L 2 STABILITY ANALYSIS OF THE CENTRAL DISCONTINUOUS GALERKIN METHOD AND A COMPARISON BETWEEN THE CENTRAL AND REGULAR DISCONTINUOUS GALERKIN METHODS

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1 ESAIM: MAN DOI: 0.05/mn:00808 ESAIM: Mtemticl Modelling nd Numericl Anlysis L STABILITY ANALYSIS OF THE CENTRAL DISCONTINUOUS GALERKIN METHOD AND A COMPARISON BETWEEN THE CENTRAL AND REGULAR DISCONTINUOUS GALERKIN METHODS Yingjie Liu, Ci-Wng Su, Eitn Tdmor 3 nd Mengping Zng Abstrct. We prove stbility nd derive error estimtes for te recently introduced centrl discontinuous Glerkin metod, in te context of liner yperbolic equtions wit possibly discontinuous solutions. A comprison between te centrl discontinuous Glerkin metod nd te regulr discontinuous Glerkin metod in tis context is lso mde. Numericl experiments re provided to vlidte te quntittive conclusions from te nlysis. Mtemtics Subject Clssifiction. 65M60. Received April 7, 007. Publised online My 7, Introduction We continue our study of te recently introduced centrl discontinuous Glerkin DG metod [8]. In tis pper we prove stbility nd derive error estimtes of centrl DG pproximtion for liner yperbolic equtions. We lso compre te centrl DG metod wit te regulr DG metod in tis context. Te centrl DG metod is bsed on two existing tecniques: te centrl sceme frmework e.g. [7,9] nd te DG frmework e.g. [,]. It uses overlpping cells nd ence duplictive informtion, but voids numericl fluxes Riemnn solvers wic is distinct dvntge of centrl scemes. Te centrl DG sceme lso voids te excessive numericl dissiption for smll time steps, common to some of te erlier centrl scemes, by suitble coice of te numericl dissiption. Being vrint of te DG metod, it sres mny of its dvntges, Rpide Note Keywords nd prses. Centrl discontinuous Glerkin metod, discontinuous Glerkin metod, liner yperbolic eqution, stbility, error estimte. Scool of Mtemtics, Georgi Institute of Tecnology, Atlnt, GA , USA. yingjie@mt.gtec.edu Reserc supported in prt by NSF grnt DMS Division of Applied Mtemtics, Brown University, Providence, RI 09, USA. su@dm.brown.edu Reserc supported in prt by NSFC grnt wile e ws visiting te Deprtment of Mtemtics, University of Science nd Tecnology of Cin, Hefei, Anui 3006, P.R. Cin. Additionl support ws provided by ARO grnt W9NF nd NSF grnt DMS Deprtment of Mtemtics, Institute for Pysicl Science nd Tecnology nd Center of Scientific Computtion nd Mtemticl Modeling CSCAMM, University of Mrylnd, College Prk, MD 07, USA. tdmor@cscmm.umd.edu Reserc supported in prt by NSF grnt nd ONR grnt N000-9-J-076. Deprtment of Mtemtics, University of Science nd Tecnology of Cin, Hefei, Anui 3006, P.R. Cin. mpzng@ustc.edu.cn Reserc supported in prt by NSFC grnt Article publised by EDP Sciences c EDP Sciences, SMAI 008

