equations via hodograph transformations Discontinuous Galerkin methods for short pulse type Journal of Computational Physics

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Journal of Computational Physics

www.elsevier.com/locate/jcp

Discontinuous Galerkin methods for short pulse type equations via hodograph transformations

Qian Zhang, Yinhua Xia

^∗,1

SchoolofMathematicalSciences,UniversityofScienceandTechnologyofChina,Hefei,Anhui230026,PRChina

a r t i c l e i n f o a b s t r a c t

Articlehistory:

Received9January2019

Receivedinrevisedform19August2019 Accepted3September2019

Availableonline11September2019

Keywords:

DiscontinuousGalerkinmethod Shortpulseequation Nonclassicalsolitonsolution Conservativescheme Hodographtransformation

In thepresent paper, weconsider the discontinuousGalerkin (DG) methodsfor solving shortpulse(SP)typeequations.Theshortpulseequationhasbeenshowntobecompletely integrable, which admits the loop-soliton,cuspon-soliton solutions as well as smooth- solitonsolutions.Throughhodographtransformations,thesenonclassicalsolutionscanbe profiledas the smoothsolutions ofthe coupleddispersionless (CD)systemorthe sine- Gordon equation. Therefore, DG methods can be developed for the CD system or the sine-Gordonequationto simulatethe loop-solitonorcuspon-solitonsolutions ofthe SP equation. The conservativenessordissipation ofthe Hamiltonianor momentumforthe semi-discrete DG schemes can be proved. Also we modify the above DG schemes and obtain an integration DG scheme. Theoretically the a-priori error estimates have been provided forthemomentumconserved DG schemeand theintegrationDG scheme.We alsoproposetheDGschemeandtheintegrationDGschemeforthesine-Gordonequation, incasetheSPequationcannotbetransformedtotheCDsystem.AlltheseDGschemes canbeappliedtothegeneralizedormodifiedSPtypeequations.Numericalexperiments areprovidedtoillustratetheoptimalorderofaccuracyandcapabilityoftheseDGschemes.

©^{2019ElsevierInc.Allrightsreserved.}

1. Introduction

Inthispaper,wemainlystudytheclassicshortpulse(SP)equationderivedbySchäferandWaynein[27]

uxt

=

^u

+

¹

6

(

u³

)

xx

.

(1.1)

The SP equation models the propagationof ultra-short light pulses in silica optical fibers. Here, u∈R is a real-valued functionwhichrepresentsthemagnitudeoftheelectricfield.Itiswell-knownthat thecubicnonlinearSchrödinger(NLS) equation derived fromthe Maxwell’sequation can describethe propagationofpulse inoptical fibers.Twopreconditions ofthisderivationneedtobe satisfied:First,theresponseofthematerial attainsaquasi-steady-stateandsecondthat the pulsewidthisaslargeastheoscillationofthecarrierfrequency.Andnowwecancreateveryshortpulsesbytheadvanced technology andthe pulsespectrum isnot narrowly localizedaround the carrierfrequency, that is,when the pulseis as shortasafewcyclesofthecentralfrequency.Numericalexperimentsmadein[8] showasthepulseshortens,theaccuracy ofthe SPequation approximatedto Maxwell’sequation increases, however,theNLS equation becomes inaccurateforthe

*

Correspondingauthor.

E-mailaddresses:gelee@mail.ustc.edu.cn(Q. Zhang),yhxia@ustc.edu.cn(Y. Xia).

1 ResearchsupportedbyNSFCgrantNo.11871449,andagrantfromLaboratoryofComputationalPhysics(No.6142A0502020817).

https://doi.org/10.1016/j.jcp.2019.108928

0021-9991/©^{2019ElsevierInc.Allrightsreserved.}

ultra-shortpulse.Therefore,weuseSPequation toapproximatetheultra-shortlightpulse.Ifthepulseisasshortasonly onecycleofitscarrierfrequency,thenthemodifiedshortpulseequationin[30] isusedtodescribethepropagationofpulse in opticalfibers. Similarto theextension ofcoupled nonlinearSchrödinger equationsfromNLS equations,itis necessary to consider itstwo-component ormulti-component generalizationsfordescribing the effectofpolarization oranisotropy [10,24,31].Forbirefringentfibers,theauthorsin[11,14] alsointroducedsomeextensionsoftheSPequationtodescribethe propagationofultra-shortpulse.WewillintroducetheseextensionsspecificallyinSection3.

