solutions Galerkinmethod for parabolic equations with blow-up Positivity preserving high-order local discontinuous Journal of Computational Physics

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Journal of Computational Physics

www.elsevier.com/locate/jcp

Positivity preserving high-order local discontinuous Galerkin method for parabolic equations with blow-up solutions

Li Guo

^a,b

, Yang Yang

^b,∗

aSchoolofMathematicalSciences,UniversityofScienceandTechnologyofChina,Hefei,Anhui 230026,PR China bDepartmentofMathematicalSciences,MichiganTechnologicalUniversity,Houghton,MI 49931,UnitedStates

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

Articlehistory:

Received13October2014

Receivedinrevisedform17January2015 Accepted18February2015

Availableonline2March2015

Keywords:

Positivity-preserving Blow-upsolutions

LocaldiscontinuousGalerkinmethod Dirichletboundaryconditions

Inthispaper,we applypositivity-preservinglocal discontinuousGalerkin(LDG)methods to solve parabolic equations with blow-up solutions. This model is commonly used in combustionproblems.However,previousnumericalmethodsaremainlybasedonasecond orderfinitedifferencemethod.Thisisbecausethepositivity-preservingpropertycanhardly be satisfied for high-order ones, leading to incorrect blow-up time and blow-up sets.

Recently,we haveapplieddiscontinuousGalerkinmethodstolinearhyperbolicequations involvingδ-singularitiesandobtainedgoodapproximations.Fornonlinearproblems,some special limiters are constructedto capture the singularities precisely. We will continue thisapproach and studyparabolic equations withblow-up solutions.We will construct speciallimiterstokeepthepositivityofthenumericalapproximations.DuetotheDirichlet boundaryconditions,we havetomodifythenumericalfluxesandthelimitersusedinthe schemes.Numericalexperimentsdemonstratethatourschemes cancapturetheblow-up sets,andhigh-orderapproximationsyieldbetternumericalblow-uptime.

©^{2015ElsevierInc.}All rights reserved.

1. Introduction

Inthispaper,we willdevelopandanalyzethepositivity-preservinghigh-orderlocaldiscontinuousGalerkinmethodsfor parabolicequationswithreactiontermsinonespacedimension

ut

= (

u^α

)

xx

+

s

(

u

),

x

∈ [

xa

,

xb

],

u₀

(

x

) =

^u

(

x

,

0

) ≥

⁰

,

x

∈ [

^xa

,

x_b

] ,

(1)

aswellasitstwodimensionalextension,where

α

≥^{1 isaparameterands}(u)≥^0.We considerDirichletboundarycondi- tions

u

(

xa

,

t

) =

g1

(

t

),

and u

(

xb

,

t

) =

g2

(

t

).

Thesourceterm s(u)ischosen tobesuperlinear,i.e.s(u)=^{um withm}≥^{1.The initialcondition u}=^u0(x) isassumedto benon-negative. Byamaximum-principle,theexactsolutionu≥^{0 fort}∈(0,T).Here(0,T)isthemaximal timeinterval

*

Correspondingauthor.

E-mailaddresses:[email protected],[email protected](L. Guo),[email protected](Y. Yang).

http://dx.doi.org/10.1016/j.jcp.2015.02.041 0021-9991/©^{2015ElsevierInc.}All rights reserved.

of existenceofthesolution u.The time T maybefiniteorinfinite.When T isinfinite,we saythat thesolution u exists globally.WhenT isfinite,thesolutionu developsasingularityinfinitetime,namely

lim

t→T⁻

u

(·,

t

)

_∞

= ∞,

where^u∞ isthestandard L^∞-normofu on[^xa,x_b]^{.In thiscase,we saythatthesolutionublowsupinfinitetimeand} thetimeT iscalledtheblow-uptimeofthesolutionu,and{^x∈ [^xa,xb]|^u(x,t)→ ∞^asT→^T−}^{iscalledtheblow-upset.}

In this paper,we assumethe boundaryisnot includedinthe blow-upset.Therefore,the exactsolutionatthe boundary shouldbesmallerthanthevaluesnearby.

