Space-Time Discontinuous Galerkin Method for Maxwell s Equations

Transcription

1 Commun. Comput. Pys. doi: /cicp a Vol. 4, No. 4, pp Octobr 203 Spac-Tim Discontinuous Galrkin Mtod for Maxwll s Equations Ziqing Xi,2, Bo Wang 3,4, and Zimin Zang 5,6 Scool of Matmatics and Computr Scinc, Guizou Normal Univrsity, Guiyang, Guizou 55000, Cina. 2 y Laboratory of Hig Prformanc Computing and Stocastic Information Procssing, Ministry of Education of Cina, Hunan Normal Univrsity, Cangsa, Hunan 4008, Cina. 3 Collg of Matmatics and Computr Scinc, Hunan Normal Univrsity, Cangsa, Hunan 4008, Cina. 4 Singapor-MIT Allianc, 4 Enginring Driv 3, National Univrsity of Singapor, Singapor 7576, Singapor. 5 Dpartmnt of Matmatics, Wayn Stat Univrsity, Dtroit, MI 48202, USA. 6 Guangdong Provinc y Laboratory of Computational Scinc, Scool of Matmatics and Computational Scinc, Sun Yat-sn Univrsity, Guangzou, 50275, Cina. Rcivd 23 April 202; Accptd in rvisd vrsion 27 Dcmbr 202 Availabl onlin 9 Marc 203 Abstract. A fully discrt discontinuous Galrkin mtod is introducd for solving tim-dpndnt Maxwll s quations. Distinguisd from t Rung-utta discontinuous Galrkin mtod RDG and t finit lmnt tim domain mtod FETD, in our scm, discontinuous Galrkin mtods ar usd to discrtiz not only t spatial domain but also t tmporal domain. T proposd numrical scm is provd to b unconditionally stabl, and a convrgnt rat O t r+ + k+/2 is stablisd undr t L 2 -norm wn polynomials of dgr at most r and k ar usd for tmporal and spatial approximation, rspctivly. Numrical rsults in bot 2-D and 3-D ar providd to validat t tortical prdiction. An ultra-convrgnc of ordr t 2r+ in tim stp is obsrvd numrically for t numrical fluxs w.r.t. tmporal variabl at t grid points. AMS subjct classifications: 35Q6, 65M2, 65M60, 65N2 y words: Discontinuous Galrkin mtod, Maxwll s quations, full-discrtization, L 2 -rror stimat, L 2 -stability, ultra-convrgnc. Corrsponding autor. addrsss: ziqingxi@yaoo.com.cn Z. Q. Xi, bowangn@gmail.com B. Wang,ag776@wayn.du Z. Zang ttp:// 96 c 203 Global-Scinc Prss

2 Z. Q. Xi, B. Wang and Z. Zang / Commun. Comput. Pys., 4 203, pp Introduction Finit lmnt mtods, including dg lmnt mtods and discontinuous Galrkin mtods, av bn widly usd to solv tim-armonic Maxwll s quations [2, 4, 28] as wll as tim-dpndnt Maxwll s quations [8, 9,, 5, 20 27], du to tir ig ordr accuracy and flxibility in andling complicatd domains. Traditionally, ty wr only usd to discrtiz t spatial domain to produc a systm of ordinary diffrntial quations in tim t, wic was tn solvd by t finit diffrnc or Rung-utta mtods [8,, 5, 25, 27]. Towards tis nd, Makridakis and Monk proposd a fully discrt finit lmnt mtod for Maxwll s quations and invstigatd t corrsponding rror stimats in [26]. Tir approac rsultd in a coupld non-symmtric and indfinit linar algbraic systm involving bot lctric and magntic filds. Latr, Ciarlt Jr. and Zou [9] analyzd a fully discrt finit lmnt approac for a scond-ordr lctric fild quation drivd from Maxwll s quations by liminating t magntic fild. Bot optimal nrgy-norm rror stimat and optimal L 2 -norm rror stimat wr obtaind. Wn disprsiv mdia wr involvd, Li proposd som fully discrt numrical scms. Bot mixd finit lmnt mtod [20 22] and intrior pnalty discontinuous Galrkin mtod [23] ar considrd for spatial discrtization. Sinc Maxwll s quations ar a coupld systm, a fully discrt scm was proposd by Ma [25], aimd to rduc t computational cost by dnoting t magntic fild xplicitly in t numrical scm. T ida to discrtiz t tmporal domain by finit lmnt mtod is not somting nw in t litratur. Actually it was proposd as arly as in lat 60 s by Argyris and Scarpf [], and Odn [30]. Sinc tn t spac-tim finit lmnt mtods av bn widly usd to solv a varity of diffrntial quations,.g., s [3,5,6] for t implmntation of tim-continuous Galrkin finit lmnt scms. Som works on spac-tim finit lmnt mtod for solving yprbolic quations ar availabl, s [29, 32]. Rcntly, Tu t al., proposd a spac-tim discontinuous Galrkin cll vrtx scm to solv consrvation law and tim dpndnt diffusion quations [33]. Tis scm is concptually simplr tan otr xisting DG-typ mtods. Nvrtlss, to t bst of our knowldg, t finit lmnt mtod as not bn usd to discrtiz t tmporal domain in fully discrt scm for Maxwll s quations up to now. On t otr and, tim-discontinuous Galrkin mtods wr originally dvlopd for t first ordr yprbolic quations [9, 3] and av bn succssfully applid to various yprbolic and parabolic quations s [2, 6] and t rfrncs trin. Ty usually lad to som stabl and igr-ordr accurat numrical scms. Actually in [8, 9], t tim-discontinuous Galrkin mtod was first sown to b an A-stabl, igr-ordr accurat ordinary diffrntial quation solvr. Furtrmor, t tim-discontinuous Galrkin framwork sms conduciv for t rigorous justification of t rror stimats [8]. In [34] w introducd a smi-discrt locally divrgnc-fr DG mtod for solving Maxwll s quations in disprsiv mdia undr a unifid framwork. Aftr t discrtization of t spatial domain, w obtaind a Voltrra intgro-diffrntial systm in