Integrablediscretizationsofshortpulsetypeequationshavereceivedconsiderableattentionrecently,especiallytheloop- soliton,antiloop-solitonandcuspon-solitonsolutionsin[11,13,14,12,31].Theauthorslinkedtheshortpulsetypeequations with the coupled dispersionless (CD) type systems or the sine-Gordon type equations through the hodograph transfor- mations. The key of the discretization is an introduction of a nonuniform mesh, which plays a role of the hodograph transformations asin continuous case. In this paper,we aim at solving the loop-soliton,cupson-soliton solutions of the shortpulsetypeequationsaswellassmooth-solitonsolutions.Throughthehodographtransformation(x,t)→(y,s)which wasproposedin[28],

_∂

∂x

=

¹_ρ∂^∂_y

,

∂

∂t

=

_∂^∂_s

+

^u_2ρ² _∂^∂_y

,

wecanestablishthelinkbetweentheSPequation(1.1) andtheCDsystem[19],

ρ

s

+ (

¹

2u²

)

y

=

⁰

,

uys

= ρ

u

.

There existssome shortpulsetype equationswhicharefailedto betransformedintoCDsystems.Therefore,we consider analternativeapproachbyintroducinganewvariablezanddefineanotherhodographtransformation,

_∂

∂x

= (

cosz

)

⁻¹_∂^∂_y

,

∂

∂t

=

_∂^∂_s

+

¹₂^z2_s

(

cosz

)

⁻¹_∂^∂_y

,

whichconnectstheshortpulseequation(1.1) withthesine-Gordonequation[30]

zys

=

^sinz

.

For the CD system or the sine-Gordon equation, we develop the discontinuous Galerkin (DG) schemes to obtain the high-order accuracy numericalsolution uh(y,s) orzh(y,s).Consequently,a point-to-point profile uh(x,t) ofloop-soliton, cuspon-solitonsolutionsoftheSPequationcanbeobtained,whichareshownbythenumericalexperimentsinSection 4.

TheDGmethodwasfirstintroducedin1973byReedandHillin[26] forsolvingsteadystatelinearhyperbolicequations.

The importantingredientofthismethodisthedesignofsuitableinter-elementboundarytreatments (socallednumerical fluxes) to obtainhighly accurate andstableschemes in manysituations.Withinthe DGframework, the methodwas ex- tended to deal with derivatives oforder higher than one, i.e. local discontinuous Galerkin (LDG) method.The first LDG methodwasintroducedbyCockburnandShuin[7] forsolvingconvection-diffusionequation.Theirworkwasmotivatedby the successfulnumericalexperimentsofBassiandRebay[2] forcompressibleNavier-Stokes equations.Later,YanandShu developedanLDGmethodforageneralKdVtypeequationcontainingthirdorderderivativesin[42],andtheygeneralized the LDGmethodto PDEswithfourthandfifthspatial derivativesin[43].Levy,ShuandYan[21] developedLDGmethods fornonlineardispersiveequationsthathavecompactlysupportedtravelingwavesolutions,theso-calledcompactons.More recently,XuandShufurthergeneralizedtheLDGmethodtosolveaseriesofnonlinearwaveequations[37–40,45].Werefer tothereviewpaper[36] ofLDGmethodsforhigh-ordertime-dependentpartialdifferentialequations.

Most recently, a series of schemes which called structure-preserving schemes have attracted considerable attention.

For some integrable equationslike KdV type equations [9,20,22,46], Zakharovsystem [34], Schrödinger-KdV system[35], Camassa-Holm equation [44], etc., the authors proposed various conservative numerical schemes to“preserve structure”.

Theseconservativenumericalschemeshavesome advantagesoverthedissipativeones,forexample,theHamiltoniancon- servativenesscan help reducethe phaseerroralong the longtime evolutionandhavea moreaccurate approximation to exact solutionsforKdVtypeequations[46].The CDsystemandthegeneralizedCDsystemare integrable,thusthey have aninfinitenumberofconservedquantities[18].FortheCDsystem,thefollowingtwoinvariants

H₀

=

ρ

u²dy

,

H₁

=

ρ

²

+

^u2ydy

,

arecorrespondingtotheHamiltonianE0,E1 oftheSPequation[3,4] viathehodographtransformation, E0

=

u²dx

,

E1

=

1

+

u²_xdx

.

In thispaper, we first construct E₀ conserved DG schemefor theSP equation directly. Forthe loop-solitonandcuspon- soliton solutions, the H0, H1 conserved DG schemes for CD system are developed respectively, to profile the singular

solutions of the SP equation. Alsowe modify the above DG schemes and propose an integration DG schemewhich can numerically achieve the optimal convergence rates for

ρ

,u, and uy. Theoretically, we prove that the H1 conserved DG schemehastheoptimalorderofaccuracyfor

ρ

,uanduyin L² norm.TheintegrationDGschemecanbeprovedtheopti- malorderofaccuracyfor

ρ

,u_y inL² normandthesuboptimalorderofaccuracyforu inL^∞ norm.AlltheseDGschemes canbeadoptedtothegeneralizedormodifiedSPtypeequations.

