nonlinear Serre equations Conservative discontinuous Galerkin methods for the Journal of Computational Physics

Contents lists available atScienceDirect

Journal of Computational Physics

www.elsevier.com/locate/jcp

Conservative discontinuous Galerkin methods for the nonlinear Serre equations

Jianli Zhao

^a

, Qian Zhang

^a,1,2

, Yang Yang

^b,3

, Yinhua Xia

^a,∗,4

aSchoolofMathematicalSciences,UniversityofScienceandTechnologyofChina,Hefei,Anhui230026,PRChina bDepartmentofMathematicalSciences,MichiganTechnologicalUniversity,Houghton,MI49931,USA

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

Articlehistory:

Received10January2020

Receivedinrevisedform17July2020 Accepted19July2020

Availableonline24July2020

Keywords:

Serreequations

DiscontinuousGalerkinmethods Conservativeschemes Well-balancedschemes

Inthispaper,wedevelopthreeconservativediscontinuousGalerkin(DG)schemesforthe one-dimensional nonlineardispersiveSerre equations,includingtwo conserved schemes for the equations in conservative form and a Hamiltonian conserved scheme for the equationsinnon-conservativeform.Oneoftheschemesownsthewell-balancedproperty viaconstructing ahighorderapproximation tothe source termforthe Serreequations with anon-flatbottom topography.By virtueofthe Hamiltonianstructure ofthe Serre equations,weintroduceanHamiltonianinvariantandthendevelopaDGschemewhich canpreservethediscreteversionofsuchaninvariant.Numericalexperimentsindifferent casesareperformedtoverifytheaccuracyandcapabilityoftheseDGschemesforsolving theSerreequations.

©^{2020ElsevierInc.Allrightsreserved.}

1. Introduction

WeintroduceandstudythediscontinuousGalerkin(DG) methodsfortheSerreequations

⎧ ⎪

⎨

⎪ ⎩

h_t

+ (

hu

)

_x

=

⁰

,

u_t

+

^uux

+

^ghx

+ β

h⁻¹

(

h²

γ )

_x

=

⁰

, γ ₌

h

(

ux2

−

^uxt

−

^uuxx

),

(1.1)

whereh(x,t)representsthetotalwaterdepth,u(x,t)istheaveragedvelocityofthefluid, gisthegravitationalconstant,β isaparameterand

γ

isthefluidverticalaccelerationonthefreesurface.

TheSerreequations(1.1) can beused todescribethe shallowwaterwithlarge amplitudeonthefree surface. In[40], weknowthatEulerequationsoftheincompressiblefluidflowdescribethepropagationofthefullwaterwavesonthefree surface. Sincethe difficulty insolving thisproblemis mainlycaused bythe freesurface, various approximatemodelsfor thiscasearepresented,andthecorresponding analyticaltheoriesandnumericalsimulations arealsodeveloped.Inthese

*

Correspondingauthor.

E-mailaddresses:zjl88914@mail.ustc.edu.cn(J. Zhao),gelee@mail.ustc.edu.cn,qian77.zhang@polyu.edu.hk(Q. Zhang),yyang7@mtu.edu(Y. Yang), yhxia@ustc.edu.cn(Y. Xia).

1 Currentaddress:DepartmentofAppliedMathematics,TheHongKongPolytechnicUniversity,HongKong.

2 Researchsupportedbythe2019HongKongScholarProgram.

3 ResearchsupportedbyNSFgrantDMS-1818467.

4 ResearchsupportedbyNationalNumericalWindtunnelgrantNNW2019ZT4-B08andNSFCgrantNo.11871449.

https://doi.org/10.1016/j.jcp.2020.109729

0021-9991/©^{2020ElsevierInc.Allrightsreserved.}

models,thepropagationofthewavesundertheassumptionofshallowwaterwithlargeamplitudeisgovernedby theSerre equations,whileforsmallamplitudeorweaklynonlinearshallowwaterwavesthemodelreducestotheclassicalBoussinesq system[40].TheSerreequationscanberegardedasanasymptoticapproximationofthetwo-dimensionalEulerequations.

Itisafullynonlineardispersivesystemthatmodelstwo-waypropagationoflongwavesonthesurfaceofanidealfluidina channel.Andifweignoreallthedispersiveterms,itwillbecomeahyperbolicsystemofshallowwaterwave.Thedispersive termscombinedbythetemporalandspatialderivativesarethemajordifficultyinsolvingtheSerreequations[16].

TheSerreequationswere firstderivedbyFrancoisSerre[36] in1953andthenrestudiedbySuandGardner[39],Green [22],Seabra-Santosetal.[35],GreenandNaghdiin[23] derivedthesystemwithtwospatialdimensionsforgeneraluneven bottomsin1976.LannesandBonnetonin[24] providedtheformalderivationoftheequations fromEulerequationsandthe justificationforseveralapproximativewaterwavemodelsofEulerequations.Furtherstudiescanbegotfromsomerecent literaturesbyBarthélemy[3],DiasandMilewski[16],CarterandCienfuegos[9],andthereferencestherein.