3 98 Z. Q. Xi, B. Wang and Z. Zang / Commun. Comput. Pys., 4 203, pp tim t. Tn a continuous Galrkin mtod was usd to solv tis rducd systm. T numrical rsults ar surprisingly good! T scm is stabl vn wn t tim stp siz t is largr tan t spatial ms siz! Indd, t scm is ssntially implicit and placs no rstriction on t tim stp siz. Tis advantag ovr many xplicit scms, wic av a so-calld CFL condition on t tim stp siz t, maks t scm wortwil to b studid. Unfortunatly, tortical analysis of tis mixd scm DG mtod in spac and continuous Galrkin finit lmnt mtod in tim sms to b vry difficult. Trfor, w want to sk for a numrical scm, wic not only is unconditionally stabl, but also is accssibl for tortical justification. Sinc t DG mtod is applid to discrtiz t spatial domain in our arlir work [34], a natural considration is to us it to trat t tmporal domain as wll. Hnc, w propos r a spac-tim DG scm, in wic two diffrnt DG mtods ar usd to discrtiz t spatial and tmporal domains, rspctivly. Fortunatly, w ar abl to prov tat t nw scm is unconditionally stabl. Again, w obtain vry accurat numrical solutions vn wn t tim stp siz t is largr tan t spatial ms siz, as xpctd. Furtrmor, w prov t convrgnc rat O t r+ + k+ 2 in t L 2 -norm wn t r-t and k-t ordr polynomials ar usd in tmporal discrtization and spatial discrtization rspctivly. Comparing wit finit diffrnc mtods usd in [9, 4, 20 23, 25, 26], our spac-tim DG mtod is a ig-ordr scm in tmporal variabl. T situation is vn bttr in our numrical xprimnts, wr t optimal convrgnc rat O k+ in t spatial stp is sown. Morovr, an ultra-convrgnc rat O t 2r+ in t tim stp is obsrvd for t numrical fluxs wit rspct to t tmporal variabl at t grid points. Tis is anotr significant advantag of our approac ovr many xisting numrical mtods. T outlin of tis papr is as follows. T modl problm and our proposd spactim DG scm ar introducd in Sction 2. Bot t L 2 -stability and L 2 -rror stimat ar provd in Sction 3, wr an oprator splitting is introducd to dcompos t rror into tmporal part and spatial part. Numrical xampls ar providd in Sction 4. Finally, som possibl futur works and concluding rmarks ar prsntd in Sction 5. 2 Spac-tim discontinuous Galrkin mtod 2. Modl problm W considr Maxwll s quations in simpl omognous mdia as follows: µ H t = E, x,t Ω I, 2. ǫ E = H, x,t Ω I, 2.2 t