The restofthis paperis organized asfollows.InSection 2, we develop theDG schemes forthe SPequation directly, andvia the hodographtransformations forthe CDsystemand thesine-Gordon equation. Some notations forsimplifying expressions are givenin Section 2.1. In Section 2.2, we first propose the E0 conserved DG scheme forthe SP equation.

To simulatethe loop-solitonor cuspon-soliton solutions ofthe SP equation, the H0, H1 conserved DG schemes andthe integrationDG schemeare constructedforthe CDsystemwhichlinksthe SPequation by thehodograph transformation.

Meanwhile, the a priori error estimates for H1 conserved DG and integration DG schemes are also provided. Moreover, we develop two kindsof DG schemes forthe sine-Gordon equation to introduce another resolution forthe SP equation inSection 2.3.Section3 isdevotedto summarizethegeneralizedshortpulseequationsandintroducethe corresponding conservedquantitiesbriefly.SeveralnumericalexperimentsarelistedinSection4,includingthepropagationandinteraction ofloop-soliton,cuspon-solution, breathersolution oftheshortpulsetype equations.We alsoshow the accuracyandthe changeofconservedquantitiesinSection4.Finally,someconcludingremarksaregiveninSection5.

2. ThediscontinuousGalerkindiscretization

Inthissection,wepresentthediscontinuousGalerkindiscretizationforsolvingtheshortpulsetype equations.Inorder todescribethemethods,wefirstintroducesomenotations.

2.1. Notations

We denotethemesh Th onthespatial y by Ij= [^y_j−¹

2,y_j+1

2] ^{for j}=¹,. . . ,N,withthecell centerdenoted by yj=

1 2(y_j−1

2 +^y_j+¹

2).Thecellsizeisy_j=^y_j+¹

2 −^y_j−¹

2 andh= ^max

1≤^j≤^N y_j.Themeshisregularinthesensethat theratio betweenthemaximumandtheminimummeshsizesstaysboundedduringmeshrefinements.Thefiniteelementspaceas thesolutionandtestfunctionspaceconsistsofpiecewisepolynomials

V_h^k

= {

^v

:

^v

|

I_j

∈

^Pk

(

Ij

) ;

¹

≤

^j

≤

^N

} ,

where P^k(I_j) denotes the set of polynomial of degree up to k defined on the cell I_j. Notably, the functionsin V_h^k are allowed tobe discontinuous acrosscellinterfaces.The valuesofu at y_j+1

2 aredenoted bythe u⁻ j+¹2

andu⁺ j+¹2

, fromthe leftcell I_jandtherightcell I_j+1respectively.Additionally,thejumpofuisdefinedasJ^uK=^u+−^{u−,theaverageofuas} {{^u}}=¹₂(u⁺+^u−).Tosimplifyexpressions,weadopttheroundbracketandanglebracketforthe L² inner productoncell Ijanditsboundary

(

u

,

v

)

I_j

=

Ij

uvdy

,

<

u

ˆ ,

v

>

I_j

= ˆ

^u_j+¹

2v⁻

j+¹₂

− ˆ

^u_j−¹

2v⁺

j−¹₂

,

foronedimensionalcase.

Forthespatialvariablex,wedenotethemeshT_h by I_j= [^xj−¹2,x_j+1

2]^{for j}=¹,. . . ,N.Similartothenotationsonthe mesh Th,we have x_j,x_j,andh= ^max

1≤^j≤^N x_j.We assume that the mesh onthe coordinate x isalso regular.Without misunderstanding,westilluseu⁻

j+¹₂ andu⁺

j+¹₂ denotethevaluesofuatx_j+1

2,fromtheleftcell I_j andtherightcellI_j+1 respectively.

2.2. Theshortpulseequation

Recalltheshortpulseequation uxt

=

u

+

¹

6

(

u³

)

xx

,

x

∈

I

= [

xL

,

xR

],

(2.1)

where u(x,t)∈R is a real-valued function, t denotes the temporal coordinate and x is the spatial scale. Through the hodographtransformation,itcanbeconvertedintoacoupleddispersionless(CD)system

ρ

s

+ (

¹₂u²

)

y

=

⁰

,

u_ys

= ρ

u

,

^(2.2)

wheresdenotesthetemporalcoordinate,andyisthespatialscale,y∈^I= [^yL,yR]^{.Thehodograph}transformation(y,s)→ (x,t)isdefinedby

_∂

∂y

= ρ

_∂^∂_x

,

∂

∂s

=

_∂^∂_t

−

^u₂²_∂^∂_x

.