TherearevariousnumericalmethodsdealingwiththenonlinearSerreequationsincludinghigherorderdispersiveterms.

In [26], a pseudo-spectral methodwas proposed andused to solvethe one-dimensional Serreequations forfree surface waves in a homogeneous layer of an ideal fluid. In [29,7,8,2,18], finite difference and finite volume methods were also applied in simulating all the processes of wave transformation with nonlinear dispersive effect. For the equations with non-flat bottoms (Green-Naghdiequations),the well-balancedcentralDG method was proposed in[25],inwhich theel- liptic partwas solved by the continuous finiteelement method.A localDG (LDG)scheme was presentedin[31] for the one-dimensional Green-Naghdiequationsbasedonrotationalcharacteristicsinthevelocity field.Forthetwo-dimensional problem,DuranandMarchedevelopedaLDG methodbyrewritingtheequationsasahyperbolic-ellipticsystemin[17],and ahybridizedDGmethodwasintroducedin[34] forthesplittingsystem(thehyperbolicpartanddispersivepart).Asforthe shallowwaterequationsandotherhyperbolic conservationlaws,theconservative schemeisdemanded whenthesolution isnotsmoothenough.Inthispaper,wewilldeveloptheLDG methodsfortheSerreequationsbasedontheirconservation lawsandalsowell-balancedpropertyasintheshallowwaterequations.

TheDG methodwhichisaclassoffiniteelementmethodswasfirstintroducedbyReedandHill[33] forsolvingtheneu- trontransportlinearequationin1973.Completelydiscontinuouspiecewisepolynomialspaceisusedforthetestfunctions andthenumericalsolutionsinthismethod.Later,CockburnandShuhavemadeagreatcontributionindevelopingtheDG methodinaseriesofpapers[11,13–15].TheyusedthestablehighorderRunge-Kuttamethodsfortemporaldiscretization and theexact or approximateRiemann solvers asthe interface fluxesin spatial discretization.In addition, they alsode- signedtheLDGmethodforsolvingconvection-diffusionproblemscontainingsecondderivativeterms,whichwasmotivated by BassiandRebay[4] insolving thecompressibleNavier-Stokesequations.Atpresent, theDGmethodiswidely usedin many areas,such as gasdynamics,electromagnetism, magnetohydrodynamics, modeling ofshallow water,etc. Moreover, theDGmethodenjoysmanyattractiveproperties.Firstwecangethighorderaccuracyforthenumericalsolutionbyusing element-based test functions. Second theDG methodcan be appliedon arbitraryunstructuredmesheswiththe suitable choices ofthe interelement boundary fluxes. Finally,parallel computing caneasily be applieddue to localdata commu- nications. In addition to the advantages given above, the LDG method contains auxiliary variables that approximate the gradientsoftheprimarysolutionstotheexpectedorderofaccuracy.Thentheauxiliaryvariablescanbelocallyeliminated.

TheLDGmethodshavebeengeneralizedtononlinearwaveequationsin[49,50,43–47,51],suchasKdVequation,Camassa- Holm equationandnonlinearSchrödinger equation,etc.Werefertothereviewpaper[48] ofLDGmethodsforhigh-order time-dependentpartialdifferentialequationsandthereferencestherein.

Mostrecently, aseriesof schemesso calledstructure-preservingschemesfornonlinear waveequations havebeenat- tracted considerable attention. For some integrableequations, such as KdVtype equations[6,27,52,53], Zakharovsystem [42], Schrödinger-KdV system [41], etc., the authors proposed various conservative numerical schemes to “preserve the structures”. Forthe Serreequations,thealiasing orspurious dissipativeeffects willresultinsignificant errors forsolving large amplitudeor rapidly oscillating waves, such as the dispersive shockwaves. Therefore, Mitsotakis andDutykh [30]

constructedanon-dissipativefiniteelementmethodtoapproximatetherapidlyoscillatingwavesaccurately.Afundamental propertyoftheSerreequationsisitsHamiltonianstructure.Fromthefollowingconservationlawsderivedbytheequations (1.1),

(

hu

)

t

+ (

hu²

+

¹

2gh²

+ β

h²

γ )

x

=

⁰

,

(1.2)

(

hu

− β(

h³u_x

)

x

)

t

+ (

hu²

+

¹

2gh²

−

²

β

h³u²_x

− β

h³uu_xx

−

³

β

h²h_xuu_x

)

x

=

⁰

,

(1.3)

(

u

− β

h⁻¹

(

h³ux

)

x

)

t

+ (

¹

2u²

+

gh

−

¹

2h²u²_x

− β

uh⁻¹

(

h³ux

)

x

)

x

=

0

,

(1.4)

(

¹

2hu²

+

¹

2

β

h³u²_x

+

¹

2gh²

)

t

+ ((

¹ 2u²

+

¹

2

β

h²u²_x

+

^gh

+ β

h

γ )

hu

)

x

=

⁰

,

(1.5) wehavethecorrespondingconservativequantities E0, E1,E2andHamiltonianH^{intherelevantdomain I,}

E₀

=

I

hudx

,

E₁

=

I

(

hu

− β(

h³u_x

)

x

)

dx

,

E2

=

I

(

u

− β

h⁻¹

(

h³ux

)

x

)

dx

,

H

=

I

1

2

(

hu²

+ β

h³u²_x

+

^gh2

)

dx

.