We can regard (1) as a mathematical model of combustion, andthe unknown variable u can be interpreted as the temperature.We canrewrite(1)as

ut

= (

a

(

u

)

ux

)

x

+

s

(

u

),

x

∈ [

xa

,

xb

],

u

(

x

,

0

) ≥

⁰

,

x

∈ [

^xa

,

x_b

],

⁽²⁾

witha(u)=

α

uα−¹,whichcanbeconsideredasanonlinearheatconductioncoefficientofthemedium.If

α

₌1,(2)reduces tothewell-studiedsemilinearheatequation[4].Suchproblemshavebeeninvestigatedbymanyauthorsandexistenceand uniquenessofaclassicalsolutionhavebeenproved(seee.g.[26,34,6,5,20,21,29,25,41]forsomerecentworks).Undersome assumptions,it isalsoshownthattheclassicalsolutionblowsupinfinitetimeandtheblow-up timehasbeenestimated.

Moreover,manynumericalmethodsarealsobeenstudiedin[11,8,9,28,13,22,35,2,7,31,10,1,27].However,to thebestknowl- edge ofthe authors,all the previous methodsare based onsecond-order finite differencemethods, whichguarantee the positivityofthe numericalapproximationsautomaticallyundersome specialrequirementsforthemeshes.Forhigh-order methods,strongoscillationsnearthesingularitiesmaysendphysicallypositivequantitiesnegative,andnegativecoefficient ofthediffusiontermwillmakethenumericalsolutionblowup immediately,leadingtowrongblow-up timeandblow-up sets. In this paper,we wouldlike toapply high-order positivity-preserving localdiscontinuous Galerkinmethods tosuch problems, andnumericallystudythe blow-up time,blow-up locations andthebehavior ofthe solutions beforeblow-up.

We willalsousenumericalexperimentstodemonstratethesignificanceofthepositivity-preservingtechnique.

The study of the blow-up solutions is important and challenging. Since the exact solutions may not be smooth or boundedatfinitetime,thenumericalschemesmightyieldpoorapproximations.Recently,we[38]haveapplieddiscontin- uous Galerkin(DG)methodstoobtaingoodapproximationsforPDEswithδ-singularity,onespecialunboundedsingularity.

In[38],we provedthesuperconvergenceresultsforlinearhyperbolicequationswithsingularinitialdataandsingularsource terms.Moreover,severalnumericalexampleswerealsogivenin[38–40]todemonstratetheadvantages oftheDGscheme in approximating δ-singularitiesfor both linear andnonlinear hyperbolic equations.In this paper,we will continue this approach,andusethehigh-orderDGmethodstosolveparabolicPDEswithblow-upsolutions.

The DG method was first introduced in 1973 by Reed and Hill [30], in the framework of neutron linear transport.

Later, the method was applied by Johnson and Pitkäranta to a scalar linearhyperbolic equation and the L^p-norm error estimate was proved [24]. Subsequently, Cockburn etal.developed Runge–Kuttadiscontinuous Galerkin (RKDG)methods for hyperbolic conservation lawsin a seriesof papers [16,15,14,17]. In [18], Cockburn andShu first introduced the LDG methodtosolvetheconvection–diffusionequation.TheirideawasmotivatedbyBassiandRebay[3],wherethecompressible Navier–Stokesequationsweresuccessfullysolved.

