Third order maximum-principle-satisfying DG schemes for convection-diffusion problems with anisotropic diffusivity

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Journal of Computational Physics

www.elsevier.com/locate/jcp

Third order maximum-principle-satisfying DG schemes for convection-diffusion problems with anisotropic diffusivity

Hui Yu

^a

^,∗ , Hailiang Liu

^b

aTsinghuaUniversity,YauMathematicalSciencesCenter,Beijing,100084,China bIowaStateUniversity,MathematicsDepartment,Ames,IA50011,UnitedStatesofAmerica

a rt i c l e i n f o a b s t ra c t

Articlehistory:

Received14December2018 Receivedinrevisedform10April2019 Accepted13April2019

Availableonline18April2019

Keywords:

Diffusion

DiscontinuousGalerkinmethod Maximumprinciple

Highorderaccuracy

Foraclassofconvection-diffusion equationswithvariable diffusivity,weconstructthird orderaccurate discontinuousGalerkin (DG) schemes onboth one and two dimensional rectangularmeshes. TheDG methodwith anexplicittimesteppingcanwell beapplied to nonlinear convection–diffusion equations. It is shown that under suitable time step restrictions,thescalinglimiterproposedinLiuandYu(2014)[23] whencoupledwiththe presentDGschemes preservesthe solutionbounds indicatedbytheinitialdata,i.e.,the maximumprinciple,whilemaintaining uniformthirdorderaccuracy.Theseschemescan beextendedtorectangularmeshesinthreedimension.Thecrucialforallmodelscenarios isthat aneffective test setcan be identifiedtoverify the desiredbounds ofnumerical solutions.Thisisachievedmainlybytakingadvantageoftheflexibleformofthediffusive flux and the adaptable decomposition of weighted cell averages. Numerical results are presentedtovalidatethenumericalmethodsanddemonstratetheireffectiveness.

©^{2019ElsevierInc.Allrightsreserved.}

1. Introduction

In this paper, we constructand analyze third ordermaximum-principle-satisfying (MPS) discontinuous Galerkin (DG) schemesforthefollowingproblem,

M

(

x

)∂

tu

+ ∇ ·

^f

(

u

) = ∇ · (

A

∇

^u

),

x

∈ R

^d

,

t

>

0

,

u

(

0

,

x

) =

^u0

(

x

),

x

∈ R

^d

.

^(1.1)

This equation can be considered as a model for numerous physical problems. In this equation, u=^u

(

t

,

x

)

is a time- dependent unknown scalarfunction, f

(

u

)

isthe smooth vectorflux, andd isthe spatialdimension. Weassume that the diffusiontensor A=^A

(

x

,

u

)

isasymmetricandnonnegativedefinitematrix,andM

(

x

)

isstrictlypositivescalarfunction.In thispaper,wepresentourresultsandanalysisford=¹

,

2,extensiontothreedimensioncanbedoneaswell.

Our concern for(1.1) arises fromseveral typical scenarios. The first case with f =^{0 is a model for heat conduction} in a non-uniform body with M

(

x

)

= _ρ^c(₍^x_x⁾₎^{, where c}

(

x

)

is the specific heat and

ρ (

x

)

the mass density. The heat flux is proportional to the temperature difference −^A∇^{u, known as the Fourierlaw of heat} conduction. Here A measures the abilityofthematerialtoconductheat,calledthethermalconductivity.InthesecondcasewithM=^{1,wehavetheusual} convection-diffusionequation,

*

Correspondingauthor.

E-mailaddresses:[email protected](H. Yu),[email protected](H. Liu).

https://doi.org/10.1016/j.jcp.2019.04.028

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

∂

_tu

+ ∇ ·

^f

(

u

) = ∇ · (

A

(

x

,

u

) ∇

^u

).

(1.2) There are many interpretations and derivations from fluid dynamics and other application areas that motivate the convection-diffusion equation, such as Navier-Stokes equations and the porous medium equation. The third case is the Fokker-Planckequation,

∂

t

ρ = ∇ · (∇ ρ + ∇

^V

(

x

) ρ ),

(1.3)

whichcanberewrittenintermsofu=

ρ

e^V(x) as(1.1) with A

(

x

)

=^M

(

x

)

=^{e−V(x),and f}=^{0.Thissettingallowsformany} variants.

