŠŽ~¢(Jdã4-ã6‰Ñ.

Figure5.Example2Convergenceoffourdomainsintime

ã5.~2žmo«•Âñ5

Figure6.Example2Convergenceofsixdomainsintime

ã6.~2žm8«•Âñ5

žmõ«•ž˜Ì‚ª3žm†˜mþþ÷vÌÂñ. dM= 8,N=64,K= 6 ž,~1!~

DOI:10.12677/pm.2021.1161171045nØêÆ

{JŒ

2 žm8«•Øþ'žmü«••, ÙØÑˆ10

−14

þ?. 8Œ±æ^žmõ«•

ž˜Ì•{5¦)da•§,~Ø.

6.(åŠ

Ä7‘8

I[g,‰ÆÄ7(11971016)"

ë•©z

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SivashinskyEquationsandtheIto-TypeCoupledKdVEquations.ComputerMethodsinAp-

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{JŒ

[9]Xu,Y.andShu,C.W.(2004)LocalDiscontinuousGalerkinMethodsforThreeClassesof

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