Thenumericalschemescanbeestablishedbasedontheaboveconservedquantities.Fortheform(1.2),themaindifficulty indesigning E0 conservednumericalschemeliesinthemixedspatial andtemporalderivativesin

γ

.Therefore,wedesign andimplementthreekinds ofnumericalschemeswhich areall basedon theDGmethods for theone-dimensional Serre equations.ThefirsttwoareconstructedforE1andE2 intheconservativeform,andthelastonepreservestheHamiltonian invariantHinanon-conservativeform.TheE1conservedschemecanmaintainthestill-waterstationaryforthewaterwave equationsovernon-flatbottomtopography,socalledwell-balancedscheme.FortheHamiltonian(H^{)conservedDGscheme,} we takeadvantage oftheDGmethodforHamilton-Jacobi equations[10] todeal withthe nonlinearterms.Moreover,the stabilityoftheHamiltonianfortheSerreequationswouldbeprovedtheoretically.

Thepresentpaperisorganizedasfollows.InSection2,theE₁andE₂conservedDGschemesaredesignedindetail.For the E1 conservedscheme,we alsodiscussandproveits well-balancedpropertywhenthe bottomisnotflat.Section 3is devotedtoelaborate theHamiltonian(H^{) conservedDGscheme.Moreover, thestabilitybasedon the}Hamiltonianisalso giveninthispart.InSection4,numericalexperimentsindifferentcircumstancesareprovidedtoillustratetheaccuracyand capabilityoftheseschemes.ConcludingremarksaregiveninSection5.

2. TheLDGdiscretizationinconservativeform

Inthissection,wedevotetodesignLDGschemesfortheSerreequationsby adoptingtheconservationlaws E1 andE2 in(1.3) and(1.4) respectively,andthengivethecorrespondingalgorithmflowcharts.

Todiscretizethespatial derivatives,we proceedasfollows.Foreach partitionoftheinterval I= [^a,b]^{,wesetthecells} and cell centers as Ij= [^x_j−¹

2,x_j+1

2]^{, x}j=¹₂(x_j−1 2 +^x_j+¹

2) for j=¹,· · ·,N, and denote the mesh size x= ^max

1≤j≤Nxj withxj=^xj+¹2 −^xj−¹2. The meshis assumedto be quasi-uniform, namelythere existsa positive constant c such that x≤^{c min}

1≤^j≤^Nx_j.

Weseekthenumericalapproximationsbelongingtothefiniteelementspace V_h

= {

^v

(

x

) :

^v

(

x

) ∈

^Pk

(

I_j

)

forx

∈

^Ij

,

j

=

¹

,· · · ,

N

},

where P^k(I_j) denotespolynomialsin I_j ofdegree upto k. Ittranspiresthat the functionsbelonging to V_h could bedis- continuousacross thecell interfaces.Forv∈^Vh,thevaluesofv atx_j+1

2,fromtheleftandrightcells Ij andIj+¹,canbe regardedas v⁻

j+¹2

and v⁺ j+¹2

,respectively.The notations[[^v]]=^v+−^{v− and}{{^v}}= ¹₂(v⁺+^v−)areusedto representthe jumpandthemeanofthefunctionatthecellinterfaces.

2.1. E₁conservedDGscheme

Inordertotreatthemixedspatialandtemporalderivativesin(1.1),weadopttheconservationlaw(1.3) anddevelopthe E₁ conservedDGschemeforthefollowingsystem

h_t

+ (

hu

)

_x

=

⁰

,

(2.1a)

z_t

+ (

uz

+

¹

2gh²

−

²

β

h³q²

)

x

=

⁰

,

(2.1b)

hu

− β(

h³q

)

x

=

^z

,

(2.1c)

q

−

^ux

=

⁰

,

(2.1d)

with periodic boundary conditions.To design the LDG method,we accomplish thecomplete schemes for the system as follows.Forsimplicity,westilluseh,u,zandqtodenotethenumericalsolutions.