Theabovethreetypesofproblemscanbeformulatedunderthemodelclass(1.1).Oneimportantsolutionto(1.1) isthe onethatisboundedbytwoconstantsdictatedbytheinitialdata,leadingtotheso-calledMaximumPrinciple(MP).Inother words,if

c₁

=

^min

x u₀

,

c₂

=

^max

x u₀

,

thenu

(

t

,

x

)

∈ [^c1

,

c₂]^foranyx∈R^dandt

>

0.

Fromthe analyticalviewpoint,themaximumprincipleisquite generalbutverysignificant duetoits physicalimplica- tions. Fromthe numericalviewpoint, itiswidely recognized thatmaximum principleprovides avaluable tool inproving solvabilityresults(existenceanduniqueness ofdiscrete solutions),enforcingnumericalstability,andderivingconvergence results(apriorierrorestimates)forthesequenceofapproximatesolutions.Designofahighorderschemetopreservethe maximum principle isknown a challenging task. Ourgoal isto better understand how ahigh orderDG scheme can be constructedfor(1.1) torespecttheMPSproperty.ThemaindifficultyintheanisotropiccasewithavariableweightM

(

x

)

is thederivationofsuitablesufficientconditionssothattheweightedcellaveragestaysin[^c1

,

c2]^{during thetimeevolution.}

Suchweightedcellaverageisessentiallyusedforlimitingthenumericalsolutioninto[^c1

,

c₂]^{,withoutdestroyingaccuracy.}

1.1. Relatedwork

Earlydiscussionofthediscretemaximumprinciplefortheconvection-diffusionequationsincludesthelinearfiniteele- mentsolutionsforparabolicequations[8],andrecentdevelopments[10,9,30,31,11],aswellas[25] bythePetrov–Galerkin finiteelementmethodtosolveconvectiondominatedproblems.However,theyareunderadifferentframework.

The presentinvestigation involvesthechoice ofnumericalfluxesandmonotonicityofweighted cell averagesinterms ofpoint values.In thelargest sense, theorigins oftheseideas go all theway backto monotone schemes forhyperbolic conservationlaws.

Indeed,forscalarconservationlaws,i.e., (1.2) with A=^{0,manyfirstorderclassicalschemes canbeshowntobe MPS} (othernamesofthissortincludebound-preserving,positivitypreserving,ormaximum-principle-preserving)sincesuchlow order accurate schemes are usually monotone. Onthe other hand,the Godunov Theorem states that a linear monotone schemeisatmostfirstorderaccuratefortheconvectionequation[17].Toconstructhighorderaccurate MPSschemesfor scalarconvection,weakmonotonicityinfinitevolumetypeschemesincludingDGmethodswasfirstusedin[35–37].Here byweakmonotonicityitmeansthateachcellaverageismonotonewithrespecttopointvaluesinthatcell,seee.g.[34].The mainideaintheirworkistofindsufficientconditionstopreservethedesiredboundsofcellaveragesbyrepeatedconvex combinations.AsimpleandefficientlocalMPSlimitercanthenbeusedtocontrolthesolutionvaluesattestpointswithout affectingaccuracyandconservation.Togetherwithstrongstabilitypreserving(SSP)Runge–Kuttaormultistepmethods[13], whichare convexcombinationsofseveral formalforward Eulersteps,a highorderaccurate finitevolume orDGscheme canberenderedMPSwiththelimiter.

For diffusion,a linear finite volume scheme can only be up to second order accurate in order to preserve the weak monotonicity, unless a non-conventional discretization is used in the scheme construction such asthat in [37]. For DG methods,ingeneralonlysecondorderaccuracycanbeobtainedtofeaturetheMPSproperty;see[38] forsolving(1.2) on triangularmeshes.

TheonlyDGmethodknowntosatisfytheweakmonotonicityuptothirdorderaccuracyisthedirectDG(DDG)method introducedin[21,22].Indeed,thespecialmethodparametersintheDDGdiscretizationallowedustodesignin[23] athird orderMPSmethodforthelinearFokker-Planckequation (1.3).Akeyideain[23] istheuseofthenon-logrithimic Landau formulation

M

∂

tu

= ∇ · (

M

∇

^u

)

withM

(

x

) =

^e−V(x)andu

= ρ

M

,

sothatthecorrespondingmaximumprincipleon

ρ (

t

,

x

)

:

c₁e^−V

≤ ρ (

0

,

x

) ≤

^c2e^−V

=⇒

^c1e^−V

≤ ρ (

t

,

x

) ≤

^c2e^−V

∀

^t

>

0

,

reducesto

c₁

≤

^u

(

0

,

x

) ≤

^c2

=⇒

^c1

≤

^u

(

t

,

x

) ≤

^c2

∀

^t

>

0

.