4 Z. Q. Xi, B. Wang and Z. Zang / Commun. Comput. Pys., 4 203, pp wr ǫ and µ ar t lctric prmittivity and magntic prmability rspctivly, Ω is a Lipscitz polydron and I=[0,T]. Morovr a simpl initial condition and a prfct conduct boundary condition Hx,0=H 0 x, Ex,0=E 0 x in Ω 2.3 n E = 0 on Ω I 2.4 ar imposd. Hr n is t outward normal of Ω, E 0 and H 0 ar givn functions wit H 0 satisfying [27] µh 0 =0 in Ω, H 0 n=0 on Ω. 2.5 Tn 2.5, togtr wit 2., implis µh=0 in Ω I, 2.6 wic is usually a statmnt of Maxwll s quations. Furtrmor t scond condition in 2.5, combind wit 2. and 2.2, lads to [27] As in [], w rwrit into t consrvativ form H n = 0 on Ω 0,T]. 2.7 QU t + fu=0, x,t Ω I, 2.8 Ux,0=U 0 x, x Ω, 2.9 wr H U= E µi3 3 0 Q= 0 ǫi 3 3 H0 x, U 0 x= E 0 x, f i U=, fu=[f U,f 2 U,f 3 U] T, 2.0 i E i H. 2. T following notations in Soblv spac will b usd latr. Dnot H k Ω and H k I t standard Soblv spacs quippd wit norms k,ω and k,i, rspctivly. Furtr, dfin { L 2 I,H k Ω= u u,t H k Ω, t I, and u,t 2 k,ω }, dt< 2.2 I quippd wit norm and u k,0 = u,t 2 k,ω dt 2 ; 2.3 I { H k I,L 2 Ω= u ux, H k I, x Ω, and ux, 2 k,i }, dx< 2.4 Ω

5 920 Z. Q. Xi, B. Wang and Z. Zang / Commun. Comput. Pys., 4 203, pp quippd wit norm u 0,k = ux, 2 k,i dx Ω T corrsponding vctor function spacs ar dnotd by H k Ω 3, H k I 3, L 2 I,H k Ω 3 and H k I,L 2 Ω 3. For U=H,E T and H,t,E,t H k Ω 3, t I, dfin and 2.2 Numrical scm U,t k,ω = H,t 2 k,ω + E,t 2 k,ω /2, 2.6 Q /2 U,t k,ω = µ H,t 2 k,ω +ǫ E,t 2 k,ω / Assum tat T is a triangulation of t domain Ω wit t lmnt dnotd by, t dg by, and t outward normal by n. W assum tat vry lmnt of t triangulation T is affin quivalnt, s [0, Sction 2.3]. For ac lmnt, w dnot by t diamtr of and by ρ t diamtr of t biggst ball includd in. St =max {t radius of t largst circl witin }. W also dnot bye I t union of all intrior facs of T, by E D t union of all boundary facs of T, and by E =E I ED t union of all facs of T. T triangulation w considr as to b rgular, i.. tr xists a positiv constant C suc tat ρ C, T, 2.8 s [0, Sction 3.]. Morovr, lt 0=t 0 <t < <t n =T b a uniform triangulation of t intrval I wit lmnts dnotd by =[t j,t j+ ], j=0,,,n and t tim stp siz by t=t j+ t j. Lt P k or P k dnot t spac of polynomials in or of dgr at most k. Tn t DG finit lmnt spac for t spatial discrtization is S k,ω ={ v L 2 Ω : v P k, T }. On t otr and, t DG finit lmnt spac for t tmporal discrtization is S r,i ={ v L 2 I : v Ij P r, j=0,,,n }. Sinc our approac is to discrtiz bot t spatial and tmporal domains by t DG mtods, t spac-tim discontinuous finit lmnt spac is dfind by V r,k = V r,k V r,k,

6 Z. Q. Xi, B. Wang and Z. Zang / Commun. Comput. Pys., 4 203, pp wr V r,k =Sk,Ω Sr,I 3. In fact, ac componnt of t lmnt in V r,k is t product of two lmnts from S k,ω and S r,i. It is wll known tat t coic of t numrical fluxs plays a crucial rol in t dsign of discontinuous Galrkin scms. In ordr to dfin numrical fluxs, w nd to introduc som notations first. Lt b an intrior fac blonging to lmnt. W dnot v int x= lim δ 0 vx+δn, v xt x= lim δ 0+ vx+δn x. Tn w dfin t avrag and tangntial jump of v on any intrior fac as follows: v= vint +v xt, [v] T = n v int n v xt. 2 For a boundary fac E D wic blongs to t lmnt, w dnot In addition, w dfin v int x=v int x x. vt + j = lim t t j +0 vt, vt j = lim t t j 0 vt. Now w ar rady for t dfinition of t numrical scm. Multiplying by a tst function v, intgrating ovr ac Q j =, and tn intgrating by parts, w obtain QU v t dxdt+ fu n vdsdt fu vdxdt+ QU v t j+ t j dx= Tn t DG scm basd on 2.9 is to find U V r,k suc tat QU v t dxdt+ fu n v dsdt t fu v dxdt+ QÛ v j+ t j dx= for all v V r,k and all lmnts Q j =, T, j=0,,,n. Hr fu n and Û ar t numrical fluxs on t fac E and at t nodal points t j,j=0,,,n, rspctivly. Motivatd by [] and [34], w tak fu n n = Ē Z 2 [H ] T n H + 2Z [E 2.2 ] T