Theideaofthepositivity-preservingtechniquewewouldliketoapplyinthispaperisdifferentfromtheoneusedbefore in [43–45].In[45], theauthorsconstructsecond-orderpositivity-preserving DGschemesandarguedthat itwas impossi- ble toconstructathird-order positivity-preservingDGschemefollowing thesameapproach.Recently, in[36],theauthor appliedthefluxlimiterandconstructedmaximum-principle-satisfyingfinitedifferencemethods,whichfurthergeneralized tohigh-orderpositivity-preservingDGmethodsforconvectiondiffusionproblems.TheDGmethodappliedin[36]isbased onUltra-weakDG[12].We willconsiderLDGmethodandstudythepositivity-preservingtechnique.Anotherrelatedwork canbefoundin[42],wherethepositivity-preservingtechniqueforporousmediumequationshavebeenanalyzed.However, only a second-order scheme was considered and the technique can hardly be extended to high-order schemes. Another crucialproblemistheboundarytreatment fortheLDGmethod.NumericalexperimentsdemonstratethatthegeneralLDG methodwilldegenerateaccuracywhensolvingproblemswithDirichletboundaryconditions.Following[19],we wouldlike toaddapenaltyterminthefluxattheboundary.Withthepenaltyterm,theconvergencerateisoptimal.Moreover,we will alsotheoreticallyprovethatthepenaltytermisrequiredforthepositivity-preservingtechnique.

The organization of this paper is as follows. In Section 2, we will present the positivity-preserving high-order LDG methods inoneandtwospacedimensions, andtheimplementationoftheDirichletboundarycondition.Some numerical experiments will be giveninSection 3. We willend inSection 4 withsome concludingremarks andremarks on future work.

2. Positivity-preservinghigh-orderlocaldiscontinuousGalerkin methods

Inthissection,we presentthepositivity-preservingtechniqueappliedtotheLDGmethodsforparabolicequationssubject to Dirichletboundaryconditions.We firstintroducetheLDG methodsanddiscusshowtoenforcethe Dirichletboundary conditions. Then we describe how to apply the positivity-preserving flux limiter to the numerical fluxesin the scheme.

We discusstheEuler-forwardtime discretization,andthehigh-orderonesarestraightforwardextendable. Thefluxlimiter will be applied to guarantee the positivity of the numerical cell averages at time level n+^{1, provided the numerical} approximations are positive at time level n. However, the numerical approximations may not be positive at time level n+^{1.Therefore,we wouldliketomodifythenumerical}approximationwithsomeslopelimiters,keepingthecellaverages untouched.We willshowtheprocedureinoneandtwospacedimensions.

2.1. LocaldiscontinuousGalerkinmethod

Inthissubsection,we willdeveloptheLDGmethodinonespacedimensionandstudy(2).To constructtheLDGscheme, we considerapartitionforthespatialdomain[^xa,x_b]^givenas

x_a

=

^x1

2

<

x3

2

< · · · <

x_N−1 2

<

x_N+1

2

=

^xb

,

(3)

anddefine Ij

= (

x_j−1

2

,

x_j+1 2

)

tobethecell.Forsimplicity,we useuniformmeshonly,andthemeshsize isdenotedasx.Thefiniteelementspaceis givenas

V_h

=

u

∈

^Pk

(

I_j

)|

^j

=

¹

,

2

,· · · ,

N

,

where P^k(I_j)denotesthespaceofpolynomialsofdegreeatmostkontheintervalI_j.To constructtheLDGschemefor(2), we introduceanauxiliaryvariable

q

=

^a

(

u

)

u_x

,

with a

(

u

) =

a

(

u

),

then(2)canbewrittenintoafirst-ordersystem

u_t

= (

a

(

u

)

q

)

_x

+

^s

(

u

),

(4)

q

= (

g

(

u

))

x

,

(5)

whereg(u)=u

a(

τ

)d

τ

.Forsimplicity,we omitthesubscripthinSubsections2.1and2.2,anduseuandqforuh andqh, respectively.ThenthegeneralformulationoftheLDGschemeistofindu,q∈^Vh,suchthatforanytestfunctionv,w∈^Vh

Ij

utvdx

= −

Ij

a

(

u

)

qvxdx

+ (

_a

(

u

)

qv⁻

)

_j+1

2

− (

_a

(

u

)

qv⁺

)

_j−1

2

+

Ij

s

(

u

)

vdx

,

(6)

Ij

qwdx

= −

Ij

g

(

u

)

wxdx

+ (

_g

(

u

)

w⁻

)

_j+1

2

− (

_g

(

u

)

w⁺

)

_j−1

2

,

(7)

wherea(u)qandg(u)in(6)–(7)arethenumericalfluxeswhicharedefinedatthecellinterfaces.In thispaper,we consider alternatingfluxesonly,i.e.