Withthisreformulation,onecanshowthateachweightedcellaverageismonotoneintermsofpointvaluesunderappropri- ateCFLconditions.Theresultin[23] isdirectlyapplicabletomulti-dimensionaldiffusiononrectangularmeshes.However, it getssubtletoensuretheMPS propertyonunstructuredmeshes;werefer to[2] forathird ordersuchDDGmethodto solvediffusionequationsonunstructuredtriangularmeshes.

Anotherapproachtowardsapositivity-preservingschemewithhighorderaccuracyistousethelocalDG(LDG)method [1,5], combinedwithsome specialpositivity-preservingfluxes.Suchan effortwas firstmadein[34] forconstructinghigh orderaccuratepositivity-preservingDGschemesforcompressibleNavier–Stokesequations.Concerningfurtherdevelopments in this direction, we refer to [28,29] for solving convection–diffusion problems. An MPS third-order LDG method using overlappingmesheshasbeenrecentlyproposedin[7] forconvection-diffusionequations.

One noteworthyalternative toenforce positivityinhighorderschemes istotake a convexcombinationofhighorder flux witha first order positivity-preserving one;the method has been applied to various high order schemes including finitedifference,finitevolume,andDGmethods[15,32,33],whilerigorousjustificationofaccuracyforsuchmethodsseems difficult.

1.2. Presentinvestigation

Themaindistinctionofourpresentinvestigationfromtheabovementionedworksistheuseofweightedcellaverages forbothensuringtheweakmonotonicityandapplyingthescalinglimiter,withspecialattentionondifficultycausedbythe anisotropicdiffusivity.

The spatial discretizationexplored in thisworkis theDDGmethodintroduced by LiuandYanin [21,22]. Besides the usual advantagesofaDGmethod(seee.g.[14,26,27]),onemainfeatureoftheDDGmethodliesinnumericalfluxchoices forthesolutiongradient,whichinvolvesecond orderderivativesevaluatedcrossing cellinterfaces(see(2.2) below).With thischoice,theobtainedschemesareprovablystableandoptimallyconvergentaswellassuperconvergentfor

β

1=^{0 [18,3].}

Such methodhasalsobeensuccessfullyextendedtovarious applicationproblems,includingFokker-Plancktype equations [23,24,19,20],andthethreedimensionalcompressibleNavier–Stokesequation[6].

Builtuponthework[23],wepresentthirdorderDGmethodsforsolvingtheinitialvalueproblem(1.1),withanapplica- tiontononlinearconvection-diffusionequationsofform(1.2).Letusillustratethemainideasviaasimpleone-dimensional equationsubjecttoperiodicboundarycondition:

M

(

x

)∂

tu

= ∂

x

(

A

(

x

)∂

xu

).

ThecomputationalcellisdenotedbyIj= [^x_j−¹

2

,

x_j+1

2]^,wherex_j+¹

2’sarethegridpoints.Let

τ

andhbethetimeandspace steps fora uniformmesh andn theindex forthe time stepping. The update on uⁿ_hj, the weighted cell averageof the numericalapproximationuⁿ_h,isgivenby

uⁿ_h⁺¹

j

=

^un_h

j

− τ

h A

∂

xuⁿ_h

∂Ij

where u

j

:=

¹ h

Ij

M

(

x

)

u

(

x

)

dx

.

Notethat

∂

xu_histheapproximationto

∂

xuat

∂

I_j,thecellinterfaceofI_j,givenby

∂

_xu

= β

0

h

[

^u

] + { ∂

_xu

} + β

₁h

[ ∂

_x²u

] .

As mentioned above, this formofthe diffusive flux was originally introduced in [21,22] as partof theDDG methodfor solvingthediffusionproblem.Ifthefluxparameterssatisfy

β

0

≥

1 and 1

8

≤ β

1

≤

¹ 4

,

then theprocedure developedin[23] canbe extendedto thepresentsettingtoconcludethat,undera suitable CFLcon- dition, thesimpleEuler forwardwillkeep thecell average ^u¯ⁿ_j= ^u₁^nh_j^j ∈ [^c1

,

c₂] ^{ineach time step,andthe validity ofthe} maximumprinciplewhencombinedwithascalinglimiter.