2 59 Y. LIU ET AL. suc s compct stencil, esy prllel implementtion, etc. Te centrl DG metod performs well in numericl simultions of liner nd nonliner sclr nd systems of conservtion lws [8]. We proceed wit description of te regulr DG metod nd te centrl DG metod. For simplicity, we consider sclr one dimensionl conservtion lw u t + fu x =0, x, t [, b] [0,T]. Rpide Not wit periodic or compctly supported boundry conditions. Te DG nd centrl DG metods cn be defined for nonliner, multi-dimensionl nd system cses, nd wit oter boundry conditions s well, see [,8]. Let {x j } be prtition of [, b] wit j+ = x j+ x j nd = mx j j+. Te mes is regulr, in te sense tt mx j j+ / min j j+ is upper-bounded by fixed constnt during mes refinements. Denote x j+ = x j+ + x j, I j =x j,x j+, nd I j+ =x j,x j+. V is te set of piecewise polynomils of degree k over te subintervls {I j } wit no continuity ssumed cross te subintervl boundries. Likewise, W is te set of piecewise polynomils of degree k over te subintervls {I j+ } wit no continuity ssumed cross te subintervl boundries. Te regulr DG metod is defined using te spce V only. Te semi-discrete version is s follows. Find u,t V suc tt for ny ϕ V nd for ll j, I j t u ϕ dx = fu x ϕ dx ˆf I j + ˆf u x,t j u x,t,u j+ x +,t ϕ j+ x j+,u x +,t ϕ j x + j were ˆfu,u + is monotone numericl flux, nmely it is incresing in te first rgument nd decresing in te second rgument, or symboliclly ˆf, ; it is consistent wit te pysicl flux ˆfu, u =fu; nd it is t lest Lipscitz continuous wit bot rguments. For monotone fluxes suitble for te DG metod, see, e.g. []. For systems te monotone numericl flux is replced by numericl flux obtined troug n exct or pproximte Riemnn solver, see, e.g. [0]. Te centrl DG metod is defined on overlpping cells nd uses bot spces V nd W. We strt wit te description of te metod for te fully discrete version, wit forwrd Euler time discretiztion from t n to t n+ = t n +τ ere te time step τ could cnge wit n, but for simplicity of nottions we will use constnt τ. Te sceme is defined by te following procedure: Find u n+ V nd v n+ W, suc tt for ny ϕ V nd ψ W, u n+ I j + τ ϕ dx = θ I j fv n xϕ dx f vϕ n dx + θ u n ϕ dx I j I j v n x j+ ϕ x j+ + f v n x j. ϕ x + j.3 v n+ ψ dx = θ u n ψ dx + θ vψ n dx I j+ I j+ I j+ + τ fu n x ψ dx f u n x j+ ψ x j+ +f un x j ψ x + j. I j+ were θ = τ nd is n upper bound for te time step size due to te CFL restriction, tt is, = c wit given constnt CFL number c dictted by stbility. Computtionlly, te sceme.3. isusedwit

3 CENTRAL DISCONTINUOUS GALERKIN METHOD 595 te forwrd Euler replced by Runge-Kutt metod of suitble temporl ccurcy, e.g. te SSP Runge-Kutt metods in [5,]. Te time step τ cn be cosen rbitrrily subject to te stbility restriction τ, ence 0 θ. If θ =, te sceme is similr in spirit to te originl centrl sceme [9]. We now tke te limit τ 0 to obtin te semi-discrete version of te centrl DG sceme: Find u,t V nd v,t W, suc tt for ny ϕ V nd ψ W, I j t u ϕ dx = f v u ϕ dx + fv x ϕ dx I j I j v x j+,t ϕ x + f v j+ x j,t ϕ x + j.5 I j+ t v ψ dx = u v ψ dx + I j+ I j+ fu x ψ dx f u x j+,t ψ x j+ +f u x j,t ψ x + j..6 Notice tt, unlike te regulr DG sceme., te centrl DG sceme.5.6 does not need numericl flux to define te interfce vlues of te solution, since te evlution of te solution t te interfce is in te middle of te stggered mes, ence in te continuous region of te solution. Te first term on te rigt side of.5 or.6 is numericl dissiption term. Tis will become cler wen we discuss te stbility of te sceme in Section.. In ll te DG scemes.,.3. nd.5.6, te initil condition is tken s te L projection of te PDE initil condition into te relevnt finite element spce. Te orgniztion of te pper is s follows. In Section, we first nlyze te L stbility nd give n priorierror estimte for te semi-discrete centrl DG sceme.5.6 for te liner yperbolic eqution, using similr tecniques of stbility nd error nlysis for stndrd DG metods [3,6]. We ten perform Fourier nlysis for te semi-discrete centrl DG sceme.5.6 for te liner yperbolic eqution wit uniform meses for piecewise constnt nd liner elements, using te tecniques in [3]. Tis nlysis is more explicit nd llows us to compre te errors quntittively between te centrl nd stndrd DG scemes, wic is performed in Section 3. In Section 3 we lso perform numericl experiments to verify suc quntittive conclusions. We give few concluding remrks in Section. Rpide Note. Anlysis of te centrl DG sceme For te purpose of nlysis we will consider te liner yperbolic eqution, nmely. wit fu = u for constnt. Our nlysis cn be esily generlized to multi-dimensionl liner equtions nd yperbolic liner systems. In prticulr, tese results pply to yperbolic equtions wit propgting discontinuities. Witout loss of generlity we consider te following liner yperbolic eqution u t + u x =0, x, t [, b] [0,T].. We study te L stbility of te centrl DG sceme.5.6 for te eqution. in Section., nd compre te result wit tt for te regulr DG sceme in [6]. In Section. we provide n L priorierror estimte for smoot solutions, nd compre te result wit tt for te regulr DG sceme in [3]. In Section.3 we give n quntittive error estimte for te centrl DG sceme for polynomil degree up to using Fourier nlysis, similr to te tecnique used in [,3].