Subsequently,theparametricrepresentationofthesolutionoftheshortpulseequation(2.1) is

u

=

^u

(

y

,

s

),

x

=

^x

(

y₀

,

s

) +

y y₀

ρ (ζ,

s

)

d

ζ ,

where y₀ isa realconstant.Sincetheshortpulseequation andthe equivalentCDsystemarecompletely integrable,they haveaninfinitenumberofconservationlaws.ThefirsttwoinvariantsoftheSPequationaredescribedby

E0

=

u²dx

,

E1

=

¹

+

^u2xdx

,

andthecorrespondingconservationlawsfortheCDsystemare H₀

=

ρ

u²dy

,

H₁

=

ρ

²

+

^u2ydy

.

2.2.1. E0conservedscheme

Toconstructthediscontinuous GalerkinmethodfortheSPequationdirectly,we rewritetheSPequation (2.1) asafirst ordersystem:

⎧ ⎪

⎨

⎪ ⎩

vt

=

u

+ ω

x

,

v

=

^ux

, ω = (

¹₆u³

)

x

.

(2.3)

The local DG scheme for equations (2.3) is formulated as follows: Find uh,vh,

ω

h∈^V_h^{k such that, for all test functions}

ϕ

,φ,ψ∈^V_h^kandI_j∈T_h

⎧ ⎪

⎪ ⎨

⎪ ⎪

⎩

((

vh

)

t

, ϕ )

_I

j

= (

uh

, ϕ )

_I

j

+ < ω

h

, ϕ >

_I

j

−( ω

h

, ϕ

x

)

_I

j

,

(a)

(

v_h

, φ)

_I

j

=<

^u

h

, φ >

_I

j

−(

^uh

, φ

x

)

_I

j

,

(b)

( ω

h

, ψ )

_I

j

=<

f

(

u_h

), ψ >

_I

j

−(

^f

(

u_h

), ψ

x

)

_I

j

,

(c)

(2.4)

where f(u)=¹₆^{u3.The“hat”termsintheschemearetheso-called“numericalfluxes”,whicharefunctionsdefinedonthe} cellboundaryfromintegrationbypartsandshouldbedesignedbasedondifferentguidingprinciplesfordifferentPDEsto ensurethestabilityandlocalsolvabilityoftheintermediatevariables.ToensuretheschemeisE0conserved,thenumerical fluxeswetakeare

⎧ ⎪

⎨

⎪ ⎩

ω

h

= {{ ω

h

}},

u

_h

= {{

^uh

}},

f

(

u

) =

_JF(u)K

J^uK

, J

^u

K =

⁰

,

f

({{

u

}}), J

u

K =

0

,

(2.5)

where F(u)=u

f(

τ

)d

τ

.Numerically, E0 conserved DGschemecan achieve(k+¹)-thorderofaccuracy foreven k,and k-thorder ofaccuracy foroddk on uniformmeshes. With nonuniformmeshes, the E₀ conservedDG scheme onlyhave suboptimalorderofaccuracyregardlessoftheparityofthepolynomialdegreesk.

Proposition2.1.(Energyconservation)TheDGscheme(2.4)withthenumericalfluxes(2.5)fortheshortpulseequation(2.1)satisfies theenergyconservativeness

d

dtE0

(

u_h

) =

⁰

.

Proof. Fortheequation(2.4b),wetakethetimederivativeandget

((

vh

)

t

, η )

_I

j

=< (

uh

)

_t

, η >

_I

j

−((

uh

)

t

, η

x

)

_I

j

.

(2.6)

Since(2.6),and(2.4a)-(2.4c)holdforanytestfunctionsinV_h^k,wecanchoose

ϕ = (

u_h

)

t

, η = −(

^uh

)

t

, ψ = −

^uh

,

anditfollowsthat

((

vh

)

t

, (

uh

)

t

)

_I

j

= (

uh

, (

uh

)

t

)

_I

j

+ < ω

h

, (

uh

)

t

>

_I

j

−( ω

h

, (

uh

)

tx

)

_I

j

,

(2.7)

− ((

v_h

)

t

, (

u_h

)

t

)

_I

j

= − < (

u_h

)

_t

, (

u_h

)

t

>

_I

j

+ ((

u_h

)

t

, (

u_h

)

tx

)

_I

j

,

(2.8)

− ( ω

_h

,

u_h

)

_I

j

= − <

_f

(

u_h

), (

u_h

) >

_I

j

+ (

f

(

u_h

), (

u_h

)

x

)

_I

j

.