For a two dimensional problemwith A=

a c

c b

,the DDG methodon shape-regular Cartesian meshes with

κ

⁻¹_≤

x

y≤

κ

canberenderedMPSif

β

0

≥

¹

+ κ |

c

|

L

(

L

−

1

)

2 min

{

^a

,

b

}

^and

1

8

≤ β

1

≤

¹ 4

.

Here L is the number of Gauss-Lobatto points used in the numerical evaluation of involved integrals. With f

(

u

)

and A=^A

(

x

,

y

,

u

)

,theMPS DDGschemesare analyzedinSection 3.1wheretheparameter rangeandtheCFLconditionsare established.

Themainconclusionisasfollows:byapplyingtheweightedMPSlimiterintroducedin[23] totheDDGschemedesigned herefor(1.1),withthetimediscretizationbyanSSPRunge-Kuttamethod(see[12]),weobtainathirdorderaccuratescheme solving(1.1) satisfyingthestrict maximumprinciple inthesensethat thenumericalsolutionnevergoesoutoftherange [^c1

,

c2]^{asindicatedbytheinitialdata.}

1.3. Organizationofthepaper

Therestofthepaperisorganizedasfollows.InSection2,wedesignthenumericalmethodforonedimensionalprob- lems.WefirstformulatetheDDGschemetosolvethemodelheatequation,andprovetheMPSpropertyofthethirdorder fullydiscretizedDDGscheme,wethenapplytheresulttoshowtheMPSpropertyfornonlinearconvection-diffusionequa- tions.Section3isorganizedsimilarlyfortwodimensionalproblemsonCartesianmeshes.InSection4,wepresenttheMPS limiter,withwhichthealgorithmiscomplete.InSection5,numericaltestsfortheDDGmethodarereported,includingex- amplesfromtheheatequation,porousmediaequationtheBuckley-Leverettequation,andtwodimensionaldiffusionwith theanisotropicdiffusion.ConcludingremarksaregiveninSection6.

2. MPSschemesinonedimension

WefirstinvestigatetheMPS propertyforthirdorderDDGschemestoweighted diffusionequations,andshow howto applytheschemeobtainedtononlinearconvection-diffusionequations.

2.1. Thediffusionequation

Webeginwiththeheatequationoftheform

M

(

x

)∂

tu

= ∂

x

(

A

(

x

)∂

xu

),

(2.1)

withM

(

x

) >

0 and A

(

x

)

≥^{0 onthespatial domain}

,subjecttoinitialdatau₀

(

x

)

andtheperiodicboundarycondition.It isknownthatthefollowingmaximumprincipleholds:

ifc1

≤

^u0

(

x

) ≤

^c2

∀

^x

∈ ,

thenc1

≤

^u

(

t

,

x

) ≤

^c2

∀

^x

∈ ,

t

>

0

.

Ingeneral,theproblemcanbe eitherdefinedonaconnectedcompactdomainwithproperboundaryconditions,oritcan involvethe wholereal linewithsolutions vanishingat theinfinity. Inour numericalscheme,we will always choosethe spatialdomaintobeaconnectedinterval.Forsimplicity,theperiodicboundaryconditionsareapplied.

Wepartition thedomain

by regularcells suchthat

= ^N

j=¹

Ij with Ij= [^x_j−¹

2

,

x_j+1

2]^.Denotehj=^x_j+¹

2−^x_j−¹

2 and

h=^max

j hj.Weseeknumericalsolutionsinthediscontinuouspiecewisepolynomialspace, V_h

= {

^v

∈

^L2

() |

^v

|

Ij

∈

^Pk

(

I_j

),

j

=

¹

· · · ,

N

} .

Here P^k

(

Ij

)

is the space of k-th order polynomials on Ij. Note that the functionsin V_h can be double-valued at cell interfaces. Hence notations v⁻ and v⁺ are used for the left limit and right limit of v. The jump of these two values, v⁺−^{v−,isdenotedby}[^v]^{,andtheaverageby}{^v}^.

ThroughoutthispaperweadopttheDDGnumericalfluxoftheform

∂

xv

= β

0

h_j+1 2

[

v

] + {∂

xv

} + β

1h_j+1

2

[∂

_x²v

]

withh_j+1

2

=

^hj

+

hj+¹

2

,

(2.2)

whencrossingthecellinterfacex_j+1

2,and

(β

0

, β

1

)

areintherangetobespecifiedsothattheunderlyingthirdorderscheme canweaklysatisfy themaximum-principle.Theparameterrangewas firstidentifiedin[23] forathirdorderDDGscheme tofeaturetheMPSpropertyforlinearFokker-Planckequations.