4 596 Y. LIU ET AL... L stbility Teorem.. Te numericl solution u nd v of te centrl DG sceme.5.6 for te eqution. stisfies te following L stbility condition d dt b u +v dx = b u v dx 0.. Proof. Tking te test functions ϕ = u nd ψ = v in.5 nd.6 respectively, summing up over j, observing fu = u nd te periodic or compctly supported boundry condition, we ve d b u +v dx = b v u u +u v v dx+ [ v x u dx+ u x v dx dt j I j I j+ v x j+,t u x,t + v j+ x j,t u x +,t j u x j+,t v x j+,t + u x j,t v x j,t ] Rpide Not = b u v dx + [ xj x u v dx + j x j + v x j,t v x j+,t u x,t j+ u x j+,t v x j+,t + u x j,t v x j,t ] = b u x +,t j xj+ x j x u v dx u v dx 0. Remrk.. Te proof of Teorem. is similr to te proof of te cell entropy inequlity for te regulr DG metod in [6]. However, we cnnot prove similr L stbility result for te centrl DG sceme wen pplied to te nonliner sclr conservtion lw., even toug te proof of Teorem. cn be esily generlized to multi-dimensionl centrl DG scemes for liner equtions. Tis is in contrry to te cell entropy inequlity for regulr DG scemes, wic olds for rbitrry nonliner sclr conservtion lws [6]. b Remrk.. Teorem. indictes tt te energy dissiption term is u v dx, tt is, it is directly relted to te difference of te two duplictive representtions u nd v of te solution in overlpping cells. In contrst, for te regulr DG metod, te energy dissiption term is directly relted to te jumps of te numericl solution t cell interfces... L priorierror estimte In tis subsection we use te stndrd DG tecniques [3] toobtinnprioril error estimte for te centrl DG sceme. Teorem.. Te numericl solution u nd v of te centrl DG sceme.5.6 for te eqution. witsmootinitilconditionu, 0 H k+ stisfies te following L error estimte u u + u v C k.3 were u is te exct solution of., k is te polynomil degree in te finite element spces V nd W,nd te constnt C depends on te k +-t order Sobolev norm of te initil condition u, 0 H k+ s well s on te finl time t, but is independent of te mes size.

5 CENTRAL DISCONTINUOUS GALERKIN METHOD 597 Proof. Let us first introduce te stndrd nottion B j u,v ; ϕ,ψ = t u ϕ dx v u ϕ dx v x ϕ dx I j I j I j + v x j+ x v j+ x j x +. j + t v ψ dx u v ψ dx u x ψ dx I j+ τ mx I j+ I j+ + u x j+,tψ x j+ u x j,tψ x + j. Clerly, we ve: B j u,v ; ϕ,ψ = 0.5 for ll j nd ll ϕ V nd ψ W. It is lso cler tt te exct solution u of te PDE. stisfies B j u, u; ϕ,ψ = 0.6 for ll j nd ll ϕ V nd ψ W. Subtrcting.5 from.6, we obtin te error eqution for ll j nd ll ϕ V nd ψ W. nd B j u u,u v ; ϕ,ψ = 0.7 We now define P nd Q s te stndrd L projection into V nd W respectively. Tt is, for ec j, I j Pwx wxϕ xdx =0 ϕ P k I j.8 I j+ Qwx wxψ xdx =0 ψ P k I j+.9 were P k I j ndp k I j+ denote te spces of polynomils of te degree up to k in te cell I j nd te cell respectively. Stndrd pproximtion teory [] implies, for smoot function w, I j+ Rpide Note Pwx wx + / Pwx wx Γ C k+.0 nd Qwx wx + / Qwx wx Γ C k+. were Γ denotes te set of boundry points of ll elements I j or I j+/ respectively, te norm Γ is te stndrd L norm, nd te positive constnt C, ere nd below, solely depending on wx nd its derivtives, is independent of. We lso recll tt [], for ny w V or w W, tere exists positive constnt C independent of w nd, suc tt x w C w ; w Γ C / w. were Γ is te set of boundry points of ll elements I j or I j+/. We now tke: ϕ = Pu u, ψ = Qu v.3 in te error eqution.7, nd denote ϕ e = Pu u, ψ e = Qu u.