(

_a

(

u

)

q

)

_j+1

2

=

[

^g

(

u

) ] [

^u

]

^q+

j+¹₂

+

^Cj+12

x

[

^u

]

_j+¹

2

, (

_g

(

u

))

_j+1 2

=

^g

(

u⁻

j+¹₂

),

(8)

or

(

a

(

u

)

q

)

_j+1

2

=

[

^g

(

u

)]

[

^u

]

^q−

j+¹₂

+

^Cj+12

x

[

^u

]

_j+¹

2

, (

g

(

u

))

_j+1 2

=

^g

(

u⁺

j+¹₂

),

(9)

whereu⁺

j+¹₂ istherightlimit ofuatx=^x_j+¹

2.Likewiseforu⁻ j+¹₂,q⁺

j+¹₂ andq⁻

j+¹₂.Moreover, we denote[^u]_j+¹

2=^u+_j+1 2

− u⁻

j+¹2

asthejumpofuatthecellinterfacex_j+1 2.C_j+1

2 isthecoefficientforthepenaltyterm.Duetotheboundarycondition, we havetotakeu⁻1

2=^g1(t)andu⁺

N+¹2 =^g2(t),i.e.wehavetotake(8)for j=^{0 and(9)for j}=^{N.In thispaper,we would} liketo use(8)forall theother j’s, andinpractice,we cantake C_j+1

2 =^{0, j}=⁰,1,· · ·,N−^{1 (see}Lemma 2.1).However, if we choose C_j+1

2 =^{0 for all j, the scheme is} problematic. In [19], the author studied the steady-state problem, and argued that it isnecessary to haveall the C’s to be zero expect theone at x=^x_N+¹

2, otherwisethe numericalsolution is not solvable. For time-dependent problems, even though the numerical solutions can be solved without the penalty term, numerical experiments demonstrated that, at x=^x_N+¹

2, the first-order numerical approximations are inconsistent withtheexactsolutions,eventhoughthey aresufficientlysmooth.Forhigh-orderapproximations,we cannotobservethe inconsistency,buttheorderofaccuracymaynotbe optimal.On this otherhand,withthepenalty,we will usenumerical

experiments to demonstrate theoptimality ofthe errorestimates. Following thesame idea in [19], we want C_N+1 2 >0.

Therefore,thefluxattherightboundary(x=^x_N+¹

2=^xb)isdifferentfromthoseelsewhere.We willdiscusshowtochoose theparameterlater.

2.2. Positivity-preservingtechnique

Inthissubsection,we willdemonstratethepositivity-preservingtechnique.Forsimplicitywedonotconsiderthecon- tribution from the source,and assume s(u)=^{0. We can makesuch an assumption becausethe source will never make} negative contribution to thepositivity of the cell averages forexplicit time discretizations. The whole procedure can be dividedintothreesteps.First,we needtoprovethepositivityofthefirst-orderapproximations.Thenapplythefluxlimiter tohigh-orderapproximationstoobtainpositivecellaverages.Finally,we modifythenumericalapproximations,keepingthe cellaveragesuntouched.WithEulerforwardtimediscretization,thefirst-orderLDGschemecanbewrittenas

uⁿ_j⁺¹

=

uⁿ_j

+

t

x

((

_a

(

uⁿ

)

qⁿ

)

_j+1

2

− (

_a

(

uⁿ

)

qⁿ

)

_j−1 2

),

qⁿ_j

=

¹

x

(

g

(

uⁿ

)

_j+1

2

−

g

(

uⁿ

)

_j−1 2

).

Here t isthe timemesh size.uⁿ_j,aconstant, isthenumerical approximationattime leveln incell Ij.Likewiseforqⁿ_j. Forsimplicity,if weconsiderthenumericalapproximationsattimeleveln,thenthecorrespondenceindexwillbeomitted.