For modelequation (2.1), we consider a

(

k+¹

)

th-order DG scheme: to find u_h∈^Vh such that forany test function v∈^Vh,

Ij

M

(

x

)∂

tu_hv dx

= −

Ij

A

(

x

)∂

xu_h

∂

xv dx

+

^A

∂

xu_hv

+ (

u_h

− {

^uh

})∂

xv

^xj+12

x_j−1 2

,

withthe diffusiveflux

∂

xu_h asdefinedin(2.2). ThisDDGschemewithinterface correction was proposed in[22] forthe diffusionproblem,asanimprovedversionofthatin[21].BasedontheDDGschemein[21],wewillhave

Ij

M

(

x

)∂

tu_hv dx

= −

Ij

A

(

x

)∂

xu_h

∂

xv dx

+

^A

∂

xu_hv

^xj+12

x_j−1 2

.

WiththeforwardEulertimediscretization,theweightedcellaverageforeitheroftwoDDGschemesevolvesas uⁿ_h⁺¹

j

=

^un_h

j

+ μ

h A

∂

xuⁿ_h

^xj+12

xj−¹₂

,

(2.3)

where

μ

_=h^τ2 is themesh ratio.For a concisepresentation, a uniform meshis assumed. Here andin what follows,we denotethetimesteplengthas

τ

.Notethatforperiodicboundaryconditionsconsideredinthispaper,wetake

u⁻

h,¹₂

=

u⁻

h,N+¹₂

,

u⁺

h,N+¹₂

=

u⁺

h,¹₂

intheDDGnumericalfluxformula(2.2) when j=¹₂

,

N+¹₂^. Wealsousethenotation

q

(ξ )

j

=

¹ 2

1

−1

M

(

x_j

+

^h

2

ξ )

q

(ξ )

d

ξ,

for anyq

(ξ )

on

[−

¹

,

1

].

Andthecellaverage^u¯jis^u¯j= ^u₁^h_j^{j.For j}=¹

,

· · ·

,

N,wedefine

a_j

= ξ − ξ

²

j

1

− ξ

j

,

b_j

= ξ + ξ

²

j

1

+ ξ

j

,

(2.4)

and

˜

ω

¹_j

₌ γ − ξ(

1

+ γ ) + ξ

²

j

2

(

1

+ γ ) ,

˜

ω

²_j

₌

¹

^{− ξ}

2

j

1

− γ

²

^,

^(2.5)

˜

ω

³_j

₌ ⁻ γ + ξ(

1

− γ ) + ξ

²

j

2

(

1

− γ )

withtheweightM

(

x

)

|I_j=^M

(

x_j+^h₂

ξ )

.Werecallthefollowingkeyresult.

Lemma2.1.[23,Lemma3.3]

ω

˜ⁱ_j

>

0fori=¹

,

2

,

3ifandonlyif

γ _∈ (

a_j

,

b_j

),

whereaj

,

bjsatisfy−¹

<

aj

<

bj

<

1.

Proof. Hereweonlyshowaj

<

bj,whichmeansthatselectionof

γ

isalwaysensured.Adirectcalculationgives

b_j

−

^aj

=

^{2 1}

j

ξ

²

j

− ξ

²_j 1

+ ξ

j 1

− ξ

j

≥

⁰

,

wherethenumeratorcanbereformulatedas

1

−¹

1

−¹

( η

²

− ξ η )

M

˜ (ξ )

M

˜ ( η )

d

ξ

d

η +

1

−¹

1

−¹

(ξ

²

− η ξ )

M

˜ ( η )

M

˜ (ξ )

d

η

d

ξ

=

1

−1

1

−1

(ξ − η )

²M

˜ (ξ )

M

˜ ( η )

d

ξ

d

η >

0

,

where M˜

(ξ )

:=^M

(

xj+^h₂

ξ )

hasbeenused. 2 Wethushavethefollowingresult.