Forthetime-dependentequations(2.1a) and(2.1b),assumethat uandq areknownandhandz aretobesolved,then wecanformulatetheDGmethodasfollows:Findh,z∈^Vh suchthat,foralltestfunctionsθ,

ϕ

∈^Vh,

Ij

h_t

θ

dx

−

Ij

(

hu

)θ

xdx

+ (

f₁

θ

⁻

)

_j+1

2

− (

f₁

θ

⁺

)

_j−1

2

=

⁰

,

(2.2)

Ij

z_t

ϕ

dx

−

Ij

(

uz

+

¹

2gh²

−

²

β

h³q²

) ϕ

xdx

+ (

f₂

ϕ

⁻

)

_j+1

2

− (

f₂

ϕ

⁺

)

_j−1

2

=

⁰

,

(2.3)

∀^j=¹,· · ·,N. The “hat” terms f₁ and f₂ appear on the cell boundaryterms are the so-called numericalfluxes, which are single valuedfunctionsfrom integrationby partsandshould be designedto ensurethe stability ofthe schemes.For conservationlaws,thenumericalfluxesareusuallyLipschitzcontinuousfunctionsandpreservethemonotonepropertyand consistentproperty,namedthemonotonefluxes[12].Here,wecanchoosethefollowingdissipativenumericalflux

f₁

= {{

^hu

}} − α

2

[[

^h

]],

^(2.4)

_f2

= {{

^uz

+

¹

2gh²

−

²

β

h³q²

}} − α

2

[[

^z

]] ,

(2.5)

where

α

=^max(|^u| +

gh)isthemaximumtakenintherelevantinterval.Wehaveomittedthehalf-integerindices j+¹₂ asallquantitiesin(2.4) arecomputedatthesamecellinterfaces.

Forthetime-independentequations(2.1c) and(2.1d),assumethathandz areknownandu andqaretobesolved,the LDGschemecanbeformulatedasfollows:Findu,q∈^Vh suchthat,foralltestfunctions

ρ

,φ∈^Vh,

I_j

hu

ρ

dx

+

I_j

β

h³q

ρ

xdx

− (

h³q

ρ

⁻

)

_j+1

2

+ (

h³q

ρ

⁺

)

_j−1

2

=

I_j

z

ρ

dx

,

(2.6a)

Ij

q

φ

dx

+

Ij

u

φ

xdx

− (

u

φ

⁻

)

_j+1

2

+ (

u

φ

⁺

)

_j−1

2

=

0

.

(2.6b)

Herewetakethealternatingfluxes[48] suchas

h³q

= (

h³q

)

⁻

,

u

=

u⁺

.

(2.7)

2.1.1. Algorithmflowchart(I)

Thispartshowstheimplementationaboutthefully-discrete E₁ conservedscheme.Forsimplicity,we choosetheEuler forwardmethodto illustratethenumericalevolutionfromtimeleveltⁿ totime leveltⁿ⁺¹.Innumericalexperiments,the explicitSSP/TVDRunge-Kuttamethods[37,38] areadopted,whichcanbeviewedastheconvexcombinationsoftheEuler forwardmethod.

Leth,u,zandqdenotethevectorscontainingthedegreesoffreedomforthenumericalsolutionsh,u,zandq,respec- tively.Giventhesolutionhⁿ,uⁿ,zⁿandqⁿatthetimeleveltⁿ,thealgorithmcanbedividedinto twosteps.

StepI:Fromtheequations(2.2) and(2.3) equippedwiththecorrespondingnumericalfluxes(2.4) and(2.5),wehavethe fullydiscreteschemeinthematrixform:

hⁿ⁺¹

−

^hn

tⁿ

=

^res1

(

hⁿ

,

uⁿ

),

(2.8)

zⁿ⁺¹

−

^zn

tⁿ

=

res2

(

hⁿ

,

uⁿ

,

zⁿ

,

qⁿ

),

(2.9)

bytheEulerforwardmethodwiththetimesteptⁿ=^tn+1−^tn.

StepII:After weget hⁿ⁺¹ andzⁿ⁺¹ fromthe above,basedontheschemes (2.6a) and (2.6b) withthefluxes(2.7),we cansolveuⁿ⁺¹,qⁿ⁺¹inthefollowingmatrixform

A

(

hⁿ⁺¹

)

uⁿ⁺¹

+

^B

(

hⁿ⁺¹

)

qⁿ⁺¹

=

^zn+1

,

(2.10)

qⁿ⁺¹

+

Cuⁿ⁺¹

=

0

,

(2.11)

where the matrices A, B and C are the corresponding discrete operators in the scheme. Then we get all the numerical solutionsattimeleveltⁿ⁺¹.