6 598 Y. LIU ET AL. to obtin B j ϕ,ψ ; ϕ,ψ =B j ϕ e,ψ e ; ϕ,ψ..5 For te left-nd side of.5, we use Teorem. to conclude B j ϕ,ψ ; ϕ,ψ = d dt j b ϕ +ψ dx + We ten write te rigt-nd side of.5 ssumoftreeterms b ϕ ψ dx..6 B j ϕ e,ψ e ; ϕ,ψ =B j + B j + B 3 j.7 were Bj = ϕ e ψ e ϕ dx + ψ e ϕ e ψ dx + t ϕ e ϕ dx + t ψ e ψ dx I j I j+ I j I j+ Rpide Not Bj I = ψ e x ϕ dx ϕ e x ψ dx j I j+ Bj 3 = ψe x j+,tϕ x,t ψ e x j+ j,tϕ x +,t+ϕ e x j j+,tψ x j+,t ϕe x j,tψ x + j,t nd we will estimte ec term seprtely. By using te simple inequlity αβ α + β,.8 te L projection property.0. forϕ e, ψ e, t ϕ e nd t ψ e, nd te fct tt = O, we ve: b Bj ϕ dx + j b ψ dx + C k..9 Likewise, by using te simple inequlity.8, te L projection property.0 forϕ e nd ψ e,ndtefirst inequlity in. forϕ nd ψ,weve: b Bj ϕ dx + j b ψ dx + C k..0 Finlly, by using te simple inequlity.8, te L projection property.0forϕ e nd ψ e, nd te second inequlity in. forϕ nd ψ,weve: b Bj 3 ϕ dx + j Combining.9,.0 nd. wit wit.6, we obtin from.5 d dt b ϕ +ψ dx C b b ψ dx + C k.. ϕ +ψ dx + C k. Tis, togeter wit te pproximtion result.0, implies te desired error estimte.3.

7 CENTRAL DISCONTINUOUS GALERKIN METHOD 599 Remrk.3. Te error estimte of Teorem. is sub-optiml. Numericlly we observe te optiml k +-t order ccurcy, see [8] nd lso te numericl results in next section. In contrst, te error estimte for te regulr DG metod in suc one dimensionl nd lso in multi-dimensionl tensor product cses is optiml [3]. Te mjor tecnicl difficulty leding to tis loss of optimlity in te proof is te Bj term. For te regulr DG metod tis term is zero for bot te regulr L projection nd for specil projection wic is ortogonl to polynomils of one degree lower nd cn render te boundry terms in Bj 3 lso to vnis. However, for te centrl DG metod, since it involves ψ e nd x ϕ nd tey re defined by polynomils in different cells, it is impossible to mke te Bj term vnis no mtter ow te projection is cosen, ltoug specil projection like tt used for regulr DG metods cn render Bj 3 to be zero. A better tecnique is needed to prove te optiml k +-t order ccurcy observed numericlly, owever we ve been unble to find suc tecnique so fr. Remrk.. Te error estimte of Teorem. cn be esily generlized to one dimensionl liner yperbolic systems, multi-dimensionl sclr liner yperbolic equtions, nd multi-dimensionl symmetric liner systems..3. A quntittive error estimte vi Fourier nlysis In tis subsection we perform Fourier nlysis for te semi-discrete centrl DG sceme.5.6 forte liner yperbolic eqution wit uniform meses for piecewise constnt nd liner elements, using te tecniques in [3]. Tis nlysis is more explicit nd llows us to compre te errors quntittively between te centrl nd stndrd DG scemes. For tis purpose we rewrite te sceme.5.6 for te liner eqution. s finite difference sceme on uniform mes. Towrds tis gol we coose te degrees of freedom for te k-t degree polynomil inside te cells I j nd I j+ respectively s te point vlues of te solution, denoted by u j+ i k t, k+ i =0,..., k nd v j+ i+ k t, i =0,..., k k+ t te k+ eqully spced points j + i k, i =0,..., k k + nd i + k j +, i =0,..., k. k + Te scemes written in terms of tese degrees of freedom become finite difference scemes on globlly uniform mes wit mes size /k +, owever tey re not stndrd finite difference scemes becuse ec point obeys different form of te finite difference sceme. in te group of k+ points belonging to te cell I j or I j+ To be more specific, we first concentrte on te simplest piecewise constnt k = 0 cse. For tis cse, we coose te degrees of freedom s te point vlues t te N uniformly spced points Rpide Note or Te solution inside te cell I j or I j+ u j t, j =0,..., N v j+ t, j =0,..., N. is ten represented by u x, t =u j tϕ 0 x or v x, t =v j+ tψ0 x