Theorem2.2.(k=^{2)Thescheme(2.3) with}

β

₀

≥

^{1 and 1}

8

≤ β

₁

≤

¹

4 (2.6)

ismaximum-principle-satisfying,namely,c1≤ ¯^un_j⁺¹≤^c2ifuⁿ_h

(

x

)

isin[^c1

,

c2]^on{^Sj}^N_j=1^,where

S_j

=

^xj

+

^h 2

−

¹

, γ ,

1

with

γ

satisfying

a_j

< γ <

b_j and

| γ | ≤

⁸

β

1

−

¹

,

(2.7)

undertheCFLcondition

μ

≤

μ

0,where

μ

0isgivenin(2.12)below.

Proof. Step1.Weightedintegraldecomposition.Define p_j

(ξ ) =

^uh

x_j

+

^h

2

ξ

for

ξ ∈ [−

¹

,

1

],

weseethatinthecaseof pj

(ξ )

∈^P2[−¹

,

1]^,forany

γ

∈

(

−¹

,

1

)

,theuniqueinterpolationof pjatthreepoints{−¹

, γ ,

1} givesthefollowing

p_j

(ξ ) = (ξ −

¹

)(ξ − γ )

2

(

1

+ γ )

^pj

(−

¹

) + (ξ −

¹

)(ξ +

¹

)

( γ −

¹

)( γ +

¹

)

^pj

( γ ) + (ξ +

¹

)(ξ − γ )

2

(

1

− γ )

^pj

(

1

).

(2.8)

Thisyieldsthefollowingidentityfortheweightedaverage,

u_h

j

= ˜ ω

¹_jpj

( −

¹

) + ˜ ω

²_jpj

( γ ) + ˜ ω

³_jpj

(

1

),

(2.9) where

ω

˜ⁱ_jgivenin(2.5) areensuredpositivebyLemma2.1.

Step2.Fluxrepresentation.Adirectcalculationgives h

∂

xu_h

x_j+1 2

= α

3

(− γ )

pj+¹

(−

1

) + α

2

(− γ )

pj+¹

( γ ) + α

1

(− γ )

pj+¹

(

1

)

(2.10)

−

α

1

( γ )

pj

(−

1

) + α

2

( γ )

pj

( γ ) + α

3

( γ )

pj

(

1

) ,

where

α

1

( γ ) =

⁸

β

1

−

¹

+ γ

2

(

1

+ γ ) , α

2

( γ ) =

²¹

−

⁴

β

1

1

− γ

²

^, α

3

( γ ) = β

0

+

⁸

β

1

−

³

+ γ

2

(

1

− γ ) ,

areallpositivedueto(2.6) and(2.7).

Step3.MonotonicityundersomeCFLcondition.Wenowsubstitute(2.9) and(2.10) into(2.3) toobtain uⁿ_h⁺¹

j

=

^Rn_j

(

M

(·), μ ,

h

,

A

)

with

Rⁿ_j

(

M

( · ), μ ,

h

,

A

) =

^un_h

j

+ μ

Ah

∂

_xuⁿ_h

xj+¹₂

−

^Ah

∂

_xuⁿ_h

xj−¹₂

(2.11)

= ω ˜

¹_j

− μ α

3

(− γ )

A_j−1

2

+ α

1

( γ )

A_j+1 2

p_j

(−

¹

) + ω ˜

²_j

₋ μ α

2

( − γ )

A_j−1

2

+ α

2

( γ )

A_j+1 2

pj

( γ ) + ω ˜

³_j

− μ

α

1

(− γ )

A_j−1

2

+ α

3

( γ )

A_j+1 2

p_j

(

1

) + μ

A_j+1

2

α

3

(− γ )

p_j+1

(−

¹

) + α

2

(− γ )

p_j+1

( γ ) + α

1

(− γ )

p_j+1

(

1

) + μ

A_j−1

2

α

1

( γ )

pj−¹

(−

1

) + α

2

( γ )

pj−¹

( γ ) + α

3

( γ )

pj−¹

(

1

) .