(2.9)

Toeliminateextraterms,wetaketestfunctions

ϕ

= −

ω

h in(2.4a),

η

=

ω

h in(2.6),andthenobtain

−((

^vh

)

t

, ω

h

)

_I

j

= −(

^uh

, ω

h

)

_I

j

− < ω

h

, ω

h

>

_I

j

+( ω

h

, ( ω

h

)

x

)

_I

j

,

(2.10)

((

vh

)

t

, ω

h

)

_I

j

=< (

uh

)

_t

, ω

h

>

_I

j

−((

uh

)

t

, ( ω

h

)

x

)

_I

j

.

(2.11)

Withthesechoicesoftestfunctionsandsummingupthefiveequationsin(2.7)-(2.11),weget

(

u_h

, (

u_h

)

t

)

_I

j

+ < ω

_h

, (

u_h

)

t

>

_I

j

− ( ω

_h

, (

u_h

)

tx

)

_I

j

− < (

u_h

)

_t

, (

u_h

)

t

>

_I

j

+ ((

u_h

)

t

, (

u_h

)

tx

)

_I j

+ < (

u_h

)

_t

, ω

_h

>

_I

j

− ((

u_h

)

t

, ( ω

_h

)

x

)

_I

j

− <

_f

(

u_h

), (

u_h

) >

_I

j

+ (

f

(

u_h

), (

u_h

)

x

)

_I j

− < ω

_h

, ω

_h

>

_I

j

+( ω

_h

, ( ω

_h

)

x

)

_I

j

=

0

.

(2.12)

Nowtheequation(2.12) canberewrittenintofollowingform

(

u_h

, (

u_h

)

t

)

_I

j

+

_j+1

2

−

_j−1 2

+

_j−1

2

=

⁰

,

(2.13)

wherethenumericalentropyfluxisgivenby

= ω

_h

(

u⁻_h

)

t

− ω

⁻_h

(

u_h⁻

)

t

+

¹

2

(

u⁻_h

)

²_t

− (

u_h

)

_t

(

u_h⁻

)

t

+ (

u_h

)

_t

ω

⁻_h

−

¹

2

( ω

⁻_h

)

²

+ ω

h

ω

⁻_h

−

f

(

u_h

)

u_h⁻

+

^F

(

u⁻_h

),

andtheextratermis

= − ω

_h

_J (

u_h

)

_t

K − (

u_h

)

_t

J

^wh

K + J

^wh

(

u_h

)

_t

K +

_f

(

u_h

) J

^uh

K − J

^F

(

u_h

) K + ( (

u_h

)

_t

− {{ (

u_h

)

_t

}} ) J (

u_h

)

_t

K + ( − ω

_h

_{+ {{} ω

_h

_}} ) J ω

_h

_{K =}

0

,

whichvanishesduetothechoiceoftheconservativenumericalfluxes(2.5).Summingupthecellentropyequalities(2.13) withtheperiodicorhomogeneousDirichletboundaryconditions,itimpliesthat

(

u_h

, (

u_h

)

t

)

I

=

⁰

.

Thus,theDGscheme(2.4) fortheshortpulseequationisE0 conserved. 2

The E₀ conservedschemeresolvesthesmooth solutionsfortheshortpulseequationefficiently,asshowninSection 4.

However, for the loop-soliton and cuspon-soliton solutions, this scheme can not be used because of the singularity of solutions.Therefore,weintroducetheDGschemesviahodographtransformationsinthefollowingsections.

2.2.2. H0conservedDGscheme

Aswehavementioned,theshortpulseequation canbe convertedintothecoupleddispersionless(CD)systemthrough thehodographtransformation.ToconstructthelocaldiscontinuousGalerkinnumericalmethodfortheCDsystem,wefirst rewrite(2.2) asafirstordersystem

⎧ ⎪

⎪ ⎪

⎨

⎪ ⎪

⎪ ⎩

ρ

s

+ γ

y

=

⁰

, ω

s

= ρ

u

, ω ₌

uy

, γ =

¹₂u²

.