8 600 Y. LIU ET AL. were ϕ 0 x is te constnt function wic equls inside I j, nd similrly ψ 0 x is te constnt function wic equls inside I j+. Wit tis representtion, tking te test functions ϕ s ϕ 0,ndψ s ψ 0, respectively, we obtin esily te finite difference scemes corresponding to te centrl DG sceme: v j+ u j = u j + + = v j+ + + v j + u j + v j+ u j+. for j =0,..., N. Here u nd v denote te time derivtives of u nd v. Te sceme cn be rewritten into more compct form u j uj uj uj+ v = A + B + C.3 j+ v j v j+ v j+ 3 wit A = , B = +, C = Rpide Not We now perform te following stndrd Fourier nlysis for te finite difference sceme.3.. Tis nlysis depends evily on te ssumption of uniform mes sizes nd periodic boundry conditions. We mke n nstz of te form uj t ûm t v j+ t = e ˆv m t imxj.5 nd substitute tis into te sceme.3. to find te evolution eqution for te coefficient vector s û m t ˆv mt were te mplifiction mtrix Gm, is given by ûm t = Gm, ˆv m t.6 Gm, = A e im + B + C e im.7 wit te mtrices A, B nd C defined by.. Te two eigenvlues of te mplifiction mtrix Gm, re λ = αe i m, λ = + αe i m.8 were α = + e im +. Te generl solution of te ODE.6 isgivenby ûm t = ˆv m t e λt V + e λt V,.9 were te eigenvlues λ nd λ re given by.8, nd V nd V re te corresponding eigenvectors given by e i ξ e i ξ V =, V =,.30 wit ξ = m. For ccurcy we look t te low modes, in prticulr t m =. To fit te given initil condition u j 0 = e ixj, v j+ 0 = eix j+.3

9 CENTRAL DISCONTINUOUS GALERKIN METHOD 60 wose imginry prt is te initil condition u j 0 = sinx j, v j+ 0 = sinx j+,.3 we require, t t =0, û 0 ˆv 0 = e i, ence we obtin te coefficients nd in.9 s =0, =e i..33 We remrk tt te usul wy of tking initil conditions in finite element metod is vi n L projection, not by point vlue colloction.3, owever we ve verified tt tis does not ffect te finl results in te nlysis in tis pper. We tus ve te explicit solutions of te sceme.3. wit te initil condition.3, for exmple u j t = e ixj+λt i.3 wit te eigenvlue λ given by.8 witm = nd te coefficient given by.33. By simple Tylor expnsion, we obtin te imginry prt of u j t tobe t Im{u j t} =sinx j t 8c sinx j t + O.35 were c = τmx = O is te mximum CFL number. Tis is clerly consistent wit te exct solution to first order ccurcy. We cn similrly ceck te first order ccurcy of Im{v j+ t}. We now repet tis nlysis for te piecewise liner k = cse. Te solution inside te cell I j or I j+ is ten represented by Rpide Note u x, t =u j tϕ ξ+u j+ tϕ ξ or v j+ 3 v x, t =v j+ tψ ξ+v j+ 3 ψ ξ were ϕ ξ = ξ +, ϕ ξ =ξ +, ψ ξ = ξ + 3 nd ψ ξ =ξ x xj,witξ =. Wit tis representtion, tking te test functions ϕ nd ψ s ϕ, ϕ nd ψ, ψ respectively, nd inverting te smll mss mtrix by nd, we obtin esily te finite difference sceme corresponding to te centrl DG sceme s u j u j 5 u u j u j+ 3 j+ v j+ = A u j 3 v j 3 + B u j+ v j+ + C u j v j+ 5 v j v j+ 3 v j+ 7

10 60 Y. LIU ET AL. wit A = B = C = , ,.37 Rpide Not We mke n nstz of te form u j u j+ v j+ v j+ 3 t t t = t û m, û m, ˆv m, ˆv m, 3 t t t eimxj.38 t nd substitute tis into te sceme to find te evolution eqution for te coefficient vector s û t m, û t m,+ ˆv = Gk, t m, ˆv t m, 3 û m, û m,+ ˆv m, ˆv m, 3 t t t t.39 were te mplifiction mtrix Gm, is given by Gm, = A e im + B + C e im.0 wit te mtrices A, B nd C defined by.37. Te eigenvlues of te mplifiction mtrix Gm, re λ = λ = λ 3 = λ = + 8 e im α α 8 e im α α + 8 e im α + α 8 e im α + α