Hereitisunderstoodthat p0

(ξ )

:=^pN

(ξ

+ |

|

)

andpN+¹

(ξ

+ |

|

)

=^p1

(ξ )

forincorporatingtheperiodicboundarycondi- tions.Notealsothat₃

i

α

i

( γ )

=

β

0=₃

i

α

i

(− γ )

.Usingthefact

ω

˜³_j

( γ )

= ˜

ω

¹_j

(− γ )

andformulasfor

α

2

( γ )

,

ω

˜²_j,weseethat if

μ

ischosentobesmallerthan

μ

0where

μ

0

=

max

1≤j≤NA

(

x_j+1 2

)

₋1

min

1≤j≤N

ω ˜

¹_j

(± γ )

α

3

(∓ γ ) + α

1

(± γ ) ,

¹

− ξ

²

j

4

(

1

−

⁴

β

1

)

,

(2.12)

(2.11) isnondecreasing inthepointvalues p_j

(±

¹

),

p_j

( γ ),

p_j±1

(±

¹

),

p_j±1

( γ )

,hencewhenthesevaluesare replacedwith thelowerandupperboundsc1 andc2 respectively,wehave

c₁

3 i=¹

˜

ω

ⁱ_j

_≤

uⁿ_h⁺¹

j

≤

^c2

3 i=¹

˜ ω

ⁱ_j

,

sincethetermswith

α

i’sarecanceled out.Moreoverthesumof

ω

˜ⁱ_jis 1j.Therefore c₁ 1

j

≤

^un_h⁺¹

j

≤

^c2 1

j

⇒

^c1

≤ ¯

^un_j⁺¹

≤

^c2

. 2

Remark2.1.Forothertypesofboundaryconditions,theboundaryfluxneedstobemodified.AsimilarresulttoTheorem2.2 maybeestablishedaslongasthePDEproblemsatisfiesamaximumprinciple.

Remark2.2.Theuseof

γ

isessentialforthesuccessofourschemes,inparticularwhen M

(

x

)

isafunction.Forthespecial case M

(

x

)

=^{1,we have}

γ

∈

(−

¹₃

,

¹₃

)

,hence

γ

=^{0, which}correspondstothe usualGauss quadraturepoint,isadmissible.

TheCFLnumbergivenin(2.12) maybeoptimizedbycarefullytuning

γ

∈

(

a_j

,

b_j

)

,butnotinalinearfashion.Nevertheless, itwasobservedin[23] thatthelarger|

γ

_|is,thebetterthescheme’sperformancefortheheatequation.

2.2. Applicationtononlinearconvection-diffusionequation

InthissectionwewilldemonstratehowtoapplyourMPSDGmethodinsection2.1tothenonlinearconvection-diffusion equation,

∂

tu

+ ∂

xf

(

u

) = ∂

x

(

A

(

x

,

u

)∂

xu

),

where f

(

u

)

isa smooth functionanddiffusioncoefficient A

(

x

,

u

)

≥^{0, subjectto initialdata u}

(

0

,

x

)

=^u0

(

x

)

,andperiodic boundary conditions.ByapplyingtheDGapproximation,we obtainthe followingscheme.Weseekuh∈^Vh such thatfor anytestfunction v∈^Vh,

I_j

∂

tu_hv dx

=

I_j

f

(

u_h

)∂

xv dx

− ˆ

^f

(

u⁻_h

,

u_h⁺

)

v

^xj+12

xj−¹₂

(2.13)

−

I_j

A_h

∂

xu_h

∂

xv dx

+ {

^Ah

} ∂

xu_hv

+ (

u_h

− {

^uh

})∂

xv

^xj+12

xj−¹₂

.

For diffusionpart, we adoptthe DDGdiffusive flux (2.2) For convection,anymonotone numericalflux can be used,i.e., ˆf

(

u

,

v

)

isLipschitzcontinuous,nondecreasinginuandnonincreasinginv,consistentwith f

(

u

)

inthesensethat ˆf

(

u

,

u

)

= f

(

u

)

.Forexample,theglobalLax-Friedrichsflux

ˆ

f

(

u⁻_h

,

u⁺_h

) =

¹

2

(

f

(

u⁻_h

) +

f

(

u⁺_h

) − σ (

u⁺_h

−

u⁻_h

)), σ =

max

u∈[c1,c2]

|

f

(

u

)|.

WeconsiderthefirstorderEulerforwardtemporaldiscretizationof(2.13) toobtain

Ij

uⁿ_h⁺¹

−

^un_h

τ

^{v dx}

⁼

Ij

f

(

uⁿ_h

)∂

xv dx

− ˆ

^f

((

uⁿ_h

)

⁻

, (

uⁿ_h

)

⁺

)

v

^xj+12

xj−¹₂

(2.14)

−

Ij

Aⁿ_h

∂

xuⁿ_h

∂

xv dx

+ {

^An_h

} ∂

xuⁿ_hv

+ (

uⁿ_h

− {

^un_h

} )∂

xv

^xj+12

xj−¹₂

.