7 922 Z. Q. Xi, B. Wang and Z. Zang / Commun. Comput. Pys., 4 203, pp on an intrior fac and fu n= n H int Z n Eint 2.22 on a boundary fac = E D, wr Z= µ/ǫ dnots t impdanc of t mdium. Obviously tis numrical flux is consistnt wit fu n. On t otr and, w tak Û x,t j =U x,t j, j=,2,,n, Û x,0=p U 0, 2.23 wr P is t lmnt-wis L 2 projction oprator and will b dfind latr. 3 L 2 -stability and L 2 -rror stimat In tis sction, bot t L 2 -stability and rror stimat in L 2 norm of our numrical scm will b analyzd. T fact tat t DG mtods ar usd to discrtiz spatial and tmporal domains simultanously will facilitat t tortical justification. Actually t corrsponding tortical analysis can b don undr t framwork of standard Galrkin finit lmnt mtod basd on an oprator dcomposition tcniqu. 3. L 2 -stability W first focus on t L 2 -stability of our numrical scm. St B Ij,U,v = QU v t dxdt+ fu n v dsdt and fu v dxdt+ QÛ v t j+ t j dx, 3. B Ij U,v = T B Ij,U,v. 3.2 According to t DG scm 2.20, B Ij,U,v =0, for all v V r,k. Tn t solution U satisfis B Ij U,v =0, v V r,k, j=0,,,n. 3.3 W av t following rsult for t stability of t DG scm Torm 3.. Assum tat U =H,E T is a solution of 2.20, tn Q /2 U,T 2 0,Ω +Θ T,T U Q /2 U 0 2 0,Ω, wr n Θ T,T U = j=0 Θ Ij,T U 3.4

8 Z. Q. Xi, B. Wang and Z. Zang / Commun. Comput. Pys., 4 203, pp and Θ Ij,T U = Z [U,H ] T 2 + Z [U,E] T 2 dsdt+2 E I E D Z n Uint,E 2 dsdt. Proof. Lt v = U in 3.2, w av B Ij U,U = QU U t dxdt+ fu n U dsdt T T t fu U dxdt+ QÛ U j+ t T T j dx. 3.5 By t dfinition of t numrical flux in 2.23 and Scwatz inquality, dirct calculation sows tat t QÛ U j+ t j dx QU U t dxdt+ T T 2 Q/2 U,t j+ 2 0,Ω 2 Q/2 U,t j 0,Ω, j=,2,,n 3.6 and QU U t dxdt+ T I 0 QÛ U t t0 dx T 2 Q/2 U,t 2 0,Ω 2 Q/2 U 0 0,Ω, 3.7 wr P U 0 U 0 is usd according to t dfinition of t projction oprator P in Subsction 3.2. Morovr, by following t sam stratgy as tat in t proof of Lmma 3. in [34], w obtain a similar idntity, i.., T fu n U dsdt T fu U dxdt = 2 Θ,T U. 3.8 T combination of 3.3 and lads to for j=,2,,n and 2 Q/2 U,t j+ 2 0,Ω 2 Q/2 U,t j 0,Ω + 2 Θ,T U Q/2 U,t 2 0,Ω 2 Q/2 U 0 0,Ω + 2 Θ I 0,T U Summing up 3.9 and 3.0 ovr j from 0 to n, w obtain t dsird rsult.