11 CENTRAL DISCONTINUOUS GALERKIN METHOD 603 were α = c 5c e im 5c e im ++c 5c e 3im, α = e im + e im +0c e im [ 3 c +c. + c e im + 3+c +c e im ] / wit c = τmx being te mximum CFL number. Te generl solution of te ODE.39 isgivenby û m, t û m,+ t ˆv m, t = e λt V + e λt V + 3 e λ3t V 3 + e λt V,. ˆv m, 3 t were te eigenvlues λ, λ, λ 3 nd λ re given by., nd V, V, V 3 nd V re te corresponding eigenvectors given by V = V = V 3 = V = e im α α α 7+α α 5α 6 α α+α +e im 0c +e im α 6 α α α +α 3 α 5α 6 α e im α α α 7+α α 5α 6 α α+α +e im 0c +e im α 6 α α α +α 3 α 5α 6 α e im α +α α 7+α α 5α 6+α α+α +e im 0c +e im α 6+α α α +α 3 α 5α 6+α e im α +α α 7+α α 5α 6+α α+α +e im 0c +e im α 6+α α α +α 3 α 5α 6+α,,,,.3 Rpide Note were c = τmx given by is still te mximum CFL number, α nd α re given by., nd te remining α s re α 3 = 7 + 0c + 33c 80c 3 e im c + 53c 0c 3 e im c 68c + 0c 3 e 3im 5 6c + 3c +80c 3 e im, α = 0c + 3+0ce im,. α 5 = c +3+ce im, α 6 = +0c e im + 00c e im + 0c e 3im, α 7 = +0ce im 0c e im 0c e 3im.

12 60 Y. LIU ET AL. We gin look t te low modes to determine ccurcy. In prticulr we look t m =. Tofittegiven initil condition u j 0 = eix j, u j+ 0 = eix j+, v j+ 0 = eix j+, v j+ 3 0 = eix j+ 3.5 wose imginry prt is our initil condition for., we require, t t =0, û m, 0 e i û m,+ 0 ˆv m, 0 = e i e i. ˆv m, 3 0 e i 3 Tis gives us te coefficients,, 3 nd in te solution.. We tus ve te explicit solution of te sceme wit te initil condition.5, for exmple u j = e ixj+λt i V + e ixj+λt i V + 3 e ixj+λ3t i V3 + e ixj+λt i V.6 Rpide Not wit te eigenvlues λ, λ, λ 3, λ given by. nd te eigenvectors V, V, V 3, V given by.3 wit m =,ndtecoefficients,, 3 nd obtined bove by fitting te initil condition. Troug simple Tylor expnsion, we obtin te imginry prt of u j t tobe Im{u j t} =sinx j t sinx j t + O for fixed coice of =0.. Results for oter coices of lso indicte te sme second order ccurcy. Te results for Im{u j+ t}, Im{v j+ t} nd Im{v j+ 3 t} re similr. In principle tis nlysis cn be performed for iger order polynomils in te centrl DG sceme, owever te lgebr becomes proibitively complicted. 3. A comprison between te centrl DG nd stndrd DG metods In [3], results to similr to tose in.35 nd.7 were obtined for te regulr DG sceme. pplied to te liner eqution. wit n upwind numericl flux. For te piecewise constnt k = 0 cse, te result for te regulr DG sceme is t Im{u j t} =sinx j t sinx j t + O, 3. nd for te piecewise liner k =cse,itis Im{u j t} =sinx j t sinx j t + O We re now in position to mke quntittive comprison between te regulr nd centrl DG scemes. For te piecewise constnt k = 0 cse we ve te following conclusions: Te semi-discrete versions of te regulr nd centrl DG scemes re bot stble. Wen discretized wit te first order forwrd Euler metod, te CFL numbers for te DG metod nd for te centrl DG metod re.0 nd 0.5, respectively tis cn be verified by n esy von Neumnn nlysis. Wen discretized wit te second order nonlinerly stble Runge-Kutt metod [], te CFL numbers for te DG metod nd for te centrl DG metod re.0 nd 0.87, respectively. Tus te centrl DG metod