2.1.2. Thewell-balancedscheme

In thissubsection,weturnto presentthe well-balancedpropertyof E1 conserved DGschemefortheSerreequations withanon-flatbottom,alsoknownasGreenandNaghdiequationsin[23].Thereinto,hcanberepresentedasthesumofd and

η

,whereddenotesthebottomtopographyand

η

isthefreesurfaceelevation.Inconsiderationoftheone-dimensional shallowwaterflowswithnon-flatbottomtopography,themodelcanbeformulatedasfollows:

h_t

+ (

hu

)

_x

=

⁰

,

(2.12a)

(

hu

)

t

+ (

hu²

+

¹

2gh²

+ β

h²

γ +

¹

2h²

)

x

= (

gh

+

¹

2h

γ +

^h

)

d_x

,

(2.12b)

with

= −

^dxu_t

−

^dxuu_x

−

^dxxu²

.

(2.13)

Itcanbeseenthatfortheflatbottom,meaningd(x)isaconstant,theequation(2.12b) isequivalenttotheequation(1.2).

Thewell-balancedmethods[5] areintroducedtoexactlypreserve thestill-waterstationarysolutionatadiscrete level.

FortheSerreequations,thestill-waterstationarywavesolutioncanberepresentedasfollows,

u

=

^{0 and}

η ₌

h

−

^d

=

^constant

.

(2.14)

Inordertodealwiththemixedspatialandtemporalderivativesin(2.12b),weintroduceanewunknownvariable

ω = −β(

^h3ux

)

x

+

^h

(

1

−

^hxd_x

−

¹

2hd_xx

+

^d2_x

)

u

.

(2.15)

Noticethat

ω

=^{zwiththeflatbottom.Theequation(2.15) canbe}discretizedsimilarlyastheLDGscheme(2.6a) and(2.6b) forz.Thusweomitthedetailshere.

Asin[25],werewritethetime-dependentpartofthesystem(2.12a)–(2.12b) intoabalancelaw

η

t

+ (

hu

)

_x

=

0

,

(2.16)

ω

t

+ ( ω

u

+

¹

2gh²

−

²

β

h³u²_x

−

^h2uuxdx

)

x

=

^ghdx

+

¹

2h²uuxdxx

+

^hu2dxdxx

.

(2.17) Denoting U=(

η

,

ω

)^T astheunknowns,weformulatethe E₁ conservedschemeforthebalanceequationsasfollows:Find

η

,

ω

_∈V_h suchthat,foralltestfunctionsθ1,θ2∈^Vh,

Ij

U_t

·

dx

−

Ij

F

·

xdx

+ (

F

·

⁻

)

_j+1

2

− (

F

·

⁺

)

_j−1

2

=

Ij

S

·

dx

, ∀

^j

=

¹

, · · · ,

N

,

(2.18)

whereequals(θ1,θ2)^T,thefluxistakenas F

(

U

) = (

hu

, ω

u

+

¹

2gh²

−

²

β

h³u²_x

−

^h2uuxd_x

)

^T

,

andthesourcetermisgivenby S

= (

0

,

ghdx

+

¹

2h²uuxdxx

+

^hu2dxdxx

)

^T

.

Ingeneral, the numericalscheme(2.18) does not havethewell-balanced property.Based onthe ideain [32], wedis- cretizethesourcetermasfollows:

Ij

U_t

·

dx

−

Ij

F

·

xdx

+ (

F

·

⁻

)

_j+1

2

− (

F

·

⁺

)

_j−1

2

=

Ij

S₁

·

dx

+

S_j

,

(2.19)

with

S₁

= (

0

,

¹

2h²uu_xd_xx

+

^hu2dxd_xx

)

^T

,

and

S_j

= (

0

, ˜

s_j

)

^T

,

˜

sj

=

¹

2g

( −

Ij

d²

(θ

2

)

xdx

+ {{

^d2_j+1 2

}} (θ

2

)

⁻

j+¹₂

− {{

^d2_j−1 2

}} (θ

2

)

⁺

j−¹₂

)

+

g

η ¯

j

(−

Ij

d

(θ

2

)

xdx

+ {{

d_j+1 2

}}(θ

2

)

⁻

j+¹₂

− {{

d_j−1 2

}}(θ

2

)

⁺

j−¹₂

)

+

Ij

g

( η − ¯ η

j

)

dx

θ

2dx

,

(2.20)

wheres˜j isthehighorderapproximationofthesourceterm

I_jghdxθ2dxand

η

¯j=_I¹_j

I_j

η

dxdenotesthemeanof

η

over I_j.F isamonotoneflux,e.g.theLax-Friedrichsflux.

In ordertoensuretheunique solvabilityoftheequation (2.15) in thestill-waterstationary,thefollowingassumptions areneeded[25]:

(A1) Thewaterheighthinthestill-waterstationarysolutionconsideredherehasastrictlypositivelowerbound.

(A2) 1−¹₂^hdxx>0.

Nowweturntoprovethewell-balancedpropertyofthemodifiedscheme(2.19).

Proposition2.1.(Well-balancedproperty)Undertheassumptions(A1)and(A2),themodifiedE1conservedDGschemedescribedby (2.19) fortheSerreequationswithnon-flatbottomhasthewell-balancedproperty.