9 924 Z. Q. Xi, B. Wang and Z. Zang / Commun. Comput. Pys., 4 203, pp Error stimat Now w turn to t L 2 rror stimat of t spac-tim DG solution. For tis purpos, w introduc two lmnt-wis projction oprators Π and P and giv t corrsponding approximation rsults wic will b usd in t proof of t L 2 -rror stimat latr. First w introduc a projction oprator Π : H r+ I S r,i, suc tat Π ut j+ =ut j+, 3. u Π uvdt=0, v P r, j=0,,,n, r. 3.2 Furtrmor, w av t following rror stimat [6]. Lmma 3.. For any u H r+, w av u Π u 0,Ij C t r+ u r+,ij. 3.3 Morovr, w also nd t lmnt-wis L 2 -projction oprator P : H k+ Ω S k,ω, suc tat u P u vdx=0, v P k, T. 3.4 For tis L 2 projction oprator, w av t following approximation lmma. Lmma 3.2. Lt u H k+. Tn u P u 0, C k+ u k+,, u P u 0, C k+/2 u k+,. T rror analysis of numrical mtods for tim-dpndnt problms is oftn mor difficult tan tat for t tim-indpndnt ons. Actually, t main difficulty is ow to dcompos t rror into t tmporal part and spatial part wic can b andld indpndntly. In our work w introduc an oprator dcomposition as follows: I Π P =I Π +I P I Π I P. 3.5 In fact, tis tcniqu was first introducd in t analysis of t finit lmnt mtod for multi-dimnsional lliptic problms by Douglas, Dupont and Wlr in [3]. Tn it was implmntd to analyz t convrgnc proprty of t finit lmnt mtods for parabolic and yprbolic problms by Cn s Captr 3 in [7] for mor dtails. In trms of t ortogonality rlations in 3.2 and 3.4, w av [I Π I P u]v=0, v P k, k, 3.6 [I Π I P u]v=0, v P k, T. 3.7

10 Z. Q. Xi, B. Wang and Z. Zang / Commun. Comput. Pys., 4 203, pp Actually 3.6 can b provd by a straigtforward implmntation of 3.2. On t otr and, 3.7 can b obtaind immdiatly basd on t fact tat I Π and I P ar indpndnt of ac otr. It is notd tat t numrical fluxs dfind in 2.2, 2.22 and 2.23 ar consistnt xcpt for Û0=P U 0. St =U U. According to 2.9 and 2.20, w av On t otr and, by 2.20, B Ij,,v =0, v V r,k, j=,2,,n. 3.8 B I0,U,v [ ] = QU v t dx+ fu n v ds fu v dx dt I 0 [ + QU x,t v x,t QP U 0 v x,0 + ] dx [ ] = QU v t dx+ fu n v ds fu v dx dt I 0 [ + QU x,t v x,t QU 0 v x,0 + ] dx. 3.9 Taking v=v in 2.9 and tn subtracting 3.9 from 2.9 wit j=0, w obtain Q v t dxdt+ f n v dsdt I 0 I 0 f v dxdt+ Qx,t v x,t dx= I 0 Dnot t lft-and sid of 3.20 by B I0,,v and lt B Ij,,v =B Ij,,v for j=,2,,n. Tn w obtain t following rror quation wr B Ij,v =0, T rror can b dcomposd into v V r,k, j=0,,,n, 3.2 B Ij,v = T B Ij,,v =U Π P U U Π P U=R θ, 3.23 wr R=U Π P U, θ= U Π P U V r,k Substituting =R θ into 3.2 and taking v = θ, w av B Ij R,θ= B Ij θ,θ. 3.25

11 926 Z. Q. Xi, B. Wang and Z. Zang / Commun. Comput. Pys., 4 203, pp Lmma 3.3. In trms of B Ij θ,θ, w av t following stimat for j=,2,,n and B Ij θ,θ 2 Q/2 θ,t j+ 2 0,Ω 2 Q/2 θ,t j 2 0,Ω + 2 Θ,T θ, 3.26 B I0 θ,θ= 2 Q/2 θ,t 2 0,Ω + 2 Q/2 θ, ,Ω + 2 Θ I 0,T θ Proof. By t dfinition of B Ij,v, w av B Ij θ,θ= Qθ θ t dxdt+ fθ n θdsdt T T fθ θdxdt+ Q ˆθ θ t j+ t T T j dx, 3.28 for j=,2,,n. Similar to t proof of t stability in Torm 3., w av Qθ θ t dxdt+ Q ˆθ θ t j+ t T T j dx 2 Q/2 θ,t j+ 2 0,Ω 2 Q/2 θ,t j 2 0,Ω, 3.29 for j =,2,,n. Following t sam stratgy as tos in t proof of Torm 3., it is asy to obtain fθ n θdsdt fθ θdxdt= T T 2 Θ,T θ T combination of 3.28, 3.29 and 3.30 yilds According to 3.9, w av B I0 θ,θ= Qθ θ t dxdt+ T I 0 fθ n θdsdt T I 0 fθ θdxdt+ T I 0 Qθx,t θx,t dx T T proof is complt. = 2 Q/2 θ,t 2 0,Ω + 2 Q/2 θ, ,Ω + 2 Θ I 0,T θ. 3.3 Now t ky point is to stimat B I0 R,θ and B Ij R,θ, j=,2,,n. In trms of 3., 3.20, w av B Ij R,θ= QR θ t dxdt+ fr n θdsdt T T T fr θdxdt+ T Q ˆR θ t j+ t j dx, 3.32