13 CENTRAL DISCONTINUOUS GALERKIN METHOD 605 Tble. L nd L errors nd orders of ccurcy of te first order centrl DG metod. Numericl solution Predicted by nlysis L error Order L error Order L error Order L error Order π/80.88 E-0.65 E-0.7 E E-0 π/60.0 E E E E-0.00 π/ E E E E-0.00 π/60.67 E E E E-0.00 π/80.35 E E E E-0.00 Tble. L nd L errors nd orders of ccurcy of te first order regulr DG metod. Numericl solution Predicted by nlysis L error Order L error Order L error Order L error Order π/80. E E E E-0 π/60.7 E E E E-0.00 π/30.5 E E E E-0.00 π/ E E E E-0.00 π/80. E E E E-0.00 s smller CFL number. Considering tt te centrl DG metod uses duplictive informtion nd ence is twice s costly s te regulr DG metod for te sme mes, we could compre te regulr DG sceme on mes size wit te centrl DG sceme on mes size to equlize te cost. In suc context te CFL number would be te sme for te two DG metods wit te first order forwrd Euler time discretiztion. Tey re bot first order ccurte. By comprison of.35 nd3., te leding errors for te centrl nd regulr DG metods for te first mode i.e. for te sinx initil condition ve rtio /c. Tt is, te centrl DG metod s smller error tn te stndrd DG metod on te sme mes wen c = τmx >. If we compre te regulr DG sceme on mes size wit te centrl DG sceme on mes size to equlize te cost, ten te centrl DG metod s smller error tn te stndrd DG metod wen c = τmx >. We now compute te DG nd centrl DG solutions to. witux, 0 = sinx s te initil condition nd wit periodic boundry conditions, up to t = 5 bout four periods lter in time, to verify te quntittive comprison bove. In our computtion we tke =0.8, nmelyc =0.8. We tke smll time step τ = 0.0 to reduce te effect from te time discretiztion. In order to be consistent wit te error nlysis bove, te errors re computed for u t te points x j. Te L nd L errors nd order of ccurcy of te centrl DG nd regulr DG metods re listed in Tbles nd respectively. We lso list te predicted errors by te nlysis, nmely te leding terms in te Tylor expnsions in.35 nd3. in tese tbles. We cn see tt te predicted errors nd te ctul errors re very close, vlidting our quntittive nlysis in Section.3. Likewise, for te piecewise liner cse k =, we ve te following conclusions: Te semi-discrete versions of te regulr nd centrl DG scemes re bot stble. Wen discretized wit te second order nonlinerly stble Runge-Kutt metod [], te CFL numbers for te DG metod nd for te centrl DG metod re 0.33 nd 0.5, respectively tis cn be verified by n esy von Neumnn nlysis. Tus te centrl DG metod s lrger CFL number. If we compre te regulr DG sceme on mes size wit te centrl DG sceme on mes size to equlize te cost, ten te centrl DG metodsnevenlrgercflnumber. Rpide Note

14 606 Y. LIU ET AL. Tble 3. Te leding error term b τmx sinx j t for te centrl DG metod for different c =. c b Tble. L nd L errors nd orders of ccurcy of te second order centrl DG metod. Numericl solution Predicted by nlysis L error Order L error Order L error Order L error Order π/0.0 E-0.37 E-0.0 E-03. E-03 π/0.53 E E E E-0.00 π/80.9 E E E E-0.00 π/60.57 E E E E π/ E E E E Tble 5. L nd L errors nd orders of ccurcy of te second order regulr DG metod. Rpide Not Numericl solution Predicted by nlysis L error Order L error Order L error Order L error Order π/0.06 E-0.6 E-0.0 E-03. E-03 π/0.3 E E E E π/80.57 E E E E-0.00 π/60 6. E E E E π/30.6 E E E E Tey re bot second order ccurte. Te centrl DG metod s different leding errors b sinx j t fordifferentc = τmx, wit te sinx initil condition. We list te corresponding b wit different vlues of c in Tble 3. By comprison of Tble 3 nd 3., te leding errors for te centrl nd regulr DG metods for te first mode i.e. for te sinx initil condition re equl wen c = τmx =0.5. Te centrl DG metod s smller error tn te stndrd DG metod on te sme mes wen c<0.5. If we compre te regulr DG sceme on mes size wit te centrl DG sceme on mes size to equlize te cost, ten te centrl DG metod s smller error tn te stndrd DG metod wen c<0.3. We now compute te DG nd centrl DG solutions to. witux, 0 = sinx initil condition nd periodic boundry conditions, up to t = 5 bout four periods lter in time, to verify te quntittive comprison bove. In our computtion we tke =0., nmelyc =0.. We tke smll time step τ = 0.0 to reduce te effect from te time discretiztion. In order to be consistent wit te error nlysis bove, te errors re computed for u t te points x j nd x j+. Te L nd L errors nd order of ccurcy of te centrl DG nd regulr DG metods re listed in Tbles nd 5 respectively. We lso list te predicted errors by te nlysis, nmely te leding terms in te Tylor expnsions in Tble 3 nd 3. in tese tbles. We cn see gin tt te predicted errors nd te ctul errors re very close, vlidting our quntittive nlysis in Section.3. Finlly, in Tble 6 we crry out numericl Fourier nlysis to obtin te CFL numbers wen te centrl DG sceme is coupled wit Runge-Kutt metods of vrious orders, nd compre tem wit te results of te regulr DG metod wit te Runge-Kutt time discretiztion RKDG metods, Tb.. in []. We observe tt, except for te cse of k = 0 piecewise constnt, for ll oter k te centrl DG metod s lrger CFL number tn te RKDG metod wit te sme Runge-Kutt time discretiztion.