Proof. Let Uⁿ (n∈N) be the numericalsolution of (2.19) at time level tⁿ.For brevity, herewe consider Eulerforward time discretization,whichcanbeeasilygeneralizedtotheSSP/TVDRunge-Kuttamethods.Thenweapplytheminto(2.19), leadingto

I_j

Uⁿ⁺¹

·

dx

=

I_j

Uⁿ

·

dx

+

tⁿ

(

I_j

Fⁿ

·

xdx

− (

Fⁿ

·

⁻

)

_j+1 2

+(

Fⁿ

·

⁺

)

_j−1

2

+

Ij

Sⁿ₁

·

dx

+ (

S

ⁿ

)

j

).

(2.21)

Byinduction,firstweassume

Uⁿ

= (

C₀

,

0

)

^T and uⁿ

=

⁰

,

(2.22)

whereC0 isaconstant.Fromthisassumptionwewanttodeducethatthenumericalsolutioncomputedfrom(2.21) satisfies

Uⁿ⁺¹

= (

C0

,

0

)

^T and uⁿ⁺¹

=

0

.

(2.23)

Byvirtueoftheassumptions(2.22),theequation(2.21) becomes

η

ⁿ⁺¹

₌ η

ⁿ

₌

C0

,

(2.24)

Ij

ω

ⁿ⁺¹

θ

2dx

=

tⁿ

Ij

1

2g

(

hⁿ

)

²

(θ

2

)

_xdx

− (

¹

2g

{{(

hⁿ

)

²

}}(θ

2

)

⁻

)

_j+1 2

+ (

¹

2g

{{ (

hⁿ

)

²

}} (θ

₂

)

⁺

)

_j−1 2

+ (

sⁿ

)

_j

.

(2.25)

Denotingthesumoftheright-handsideofequation((2.25))by R_j,wecanget R_j

=

tⁿ

I_j

1

2g

(

hⁿ

)

²

(θ

₂

)

_xdx

− (

¹

2g

{{ (

hⁿ

)

²

}} (θ

₂

)

⁻

)

_j+1 2

+ (

¹

2g

{{ (

hⁿ

)

²

}} (θ

₂

)

⁺

)

_j−1 2

+ (

sⁿ

)

_j

=

tⁿ

Ij

1

2g

((

hⁿ

)

²

− (

dⁿ

)

²

−

2

η ¯

ⁿ_jdⁿ

)(θ

2

)

xdx

−

¹

2g

({{(

hⁿ

)

²

j+¹₂

}} − {{(

dⁿ

)

²

j+¹₂

}} −

2

η ¯

ⁿ_j

{{

dⁿ

j+¹₂

}})(θ

2

)

⁻

j+¹₂

+

¹

2g

( {{ (

hⁿ

)

²

j−¹₂

}} − {{ (

dⁿ

)

²

j−¹₂

}} −

²

η ¯

ⁿ_j

_{{

dⁿ

j−¹₂

}} )(θ

₂

)

⁺

j−¹₂

+

I_j

g

( η

ⁿ

_{− ¯} η

ⁿ_j

)

dⁿ_x

θ

₂dx

=

tⁿ

Ij

1

2g

( η

ⁿ

)

²

(θ

2

)

xdx

−

¹ 2g

{{( η

ⁿ

)

²

j+¹₂

}}(θ

2

)

⁻

j+¹₂

+

¹ 2g

{{( η

ⁿ

)

²

j−¹₂

}}(θ

2

)

⁺

j−¹₂

=

⁰

,

(2.26)

wherewetakeuseoftherelationshiph=^d+

η

andthefact

η

ⁿ= ¯

η

ⁿ_j=C0 by(2.22).Thentheequations(2.24)–(2.25) lead to Uⁿ⁺¹=(C₀,0)^T.Astheassumptions (A1)and(A2)ensure theuniquesolvabilityofthefiniteelementdiscretization,it impliesuⁿ⁺¹=^{0.Thiscompletesourproof.}

2.2. E2conservedDGscheme

SimilartothedevelopmentoftheE1 conservedscheme,weadopt(1.1) and(1.4) toconstructtheE2conservedscheme.

TheSerreequationsarerewritten asfollows:

ht

+ (

hu

)

_x

=

⁰

,

(2.27a)

^zt

+ (

u

^z

−

¹

2u²

+

^gh

−

¹

2h²q²

)

x

=

⁰

,

(2.27b)

u

−

h⁻¹p

=

z

,

(2.27c)

p

− β(

h³q

)

x

=

⁰

,

(2.27d)

q

−

^ux

=

⁰

.