12 Z. Q. Xi, B. Wang and Z. Zang / Commun. Comput. Pys., 4 203, pp for j=,2,,n, and B I0 R,θ= T I 0 T I 0 QR θ t dxdt+ T fr θdxdt+ T I 0 Now 3.5 is usd to dcompos R into tr parts, i.., Dnotd by tn R=I Π P U fr n θdsdt QRx,t θx,t dx =I Π U+I P U I Π I P U ξ=i Π U, η=i P U, ρ= I Π I P U, R=ξ+η+ρ According to t proprtis of projction Π sown in 3., w av ξx,t j =0, ρx,t j =0, x Ω, j=,2,,n Du to θ,θ t V r,k, tus θ Hx,, θ H t x,, θ E x,, θ E t x, P k 3, θ H t,t Ij, θ E t,t Ij P r 3. According to t dfinition of t projction oprators Π and P, 3.6 and 3.7, w av t following ortogonality rlations, i.., ξx,t θ t x,tdt=0, ηx,t θ t x,tdx=0, ηx,t θx,tdx = 0, As a consqunc, ρx,t θ t x,tdt=0, x Ω, 3.37a ρx,t θ t x,tdx=0, t 0,T], 3.37b ρx,t θx,tdx = 0, t 0,T]. 3.37c QR θ t dxdt= Qξ+η+ρ θ t dxdt=0, T, 3.38 for j=0,,2,,n. On t otr and, by 3.36 and 3.37, Q ˆR θ t j+ t j dx= Q Rx,t j+ θx,t j+ Rx,t j θx,t + j dx = Q ξx,t j+ +ηx,t j+ +ρx,t j+ θx,t j+ dx Q ξx,t j +ηx,t j +ρx,t j θx,t + j dx =0, j=,2,,n. 3.39

13 928 Z. Q. Xi, B. Wang and Z. Zang / Commun. Comput. Pys., 4 203, pp Furtrmor, QRx,t θx,t dx= Q ξx,t +ηx,t +ρx,t θx,t dx= Implmnting 3.38, 3.39 and 3.40 in 3.32 and 3.33, w av [ ] B Ij R,θ= fr n θds fr θdx dt, 3.4 T for j=0,,2,,n. Lmma 3.4. In trms of B Ij R,θ, w av B Ij R,θ C 2k+ U,t 2 k+,ω dt+ Q /2 θ,t 2 0,Ω 2 dt+ 2 Θ,T θ +C t Ex, 2r+2 2 r+,ij + Hx, 2 r+,ij dx T T Ω Proof. Substituting t dcomposition 3.35 in 3.4, w obtain B Ij R,θ= fξ n θdsdt fξ θdxdt T T + fη n θdsdt fη θdxdt T + fρ n θdsdt T fρ θdxdt =M j + M2 j + M3 j Obviously, ξ=u Π U is continuous in spac. Trfor t consistncy of t numrical flux fu n implis M j = fξ n θdsdt fξ θdxdt T T = T ξ E θ H ξ H θ E dxdt Using Lmma 3.2 and t Young s inquality, w av M j Ω 2µ ξ E 2 + 2ǫ ξ H 2 + µ 2 θ H 2 + ǫ 2 θ E 2 dtdx C t Ex, 2r+2 2 r+,ij + Hx, 2 r+,ij dx Ω + Q /2 θ,t 2 0,Ωdt