15 CENTRAL DISCONTINUOUS GALERKIN METHOD 607 Tble 6. CFL numbers for te k + -t order RKDG metod nd centrl DG metod wit Runge-Kutt metod of order ν. Regulr DG Centrl DG k ν = ν = ν = ν = Concluding remrks We ve performed n L stbility nlysis nd n priorierror estimte for te recently introduced centrl discontinuous Glerkin metod wen pplied to liner yperbolic equtions. We ve lso performed Fourier type error nlysis wic is more quntittive nd llows us to mke comprison wit te regulr discontinuous Glerkin metod. It is verified tt, even toug te centrl discontinuous Glerkin metod uses duplictive representtion of te solution, ence involves twice te computtionl cost nd storge requirement tn te regulr discontinuous Glerkin metod, it is more ccurte for certin coices of dissiption prmeter for te sme mes. Te stbility nlysis nd error estimtes do not seem to be esily generlizble to nonliner yperbolic equtions. Furter nlysis in tis direction is needed. A compreensive comprison of te numericl performnce of te centrl discontinuous Glerkin metod nd te regulr discontinuous Glerkin metod for nonliner multi-dimensionl systems of yperbolic conservtion lws would lso be very useful. References [] P. Cirlet, Te Finite Element Metod for Elliptic Problem. Nort Hollnd 975. [] B. Cockburn nd C.-W. Su, TVB Runge-Kutt locl projection discontinuous Glerkin finite element metod for conservtion lws II: generl frmework. Mt. Comput [3] B. Cockburn nd C.-W. Su, Te locl discontinuous Glerkin metod for time-dependent convection-diffusion systems. SIAM J. Numer. Anl [] B. Cockburn nd C.-W. Su, Runge-Kutt discontinuous Glerkin metods for convection-dominted problems. J. Sci. Comput [5] S. Gottlieb, C.-W. Su nd E. Tdmor, Strong stbility-preserving ig-order time discretiztion metods. SIAM Rev [6] G.-S. Jing nd C.-W. Su, On cell entropy inequlity for discontinuous Glerkin metods. Mt. Comput [7] Y.J. Liu, Centrl scemes on overlpping cells. J. Comput. Pys [8] Y.J. Liu, C.-W. Su, E. Tdmor nd M. Zng, Centrl discontinuous Glerkin metods on overlpping cells wit nonoscilltory ierrcicl reconstruction. SIAM J. Numer. Anl [9] H. Nessyu nd E. Tdmor, Non-oscilltory centrl differencing for yperbolic conservtion lws. J. Comput. Pys [0] J. Qiu, B.C. Koo nd C.-W. Su, A numericl study for te performnce of te Runge-Kutt discontinuous Glerkin metod bsed on different numericl fluxes. J. Comput. Pys [] C.-W. Su nd S. Oser, Efficient implementtion of essentilly non-oscilltory sock-cpturing scemes. J. Comput. Pys [] M. Zng nd C.-W. Su, An nlysis of tree different formultions of te discontinuous Glerkin metod for diffusion equtions. Mt. Models Metods Appl. Sci [3] M. Zng nd C.-W. Su, An nlysis of nd comprison between te discontinuous Glerkin nd te spectrl finite volume metods. Comput. Fluids Rpide Note