Bytakingthetestfunctionv=^{1 onI}j and0 elsewhere,weobtaintheevolutionaryupdateforthecellaverage,

¯

uⁿ_j⁺¹

= ¯

^un_j

− λ ˆ

_fⁿ

^xj+12

xj−¹₂

+ μ

h

{

^An_h

} ∂

xuⁿ_h

^xj+12

xj−¹₂

,

where

λ

=^τ_h ^and

μ

=_h^τ2 arethemeshratios.Assumingthat^u¯ⁿ_j∈ [^c1

,

c₂]^{forall j’s,wewouldliketoderivesomesufficient} conditionssuchthatu¯ⁿ_j⁺¹∈ [^c1

,

c2]^undercertainrestrictionson

λ

and

μ

.

Forpiecewisequadraticpolynomials,themainresultcanbestatedasfollows.

Theorem2.3.(k=^2)Thescheme(2.14) with

β

0

≥

^{1 and 1}

8

≤ β

1

≤

¹ 4

ismaximum-principle-satisfying;namely,^u¯ⁿ_j⁺¹∈ [^c1

,

c2]^ifun_h

(

x

)

∈ [^c1

,

c2]^onthesetSj’swhere

S_j

=

^xj

+

^h 2

−

¹

, γ ,

1

with

γ

satisfying

−

¹

3

< γ <

¹

3 and

| γ | ≤

8

β

1

−

1

,

(2.15)

undertheCFLcondition

λ ≤ λ

0

, μ ≤ μ

0

forsome

λ

0and

μ

0definedin(2.17) and(2.18),respectively.

Proof. Wepresenttheproofinfoursteps:

Step1.Split:wesplittheaverageu¯ⁿ_j intotwohalvessothat

¯

uⁿ_j⁺¹

=

¹

2Cⁿ_j

+

¹ 2Dⁿ_j

,

wheretheconvectiontermis Cj

= ¯

^uj

−

²

λ ˆ

_f

^xj+12

xj−¹₂

(2.16)

andthediffusiontermis

Dj

= ¯

^uj

+

²

μ

h

{

^An_h

} ∂

xu_h

^xj+12

xj−¹₂

.

ThissplitisforconvenientpresentationandmaynotleadtoanoptimalCFLcondition.

Step2.Theintegraldecomposition.From(2.9) itfollows

¯

u_j

= ω

¹p_j

(−

¹

) + ω

²p_j

( γ ) + ω

³p_j

(

1

),

where

ω

ⁱ= ˜

ω

ⁱ

,

uhj= ¯^ujforM

(

x

)

≡^1,and

ω

¹

=

¹

+

³

γ

6

(

1

+ γ ) , ω

²

=

²

3

(

1

− γ

²

) , ω

³

=

¹

−

³

γ

6

(

1

− γ ) .

Thesecoefficientsarepositivefor

γ

satisfying(2.15).

Step3.Theconvectionterm.

Usingthecellaveragedecompositionandthefluxformula,werewrite(2.16) as Cj

= ¯

^un_j

−

²

λ

f

xj+¹₂

−

f

xj−¹₂

= ω

³p_j

(

1

) −

²

λ ˆ

f

(

p_j

(

1

),

p_j+1

(−

¹

)) + ω

²p_j

( γ ) + ω

¹p_j

(−

¹

) +

²

λ ˆ

f

(

p_j−1

(

1

),

p_j

(−

¹

))

=:

G

(

pj−¹

(

1

),

pj

(−

1

),

pj

( γ ),

pj

(

1

),

pj+¹

(−

1

)).

For a monotone flux ˆf

(

u

,

v

)

being Lipschitz continuous with Lipschitz constant L^{, G is} non-increasing in all the four argumentsprovidedthefollowingconditionismet,

2

λ

L

≤

^min

{ ω

¹

, ω

³

_} .

NotethatfortheLax-Friedrichsflux,L= ^max

u∈[c1,c2]|^f

(

u

)|

^{.Moreover,the}consistencyoftheflux ˆf

(

u

,

u

)

= ^f

(

u

)

impliesthat G

(

u

,

u

,

u

,

u

)

=^{u.Hencewehave}

Cj

∈ [

^G

(

c₁

,

c₁

,

c₁

,

c₁

,

c₁

),

G

(

c₂

,

c₂

,

c₂

,

c₂

,

c₂

)] = [

^c1

,

c₂

],