If we choose the penalty parameters to be large enough, then there exists sufficient small t such that the first-order approximationsarepositive.Theresultisgivenbelow.

Lemma2.1.WecantakeC_N+1

2 andC_N−1

2 satisfy(12)and(13),andλ= _x^t2 satisfies(10),(11)and(14),thenthenumerical approximationfromthefirstorderschemeispositive.

Proof. Weconsiderthreedifferentcases.Forconvenience,we defineu0=^g1(t)anduN+¹=^g2(t). 1. For j=¹,2,· · ·,N−^2,g(u)_j−1

2 =^g(uj−¹)andqj=¹_x(g(uj)−^g(uj−¹)).Therefore, uⁿ_j⁺¹

=

^uj

+

t

x

g

(

uj+¹

) −

g

(

uj

)

uj+¹

−

^uj

qj+¹

−

^g

(

uj

) −

g

(

uj−¹

)

uj

−

^uj−¹ qj

=

uj

+ λ (

g

(

uj+¹

) −

g

(

uj

))

²

uj+¹

−

uj

− (

g

(

uj

) −

g

(

uj−¹

))

² uj

−

uj−¹

=

uj

+ λ

f²

j+¹₂

(

uj+¹

−

uj

) −

f²

j−¹₂

(

uj

−

uj−¹

)

= λ

f²

j+¹₂u_j+1

+

1

− λ(

f²

j+¹₂

+

^f2_j−1 2

)

u_j

+ λ

f²

j−¹₂u_j−1

,

where f_j+1

2

=

^g

(

u_j+1

) −

^g

(

u_j

)

u_j+1

−

^uj ispositive.Therefore,if wetake

λ(

f²

j+¹₂

+

^f2_j−1 2

) ≤

¹

,

(10)

wehaveuⁿ_j⁺¹≥^0.

2. For j=^N−^1,C_j+¹

2 maynotbezero.Followthesameanalysisabove,we have uⁿ_j⁺¹

= λ

f²

j+¹₂u_j+1

+

1

− λ(

f²

j+¹₂

+

^f2_j−1 2

)

u_j

+ λ

f²

j−¹₂u_j−1

+ λ

C_j+1

2

(

u_j+1

−

^uj

)

= λ

f²

j+¹₂

+

^C_j+¹

2

u_j+1

+

1

− λ(

f²

j+¹₂

+

^f2_j−1 2

+

^C_j+¹

2

)

u_j

+ λ

f²

j−¹₂u_j−1

.

Therefore,if wetake

λ(

f²

j+¹₂

+

^f2_j−1 2

+

^C_j+¹

2

) ≤

¹

,

(11)

wehaveuⁿ_j⁺¹≥^0.

3. For j=^{N,theanalysisisquitedifferent.}

qN

=

^g

(

uN+¹

) −

^g

(

uN−¹

)

x

=

¹

x

f_N+1

2

(

uN+¹

−

uN

) +

f_N−1

2

(

uN

−

uN−¹

)

and

uⁿ_N⁺¹

=

uN

+ λ

f_N+1 2

−

f_N−1

2

qN

x

+

C_N+1

2

(

uN+¹

−

uN

) −

C_N−1

2

(

uN

−

uN−¹

)

=

^AN+1u_N+1

+

^ANu_N

+

^AN−1u_N−1

,

where

A_N+1

= λ

f_N+1 2

(

f_N+1

2

−

^f_N−¹

2

) +

^C_N+¹

2

,

A_N

=

¹

− λ

(

f_N+1

2

−

^f_N−¹

2

)

²

+

^C_N−¹

2

+

^C_N+¹

2

,

A_N−1

= λ

C_N−1

2

−

^f_N−¹

2

(

f_N+1 2

−

^f_N−¹

2

)

.