ThenwecanformulatetheLDGnumericalmethodasfollows:Findu_h,

ρ

h,

ω

h,

γ

h∈^V_h^ksuchthat

⎧ ⎪

⎪ ⎪

⎨

⎪ ⎪

⎪ ⎩

(( ρ

h

)

s

, φ)

Ij

+ < γ

h

, φ >

Ij

−( γ

h

, φ

y

)

Ij

=

⁰

,

(a)

(( ω

h

)

s

, ϕ )

Ij

= ( ρ

hu_h

, ϕ )

Ij

,

(b)

( ω

h

, ψ )

Ij

=<

u

_h

, ψ >

Ij

−(

^uh

, ψ

y

)

Ij

,

(c)

( γ

h

, η )

Ij

= (

¹₂u²_h

, η )

Ij (d)

(2.14)

foralltestfunctionsφ,

ϕ

,ψ,

η

∈^V_h^{k andI}j∈Th.ToguaranteetheconservativenessofH₀,weadoptthecentralnumerical fluxes

γ

h

= {{ γ

h

}},

u

_h

= {{

^uh

}}.

^(2.15)

Here,theDirichletboundaryconditionisimposed.Numerically,onuniformmeshes,wewillseethattheoptimal(k+¹)-th order ofaccuracy canbeobtainedforu_h,

ρ

h whenk iseven,however,thenumericalsolutions u_h,

ρ

_h havek-thorderof accuracywhenkisodd.Ifwemodifynumericalfluxesasbelow:

γ

_h

_{= {{} γ

_h

_{}} −} α _J ρ

_h

_{K −} β J γ

_h

_K ,

u

_h

= {{

^uh

}} + μ _J

u_h

K ,

(2.16) thentheschemeisdissipativeon H0 withtheappropriateparametersinProposition2.2andtheoptimalorderofaccuracy canbeachievednumericallyforthisH0 dissipativescheme.

Proposition 2.2.(H₀ conservation/dissipation) The semi-discrete DG numerical scheme (2.14), (2.15) can preserve quantity H0(

ρ

_h,u_h)=

I

ρ

_hu²_h dy spatially.Thescheme(2.14)with(2.16)composesadissipativeDGschemeon H0 iftheparameters in (2.16)satisfytheconditions

α ₌

0

, β ≥

⁰

, μ _≥

0and

β + μ ₌

0

.

Proof. First, we take time derivative of equation (2.14c), and the test functionsare chosen as φ=

γ

h,

ϕ

= −(u_h)s,ψ = (uh)s,

η

_{= −}(

ρ

h)s.Thenwehave

(( ρ

_h

)

_s

, γ

_h

)

_Ij

+ < γ

_h

, γ

_h

>

_Ij

− ( γ

_h

, ( γ

_h

)

_y

)

_Ij

=

⁰

,

(2.17)

− (( ω

_h

)

s

, (

u_h

)

s

)

I_j

+ ( ρ

_hu_h

, (

u_h

)

s

)

I_j

=

⁰

,

(2.18)

(( ω

_h

)

s

, (

u_h

)

s

)

I_j

− < (

u_h

)

s

, (

u_h

)

s

>

I_j

+ ((

u_h

)

s

, (

u_h

)

sy

)

I_j

=

⁰

,

(2.19)

−( γ

h

, ( ρ

h

)

s

)

I_j

+ (

¹

2u²_h

, ( ρ

h

)

s

)

I_j

=

0

.

(2.20)

Summingupallequalities(2.17)-(2.20),weobtain

( ρ

hu_h

, (

u_h

)

s

)

Ij

+ (

¹

2u²_h

, ( ρ

h

)

s

)

Ij

+

< γ

_h

, γ

_h

>

I_j

− ( γ

_h

, ( γ

_h

)

y

)

I_j

− < (

u_h

)

s

, (

u_h

)

s

>

I_j

+ ((

u_h

)

s

, (

u_h

)

sy

)

I_j

=

⁰

,

whichcanbewrittenas 1

2 d ds

I_j

ρ

hu²_hdy

+

_j+1 2

−

_j−1

2

+

_j−1

2

=

⁰

,

(2.21)

wherethenumericalentropyfluxesaregivenby

= γ

h

γ

_h⁻

−

¹

2

( γ

_h⁻

)

²

− (

uh

)

s

(

uh

)

⁻_s

+

¹

2

((

uh

)

⁻_s

)

²

andtheextratermis

= ( − γ

_h

_{+ {{} γ

_h

_}} ) J γ

_h

_{K +} ( (

u_h

)

s

− {{ (

u_h

)

s

}} ) J (

u_h

)

s

K .

Therefore the choices of

γ

h,uh in (2.16) concern the conservativeness of the DG scheme. According to the parameters

α

,β,

μ

,wegivetwocases:

H₀conserved DG scheme

α = β = μ =

⁰

: =

⁰

;

^(2.22)

H0dissipative DG scheme

α =

0

, β =

¹ 2

, μ =

¹

2

: =

¹

2

(J γ

²

K + J

u²_s

K) ≥

0

.