(2.27e)

Without ambiguity, we still use h,^z,u,p,q to denote the numerical approximations. The E2 conserved DG scheme for (2.27a)-(2.27e) canbeformulatedasfollows:Findh,u,q,z,p∈^Vh,suchthat,foralltestfunctionsθ,

ϕ

,

ρ

,ψ andφ∈^Vh,

I_j

h_t

θ

dx

−

I_j

(

hu

)θ

xdx

+ (

_f1

θ

⁻

)

_j+1

2

− (

_f1

θ

⁺

)

_j−1

2

=

⁰

,

(2.28)

Ij

zt

ϕ

dx

−

Ij

(

u

z

−

¹

2u²

+

gh

−

¹

2h²q²

) ϕ

xdx

+ (

_f4

ϕ

⁻

)

_j+1

2

− (

_f4

ϕ

⁺

)

_j−1

2

=

0

,

(2.29)

I_j

p

ρ

dx

+

I_j

β

h³q

ρ

xdx

− (

h³q

ρ

⁻

)

_j+1

2

+ (

h³q

ρ

⁺

)

_j−1

2

=

⁰

,

(2.30)

I_j

u

ψ

dx

−

I_j

h⁻¹p

ψ

dx

=

I_j

z

ψ

dx

,

(2.31)

Ij

q

φ

dx

+

Ij

u

φ

xdx

− (

^u

φ

⁻

)

_j+1

2

+ (

^u

φ

⁺

)

_j−1

2

=

⁰

,

(2.32)

wherethenumericalfluxes arechosenas

_f1

= {{

^hu

}} − α

2

[[

^h

]] ,

(2.33)

f₄

= {{

^u

^z

−

¹

2u²

+

^gh

−

¹

2h²q²

}} − α

2

[[

^z

]],

^(2.34)

h²q

= (

h³q

)

⁻

,

u

=

^u+

,

(2.35)

where

α

=^max(|^u| +

gh)isthemaximumtakenintherelevantinterval.

2.2.1. Algorithmflowchart(II)

Similarto thefullydiscrete E1 conservedscheme,the E2 conservedschemecoupledwiththeEulerforwardtimedis- cretizationcanbedividedintwosteps.Giventhesolutionhⁿ,uⁿ,z˜ⁿ,pⁿ andqⁿ atthetimeleveltⁿ,thealgorithmcanbe dividedinto twosteps.

StepI:Fromthetimedependentequations(2.28) and(2.29) withthefluxes(2.33) and(2.34) wehavethefullydiscrete schemeinthematrixform:

hⁿ⁺¹

−

^hn

tⁿ

=

^res1

(

hⁿ

,

uⁿ

),

(2.36)

˜

zⁿ⁺¹

− ˜

^zn

tⁿ

=

^res4

(

hⁿ

,

uⁿ

,

^z

˜

ⁿ

,

qⁿ

),

(2.37)

bytheEulerforwardmethodwiththetimesteptⁿ=^tn+1−^tn.

StepII:Afterwegethⁿ⁺¹ and^z˜^{n+1fromthelaststep,basedonthetime}independentschemes(2.30),(2.31) and(2.32) withthenumericalfluxes(2.35),wecansolvethecorrespondinglinearsystemtogetuⁿ⁺¹,pⁿ⁺¹,qⁿ⁺¹.

Thusweobtainallthenumericalsolutionsatthenexttimesteptⁿ⁺¹.

3. TheDGdiscretizationinnon-conservativeform

Inthissection,wedesignandproveaHamiltonian(H^{)conservedDGschemeandgiveitsalgorithmflowchart.}

Combining (1.1) and (1.3) and after some manipulation, we can obtain the Serre equations in the following non- conservativeform:

h_t

+ (

hu

)

_x

=

⁰

,

(3.1)

hu_t

− β(

h³u_x

)

xt

+

^huux

− β(

u

(

h³u_x

)

x

)

x

+

¹

2g

(

h²

)

x

−

²

β(

h³u²_x

)

x

=

⁰

.

(3.2) Introducingsomeauxiliaryvariablesm,p,q,s,w,k,v,r,

m

−

^hu

=

⁰

,

p

−

¹

2u²

=

⁰

,

s

−

^h3q

=

⁰

,

q

−

^ux

=

⁰

,

w

−

^sx

=

⁰

,

k

− (

hu

)

_x

=

⁰

,

v

−

³

2h²qk

=

⁰

,

r

−

¹

2h²qm_x

−

^h2mqx

−

^hqmhx

=

⁰

,

(3.3)

theSerreequationscanbewrittenasfollows:

ht

+ (

hu

)

x

=

⁰

,

(3.4)

hut

− β

wt

+

^hpx

+

¹

2g

(

h²

)

x

− β(

v

+

^r

)

x

=

⁰

.