Therefore,we havetotake C_N−1

2

≥

^f_N−¹

2

(

f_N+1 2

−

^f_N−¹

2

),

(12)

C_N+1

2

≥ −

^f_N+¹

2

(

f_N+1 2

−

^f_N−¹

2

),

(13)

and

λ

(

f_N+1 2

−

^f_N−¹

2

)

²

+

^C_N−¹

2

+

^C_N+¹

2

≤

¹

,

(14)

toobtainuⁿ_j⁺¹>0. 2 Remark1.Since f_j+1

2 ispositive,thereforein(12)and(13),we onlyneedtohaveonenonzeroparameterC.Basedonthe assumptionthat theblow-upneveroccursattheboundary, theexactsolutionattheboundaryshould beboundedandis smaller thanthe valuesnearby. Therefore,we can take C_N−1

2 =^{0 and C}_N+¹

2 tobe a boundedconstant, andthiswill be verifiedbynumericalexperimentsinSection3.

Remark2.If

α

=^{1,then f}_j+¹

2 =^{1 forall j}=⁰,1,· · ·,N.Therefore,toconstructpositive numericalcellaverages, we can takeallthepenaltyparameterstobezero,andλ≤¹₂^{.However,duetotheboundaryeffect,C}_N+¹

2 >0 isalsorequired.

With the above lemma, we can proceed to construct positive numerical cell averages for high-order approximations.

We takethetestfunction v=^{1 anddividedby} xonbothsidesof(6)toobtaintheequationsatisfiedby thenumerical cellaveragesu¯j

¯

uⁿ_j⁺¹

= ¯

^un_j

+

t

x

(

H

ˆ

_j+1

2

− ˆ

^H_j−¹

2

),

(15)

wherethe flux Hˆ =a(u)q. Thenumerical cellaverages computedform (15)mightbe negative, andwe haveto apply a positivity-preservingfluxlimiterontheflux H.ˆ Theideaistoapply Lemma 2.1toconstructafirst-ordernumericalfluxhˆ basedonthenumericalcellaverageattimelevelnandx=^x_j+¹

2.Thenextstepistomodifythenumericalfluxtoobtaina newone

H

˜

_j+1 2

= θ

_j+1

2

(

H

ˆ

_j+1 2

− ˆ

^h_j+¹

2

) + ˆ

^h_j+¹

2

.

(16)

Theparameterθ_j+1

2 isdesignedtoensure^u¯ⁿ_j^{+1 tobepositive.Actually,we canchoose}θ_j+1

2 tobenonnegative,sinceifwe takeθ_j+1

2=^0,thenu¯ⁿ_j⁺¹ isobtainedfromthefirst-ordernumericalscheme,whichisproved toyieldapositivenumerical solutioninLemma 2.1.In[37],theauthorgaveanideatochooseθ_j+1

2.Thegoalistohave

¯

uⁿ_j⁺¹

= ¯

^un_j

+

t

x

(

H

˜

_j+1

2

− ˜

^H_j−¹

2

) > ε ,

where

ε

=^{10−13isasmallpositivenumberrelatedtomachineprecision.Theaboveequationfurtheryields}

λθ

_j+1

2F_j+1

2

− λθ

_j−1 2F_j−1

2

−

j

≥

0

,

(17)

whereλ=^t_x^{, F}j±¹₂ = ˆ^Hj±¹₂ − ˆ^hj±¹₂ and j=

ε

₋(¯uⁿ_j+λ(_hˆ_j+1

2 − ˆ^hj−¹₂)).The parameters θ_j±1

2 can be obtainedas fol- lows[37].We want

θ

_j−1

2

∈ [

⁰

,

₋1

2,Ij

], θ

_j+1

2

∈ [

⁰

,

1 2,Ij

],

where_±1

2,Ij aredesignedtokeep(17)correct.

•^{If F}_j+¹

2≥^{0 and F}_j−¹

2≤^0,then

(

1

2,I_j

,

₋1

2,I_j

) = (

1

,

1

).

•^{If F}_j+¹

2<0 and F_j−1

2 ≤^0,then

(

1

2,I_j

,

₋1 2,I_j

) =

min

1

,

_j

λ

F_j+1

2

,

1

.

•^{If F}_j+¹

2≥^{0 and F}_j−¹

2>0,then

(

1 2,Ij

,

₋1

2,Ij

) =

1

,

min

1

,−

j

λ

F_j−1 2

.