(2.23) Summingupthecellentropyequalities(2.21) and(2.22),(2.21) and(2.23),respectively,thenweget

1 2

d ds

I

ρ

hu²_hdy

=

⁰

,

H₀conserved DG scheme

,

1

2 d ds

I

ρ

hu²_hdy

≤

⁰

,

H₀dissipative DG scheme

. 2

InthenumericaltestExample4.2,itshowsthatthedissipativeschemewithparameters(2.23) canachievetheoptimal convergencerateforboth uand

ρ

nomatterk isodd oreven.However,the orderofaccuracyforthe H0 conservedDG scheme(2.22) isk-thforoddk,(k+¹)-thforevenkonuniformmeshes.Withnonuniformmeshes,the H0 conservedDG schemeissuboptimalorderofaccuracyregardlessoftheparityofthepolynomialdegrees.Thechoicesoftheseparameters arenotunique,buttheabovenumericalfluxesinthedissipativeschemecanminimizethestencilasin[46].

2.2.3. H1conservedDGscheme

Inthissection,weconstructanotherdiscontinuousGalerkinschemewhichpreservesthequantity H₁ oftheCDsystem (2.2) which linkstheHamiltonian E1 oftheshortpulseequationthroughthehodographtransformation.First,werewrite theCDsystemasafirstordersystem

⎧ ⎪

⎨

⎪ ⎩

ρ

s

+

^u

ω =

⁰

,

w_s

= ρ

u

, ω =

^uy

.

(2.24)

Thenthesemi-discreteLDGnumericalschemecanbeconstructedas:Findu_h,

ω

h,

ρ

h ∈^V_h^ksuchthat

⎧ ⎪

⎨

⎪ ⎩

(( ρ

h

)

s

, φ)

I_j

+ (

u_h

ω

h

, φ)

I_j

=

⁰

,

(a)

(( ω

h

)

s

, ϕ )

I_j

= ( ρ

hu_h

, ϕ )

I_j

,

(b)

( ω

h

, ψ )

Ij

=<

^u

h

, ψ

y

>

Ij

−(

^uh

, ψ

y

)

Ij

,

(c)

(2.25)

foralltestfunctionsφ,

ϕ

,ψ ∈^V_h^kandIj∈Th.Thenumericalfluxistakenas^uh=^u+_h^.Numerically,theoptimal(k+¹)-th orderofaccuracycanbeobtainedforbothu_h,

ρ

h.

Proposition2.3.(H1conservation)Thesemi-discreteDGnumericalscheme(2.25)canpreservethequantityH1(

ρ

h,

ω

h)=

I(

ρ

_h²+

ω

_h²)dy spatially.

Proof. Bytakingthetestfunctionsφ=

ρ

h,

ϕ

=

ω

h in(2.25),weobtain

(( ρ

h

)

s

, ρ

h

)

Ij

+ (

u_h

ω

h

, ρ

h

)

Ij

=

⁰

, ((

w_h

)

s

, ω

h

)

Ij

= ( ρ

hu_h

, ω

h

)

Ij

.

Summingupoverall I_j∈Th,itimpliesthat 1

2 d ds

I

ρ

_h²

+ ω

²_hdy

=

0

. 2

In what follows, we prepare to give the a priori error estimate for the H1 conserved DG scheme. The standard L² projectionofafunctionζ withk+1 continuousderivativesintospaceV_h^k,isdenotedbyP,i.e.,foreach Ij

(

P

ζ − ζ, φ)

Ij

=

⁰

, ∀φ ∈

^Pk

(

I_j

),

andthespecialprojectionsP^±intoV_h^{ksatisfy,foreach I}j

(

P⁺

ζ − ζ, φ)

Ij

=

⁰

, ∀φ ∈

^Pk−1

(

I_j

),

andP⁺

ζ (

y⁺

j−¹2

) = ζ (

y_j−1 2

), (

P⁻

ζ − ζ, φ)

I_j

=

0

, ∀φ ∈

P^k−1

(

Ij

),

andP⁻

ζ (

y⁻

j+¹₂

) = ζ (

y_j+1 2

).