14 Z. Q. Xi, B. Wang and Z. Zang / Commun. Comput. Pys., 4 203, pp By t dfinition of t numrical fluxs in 2.2 and 2.22, a straigtforward calculation lads to M 2 j = n η E Z T E 2 [η H] T θ int H n η H + 2Z [η E] T θ int E dsdt I + n η T H int + E Z n ηint E θe I int dsdt fη θdxdt j D T = η H [θ E ] T + Z E 2 [η H] T [θ H ] T dsdt I Similarly, + + E I E D T η E [θ H ] T + n η int M 3 j I = j H θe int 2Z [η E] T [θ E ] T dsdt + Z n ηint E n θe int dsdt θ E η H θ H η E dxdt E I + + ρ H [θ E ] T + Z 2 [ρ H] T [θ H ] T dsdt E I E D T ρ E [θ H ] T + n ρ int H θe int 2Z [ρ E] T [θ E ] T dsdt + Z n ρint E n θe int dsdt θ E ρ H θ H ρ E dxdt Sinc θ V r,k, by t ortogonality proprty of t projction oprator P, w av θ E η H θ H η E dx=0, θ E ρ H θ H ρ E dx= Implmnting Lmma 3.3 and t Young s inquality, w obtain M 2 j I 2Z η H j E Z η E 2 + Z 2 [η H] T 2 + 2Z [η E] T 2 I + Z ηh int 2 + E Z ηint E 2 dsdt+ 2Z I D j E D + 4 Z [θ H ] T 2 + Z [θ E] T 2 dsdt E I dsdt n θ int E 2 dsdt

15 930 Z. Q. Xi, B. Wang and Z. Zang / Commun. Comput. Pys., 4 203, pp C 2k+ + 2Z M 3 j I j E I U,t 2 k+,ω dt+ 4 E D E I Z [θ H ] T 2 + Z [θ E] T 2 dsdt n θ int E 2 dsdt, Z ρ H Z ρ E 2 + Z 2 [ρ H] T 2 + E D C 2k+ + 2Z E I Z ρ int H 2 + Z ρint E 2 dsdt+ 2Z Z [θ H ] T 2 + Z [θ E] T 2 dsdt U,t 2 k+,ω dt+ 4 E D According to 3.45, 3.49 and 3.50, w av B Ij R,θ C 2k+ T proof is complt. E I 2Z [ρ E] T 2 E D dsdt n θ int E 2 dsdt Z [θ H ] T 2 + Z [θ E] T 2 dsdt n θ int E 2 dsdt U,t 2 k+,ω dt+ Q /2 θ,t 2 0,Ω 2 dt+ 2 Θ,T θ. +C t Ex, 2r+2 2 r+,ij + Hx, 2 r+,ij dx. 3.5 Ω Now w giv t main rsult in trms of t L 2 -rror stimat. Torm 3.2. Lt U =H,E T b t solution of2.20 andh,e T t xact smoot solution of Assum tat Tn 3, 3. H,E L 2 [0,T],H k+ Ω E, H H r+ [0,T],L Ω 2 µ Hx,T H x,t 0 +ǫ Ex,T E x,t 0 C t r+ E 0,r+ + H 0,r+ +C k+ 2 E k+,0 + H k+,0 +C k+ Ex,T k+,ω + Hx,T k+,ω, wr C is a constant rlying on T, but indpndnt of t and.

16 Z. Q. Xi, B. Wang and Z. Zang / Commun. Comput. Pys., 4 203, pp Proof. By Eq. 3.25, Lmma 3.3 and Lmma 3.4, w av for j=,2,,n, and Tus w obtain 2 Q/2 θ,t j+ 2 0,Ω 2 Q/2 θ,t j 2 0,Ω + 2 Θ,T θ Q /2 θ,t 2 0,Ω 2 dt+ 2 Θ,T θ+c 2k+ U,t 2 k+,ω dt +C t Ex, 2r+2 2 r+,ij + Hx, 2 r+,ij dx, 3.52 Ω 2 Q/2 θ,t 2 0,Ω + 2 Q/2 θ, ,Ω + 2 Θ I 0,T θ Q /2 θ,t 2 0,Ω 2 dt+ I 0 2 Θ I 0,T θ+c 2k+ U,t 2 k+,ω dt I 0 +C t 2r+2 Ex, 2 r+,i0 + Hx, 2 r+,i 0 dx Ω Q /2 θ,t j+ 2 0,Ω t Q /2 θ,t j 2 0,Ω +C t 2k+ U,t 2 k+,ω dt +C t t 2r+2 for j=,2,,n, and Ω Ex, 2 r+,ij + Hx, 2 r+,ij dx, 3.54 Q /2 θ,t 2 0,Ω C t I 2k+ U,t 2 k+,ω dt 0 +C t t 2r+2 Ex, 2 r+,i0 + Hx, 2 r+,i 0 dx, 3.55 by using Gronwall s inquality. Hnc w obtain Ω Q /2 θx,t 2 0 C T t 2r+2 E 2 0,r+ + H 2 0,r+ +C T 2k+ E 2 k+,0 k+,0 + H On t otr and, Q /2 R,T 2 0,Ω = Q/2 η,t 2 0,Ω C2k+2 U,T 2 0,Ω. 3.57