•^{If F}_j+¹

2<0 and F_j−1

2 >0,then – IfEq.(17)issatisfiedwith(θ_j+1

2,θ_j−1

2)=(1,1),then

(

1 2,Ij

,

₋1

2,Ij

) = (

1

,

1

).

– Ifnot,then

(

1 2,I_j

,

₋1

2,I_j

) =

j

( λ

F_j+1

2

− λ

F_j−1

2

) ,

j

( λ

F_j+1

2

− λ

F_j−1 2

)

.

Thelocalparameterθ_j+1

2 canbechosenas

θ

_j+1

2

=

^min

(

1

2,I_j

,

₋1 2,I_j+1

).

Followtheabove steps,we haveu¯ⁿ_j⁺¹>0,provideduⁿ_j>0 whereuⁿ_j isthepolynomialatmostdegreek inthecell Ij at thetimeleveln.

Finally,we havetomodifythenumericalapproximationsattimeleveln+^{1,sincetheymaynotbepositive,eventhough} thecellaveragesarepositive.Theideaistousethemodified u˜ⁿ_j⁺¹toreplacethenumericalsolutionuⁿ_j⁺¹,andu˜ⁿ_j⁺¹canbe obtainedasfollows.In thecellI_j

˜

uⁿ_j⁺¹

= ¯

^un_j⁺¹

+

_j

(

uⁿ_j⁺¹

− ¯

^un_j⁺¹

) ≥ ε ,

where

j

=

min

u

¯

ⁿ_j⁺¹

− ε

¯

uⁿ_j⁺¹

−

mⁿ_j⁺¹

,

1

,

with

mⁿ_j⁺¹

=

min

x∈Ij

uⁿ_j⁺¹

(

x

).

In[44],theauthorarguedthatsuchalimiterdoesnotdegenerateaccuracy.

2.3. High-ordertimediscretizations

AllthepreviousanalysesarebasedonEuler-forwardtimediscretization.However,we canalsoapplyhigh-orderStrong- Stability-Preserving(SSP)timediscretizationstosolvetheODEsystemu_t=^L(u).Moredetailsofthesetime discretizations canbefoundin[33,32,23].In thispaper,we usethethird-orderSSPRunge–Kuttamethod[33]

u⁽¹⁾

=

^un

+

tL

(

uⁿ

),

u⁽²⁾

=

³

4uⁿ

+

¹ 4

u⁽¹⁾

+

tL

(

u⁽¹⁾

)

,

uⁿ⁺¹

=

¹

3uⁿ

+

² 3

u⁽²⁾

+

tL

(

u⁽²⁾

)

,

(18)

andthethird-orderSSPmulti-stepmethod[32]

uⁿ⁺¹

=

¹⁶ 27

uⁿ

+

3

tL

(

uⁿ

) +

¹¹

27

uⁿ⁻³

+

¹²

11

tL

(

uⁿ⁻³

)

.

(19)

SinceanSSP timediscretizationisaconvexcombinationofEulerforwards,byusingthelimitermentionedinSection2.2, thenumericalsolutionsobtainedfromthefully-discreteschemearealsopositive.

2.4. Twodimensionalcase

Inthissubsection,we willdevelop thepositivity-preserving LDGmethodin twospacedimensions, andstudythefol- lowingproblem:

u_t

= (

u^α

)

xx

+ (

u^β

)

y y

+

^um

, (

x

,

y

) ∈ [

^xa

,

x_b

] × [

^ya

,

y_b

],

u

(

x

,

y

,

0

) ≥

⁰

, (

x

,

y

) ∈ [

^xa

,

x_b

] × [

^ya

,

y_b

].

⁽²⁰⁾

WeconsiderDirichletboundaryconditions

u

(

x_a

,

y

,

t

) =

^gxa

(

y

,

t

),

u

(

x_b

,

y

,

t

) =

^g_b^x

(

y

,

t

),

u

(

x

,

y_a

,

t

) =

^ga^y

(

x

,

t

),

u

(

x

,

y_b

,

t

) =

^g_b^y

(

x

,

t

).