(3.5)

We willintroduce thestandard L² projection operatorP ofafunction

ω

intothe finiteelementspace V_h,i.e., forall v∈^Pk(I_j),

Ij

(P ( ω ) − ω )

vdx

=

⁰

, ∀

^j

=

¹

,· · · ,

N

,

(3.6)

anddefinethediscretederivativeoperator Dj

( ω ; ρ ) =

Ij

ωρ

xdx

− ( ωρ

⁻

)

_j+1

2

+ ( ωρ

⁺

)

_j−1 2

,

where the numerical flux

ω

ˆ is to be determined. Then for the equations (3.4), (3.5) and the auxiliary equations (3.3), the H ^{conserved DG scheme can be} constructed as follows: Find h,u,w,q,k,r∈^Vh such that, for all test functions θ,φ,ζ,

τ

,

γ

,

μ

∈^Vh,

Ij

h_t

θ

dx

−

^Dj

(

hu

; θ ) =

⁰

,

(3.7)

I_j

hu_t

φ

dx

− β

I_j

w_t

φ

dx

+

^Dj

(

h

φ ;

^p

) −

¹

2g D_j

(

h²

; φ) + β

D_j

(

v

+

^r

; φ) =

⁰

,

(3.8)

I_j

q

τ

dx

+

^Dj

(

u

; τ ) =

⁰

,

(3.9)

Ij

k

γ

dx

+

^Dj

(

hu

; γ ) =

⁰

,

(3.10)

Ij

w

ζ

dx

+

^Dj

(

s

; ζ ) =

⁰

,

(3.11)

Ij

r

μ

dx

−

¹

2D_j

(

h²q

μ ;

^m

) −

^Dj

(

h²m

μ ;

^q

) −

^Dj

(

hmq

μ ;

^h

) =

⁰

,

(3.12)

withm=P(hu),p=P(¹₂u²),s=P(h³q)andv=P(³₂h²qk)andthenumericalfluxes

hu

= {{

^hu

}} ,

_h

φ = {{

^h

φ }} ,

_h

²

=

^h+h−

,

(3.13)

v

+

^r

=

^v+

+

^r+

,

s

=

^s+

,

u

=

^u−

,

(3.14)

h

²q

μ = {{

^h2q

μ }},

_h

2m

μ = {{

^h

}}{{

^h

μ }}{{

^m

}},

hmq

μ = {{

^h

μ }}{{

^m

}}{{

^q

}}.

^(3.15)

Herethechoicesofthenumericalfluxesintheschemearenotunique,whichcanbeseenfromtheproofofitsconservation ofHamiltonian.

3.1. H-stability

Inthispart,weprovetheHamiltonianconservationoftheabovescheme.Forbrevity,wedefine

(

v

,

w

) :=

N j=¹

I_j

v wdx

,

D

(

v

;

^w

) :=

N j=¹

D_j

(

v

;

^w

).

(3.16)

Proposition3.1.(H-stability)Thesolutiontothescheme(3.7)–(3.12) withthenumericalfluxes(3.13)–(3.15) satisfiesthefollowing H-stability:

d dt

I

1

2

(

hu²

+

^gh2

+ β

h³q²

)

dx

=

⁰

.

(3.17)

Proof. SumminguptheDGschemes(3.7)–(3.12) over jfrom1 toN,wehave

(

h_t

, θ ) −

^D

(

hu

; θ ) =

⁰

,

(3.18)

(

hut

, φ) − β(

wt

, φ) +

D

(

h

φ;

p

) −

¹

2g D

(

h²

; φ) + β

D

(

v

+

r

; φ) =

0

,

(3.19)

(

w

, ζ ) +

^D

(

s

; ζ ) =

⁰

,

(3.20)

(

q

, τ ) +

^D

(

u

; τ ) =

⁰

,

(3.21)

(

k

, γ ) +

^D

(

hu

; γ ) =

⁰

,

(3.22)

(

r

, μ ) −

^D

(

¹

2h²q

μ ;

^m

) −

^D

(

h²m

μ ;

^q

) −

^D

(

hmq

μ ;

^h

) =

⁰

.

(3.23) Byintroducinganewauxiliaryvariable

ν

₌P(³₂h²q²)andtakingthetestfunctionsθ=^p+^gh−β

ν

in(3.18) andφ=^{u in} (3.19),itfollowsthat

(

ht

,

p

+

gh

− β ν ) −

D

(

hu

;

p

+

gh

− β ν ) =

0

, (

hut

,

u

) − β(

wt

,

u

) +

^D

(

hu

;

^p

) −

¹

2g D

(

h²

;

^u

) + β

D

(

v

+

^r

;

^u

) =

⁰

.

Ifthefollowingidentitieshold D

(

hu

;

^gh

) +

¹

2g D

(

h²

;

^u

) =

⁰

,

(3.24)

D

(

hu

; ν ) +

^D

(

v

;

^u

) =

⁰

,

(3.25)

D

(

r

;

^u

) =

⁰

,

(3.26)

thenwehave

(

h_t

,

p

+

^gh

− β ν ) + (

hu_t

,

u

) − β(

w_t

,

u

) =

⁰

.

(3.27)