Fortheprojectionsmentionedabove,itiseasytoshow[6] that

ζ

^e

L²(I)

+

^h12

ζ

^e

L^∞(I)

+

^h12

ζ

^e

L²(∂I)

≤

^Chk+1

,

(2.26)

whereζ^e=ζ−Pζ ^orζ^e=ζ−P^±ζ,andthepositiveconstantConlydependsonζ.Thereisaninverseinequalitywewill useinthesubsequentproof.For∀^uh∈^V_h^{k,thereexistsapositiveconstant}

σ

(wecallittheinverseconstant),suchthat

^uh

L²(∂I)

≤ σ

h⁻¹²

^uh

L²(I)

,

(2.27)

where^uhL²(∂I)= N+¹

j=¹((u_h)⁻

j+¹₂)²+((u_h)⁺

j−¹₂)².

First,wewritetheerrorequationsoftheH1conservedDGschemeasfollows:

(( ρ − ρ

h

)

s

, ϕ )

I_j

= −(

u

ω −

uh

ω

h

, ϕ )

I_j

,

(2.28)

(( ω − ω

h

)

s

, φ)

I_j

= ( ρ

u

− ρ

huh

, φ)

I_j

,

(2.29)

( ω − ω

_h

, ψ )

I_j

=<

u

−

u_h

, ψ >

I_j

−(

u

−

u_h

, ψ

y

)

I_j

,

(2.30)

anddenote

η

^u

₌

u

−

P⁺u

, ξ

^u

=

P⁺u

−

^uh

, η

^ρ

= ρ −

P

ρ , ξ

^ρ

=

P

ρ − ρ

_h

, η

^ω

= ω −

P

ω , ξ

^ω

=

P

ω − ω

h

.

(2.31)

Todealwiththetermξ^u,weneedtoestablisharelationshipbetweenξ^u andξ^ω infollowinglemma.

Lemma2.4.Theξ^u,ξ^ω,

η

^ωaredefinedin(2.31),thenthereexistsapositiveconstantCσ,pindependentofh butdependingoninverse constant

σ

andPoincaréconstantCp,suchthat

ξ

^u

L²(I)

≤

C_σ,p

( ξ

^ω

L²(I)

+ η

^ω

_L2(I)

).

Proof. ByPoincaréFriedrichsinequalityinChapter10 of[5],wehave

ξ

^u

L²(I)

≤

^Cp

ξ

^u_y

L²(I)

+

^h−12

Jξ

^u

K

L²(∂I)

where

ξ

^u_y

L²(I)

=

I_j∈Th

Ij

(ξ

_y^u

)

²dy

¹₂

, Jξ

^u

K

L²(∂I)

=

^N

⁺¹

j=¹

Jξ

^u

K

²_j−1 2

¹₂

.

Theinequality(4.17)in[33] gives

ξ

_y^u

L²(I)

+

^h−12

Jξ

^u

K

L²(∂I)

≤

^Cσ

( ξ

^ω

L²(I)

+ η

^ω

_L2(I)

),

whichimpliesthat

ξ

^u

L²(I)

≤

^Cσ,p

(ξ

^ω

L²(I)

+ η

^ω

L²(I)

). 2

Theorem2.5.Itisassumedthatthesystem(2.24)withtheDirichletboundaryconditionhasasmoothsolutionu,

ρ

,

ω

.Letuh,

ρ

h,

ω

h bethenumericalsolutionofthesemi-discreteDGscheme(2.25).Andthereexiststhatinitialconditionsu⁰_h,

ω

⁰_h,

ρ

_h⁰satisfythefollowing approximationproperty

^u0

−

u⁰_h

L²(I)

+ ρ

⁰

− ρ

_h⁰

L²(I)

+ ω

⁰

− ω

_h⁰

L²(I)

≤

Ch^k+1

.

ThereafterforregularpartitionsofI=(yL,yR),andthefiniteelementspaceV^k_hwithk≥0,thereholdsthefollowingerrorestimate

^u

−

^uh

L²(I)

+ ω ₋ ω

_h

_L2(I)

+ ρ ₋ ρ

_h

_L2(I)

≤

^Chk+1

,

wherethepositiveconstantC dependsonthefinaltimeT andtheexactsolutions.

Proof. Werewritetheerrorequation(2.28),(2.29) as

((ξ

^ρ

+ η

^ρ

)

s

, ϕ )

I_j

= ( −

^u

ω ₊

u_h

ω

_h

, ϕ )

I_j

= ( − ω (ξ

^u

+ η

^u

) −

^uh

(ξ

^ω

+ η

^ω

), ϕ )

I_j

, ((ξ

^ω

+ η

^ω

)

s

, φ)

I_j

= ( ρ

u

− ρ

_hu_h

, φ)

I_j

= ( ρ (ξ

^u

+ η

^u

)

I_j

+

u_h

(ξ

^ρ

+ η

^ρ

), φ)

I_j

.