Therectangulardomain[^xa,xb]× [^ya,yb]^canbediscretizedintoNx×^Ny rectangularmeshes xa

=

x1

2

<

x3

2

< · · · <

x_N

x−¹2

<

x_N

x+¹2

=

x_b

,

ya

=

y1

2

<

y3

2

< · · · <

y_N

y−¹2

<

y_N

y+¹2

=

y_b

,

(21)

withthecellK_i,j=^Ii× ^Jj,whereI_i= [^x_i−¹

2,x_i+1

2]^{and J}j= [^y_j−¹

2,y_j+1

2]^{.We alsoconsideruniformmeshesonly,andthe} meshsize inxand y directionsare denotedasxandy,respectively.Forsimplicity,if weconsiderthe cell K_i,j,then thesubscriptwillbeomitted.We definethefiniteelementspaceas

V_h

= {

^uh

:

^uh

|

K

∈

P^k

(

K

),

1

≤

ⁱ

≤

^Nx

,

1

≤

^j

≤

^Ny

} ,

whereP^k(K)denotesthepolynomialsofdegreeatmostkontheelement K.

ToconstructthelocaldiscontinuousGalerkinmethod,firstlyweneedtorewrite(20)intoafirst-ordersystemasfollows

u_t

= (

a

(

u

)

p

)

_x

+ (

b

(

u

)

q

)

_y

+

^um

,

(22)

p

= (

f

(

u

))

x

,

(23)

q

= (

g

(

u

))

y

,

(24)

wherea(u)=√

α

uα−¹,b(u)=

βu^β−1, f(u)=u

a(s)dsand g(u)=u

b(s)ds.In thissubsection,we alsodenotethe numericalsolution uh, ph andqh by u, p andq,respectively.Then thegeneralformulationoftheLDG schemeisto find u,p,q∈^Vh,suchthat

K

u_tvdxdy

= −

K

a

(

u

)

pv_xdxdy

+

J_j

((

a

(

u

)

pv⁻

)

_i+1

2,j

− (

a

(

u

)

pv⁺

)

_i−1 2,j

)

dy

−

K

b

(

u

)

qv_ydxdy

+

Ii

((

b

(

u

)

qv⁻

)

_i,j+1

2

− (

b

(

u

)

qv⁺

)

_i,j−1 2

)

dx

,+

K

u^mvdxdy (25)

K

p wdxdy

= −

K

f

(

u

)

w_xdxdy

+

J_j

((

_f

(

u

)

w⁻

)

_i+1

2,j

− (

_f

(

u

)

w⁺

)

_i−1

2,j

)

dy

,

(26)

K

q

τ

dxdy

= −

K

g

(

u

) τ

ydxdy

+

I_i

((

g

(

u

) τ

⁻

)

_i,j+1

2

− (

g

(

u

) τ

⁺

)

_i,j−1

2

)

dx

,

(27)

foranytestfunction v,w,

τ

_∈P^k(K)andi=¹,· · ·,Nx, j=¹,· · ·,Ny.Herea(u)p,b(u)q,f(u)andg(u)in(25)–(27)are thenumericalfluxeswhicharefunctionsdefinedonthecell interfacesandshouldbe designedto ensurestability.In this paper,thefluxesarechosentobe

(

a

(

u

)

p

)

_i+1 2,j

=

[

^f

(

u

) ] [

^u

]

^p+

i+¹₂,j

+

^Ci+12,j

x

[

^u

]

_i+¹

2,j

, (

_f

(

u

))

_i+1

2,j

=

^f

(

u⁻

i+¹₂,j

),

(28)

(

_b

(

u

)

q

)

_i,j+1

2

=

[

^g

(

u

) ] [

u

]

^q+

i,j+¹2

+

^Ci,j+12

y

[

u

]

_i,j+¹

2

, (

_g

(

u

))

_i,j+1 2

=

g

(

u⁻

i,j+¹₂

),

(29)