17 932 Z. Q. Xi, B. Wang and Z. Zang / Commun. Comput. Pys., 4 203, pp By using a triangular inquality, w av Q /2,T 2 0,Ω Q C /2 R,T 2 0,Ω + Q/2 θ,t 2 0,Ω C T t 2r+2 E 2 0,k+ + H 2 0,k+ wic complts t proof. +C T 2k+ E 2 k+,0 + H 2 k+,0 +C 2k+2 U,T 2 0,Ω, 3.58 Rmark 3.. It is notd tat only smi-norms ar usd in Lmmas 3. and 3.2. Hnc all norms in t rigt and sid of t rror stimat in Torm 3.2 can b rplacd by t corrsponding wakr smi-norms. 4 Numrical rsults In tis sction, som numrical xampls ar givn to justify our tortical prdiction. T uniform Cartsian ms is usd in all numrical xampls. According to t tortical analysis abov, w know tat our numrical scm is stabl witout any rstriction on t tim stp siz t. Actually w obtain accurat numrical solutions vn wn t is largr tan. Morovr, in tis sction, t L 2 rrors ar computd in t following way, 4. 2-D numrical xampl u u 0 = T u u 2 dω 2. T similar rror stimat for 2-D Maxwll quations can b obtaind in t sam way as w av don for 3-D cas, by introducing t scalar and vctor curl oprators curle= E 2 x E y, E= E E, y x T. To justify our tortical analysis, w first giv a 2-D numrical xampl. Considr t following 2-D modl problm H x t + E z y = R, H y t E z x = R 2, E z t H y x H x y = R 3,

18 Z. Q. Xi, B. Wang and Z. Zang / Commun. Comput. Pys., 4 203, pp Tabl : T convrgnc rat of L 2 rror for r=k= 2-D cas. tim tim stp ms E,T E,T 0 ordr H,T H,T 0 ordr T=0.5 N= N= N= N= T=5 N= N= N= N= T=50 N= N= N= N= Tabl 2: T convrgnc rat of L 2 rror for r=k= 2 2-D cas. tim tim stp ms E,T E,T 0 ordr H,T H,T 0 ordr T= N= N= N= T=0 N= N= N= T=00 N= N= N= in [0,] 2, wr R i, i=,2,3 ar cosn suc tat t xact solution is H x H y E z =00 x x 2ytsint y y 2xtsint xy x ytsint+x+y First w coos t tim stp siz t qual to t spatial ms siz and t polynomials of t sam dgr for bot tmporal and spatial variabls, i.. r=k. T rsults and tir corrsponding convrgnc ordr ar sown in Tabls and 2 for r=k= and r=k=2 rspctivly. It is obsrvd tat t convrgnc rat of bot E and H in L 2 -norm is O k+, wic is bttr tan t tortical prdiction. As mntiond abov, tis spac-tim DG scm is unconditionally stabl. Hr w us a numrical xprimnt to vrify tis claim. W tak t tim stp siz t largr tan t spatial ms siz and r=, k=2. T L 2 -rrors ar listd in Tabl 3. It is noticd tat, vn wn t tim stp siz t=0.2,0.3, wic is largr tan t spatial ms siz = 0.25, t rlativ rrors do not incras wn T = 20,30..

19 934 Z. Q. Xi, B. Wang and Z. Zang / Commun. Comput. Pys., 4 203, pp Tabl 3: T unconditional stability of t spac-tim DG mtod for r=, k= 2, 8 8 ms 2-D cas. t E,T E,T 0 Rlativ rror H,T H,T 0 Rlativ rror in prcntag in prcntag 0.2 T= T= T= T= T= T= Tabl 4: T ultra-convrgnc of ordr 2r+ in t, = t 2, r=k, T= 2-D cas. k t ms E,T E,T 0 ordr H,T H,T 0 ordr Tabl 5: T ultra-convrgnc of ordr 2r+ in t, = t, r=, k= 2 2-D cas. tim tim stp ms E,T E,T 0 ordr H,T H,T 0 ordr T= N= N= N= T=0 N= N= N= T=00 N= N= N= Bsids t unconditional stability, our approac as anotr important advantag ovr many xisting numrical scms, i.., t implmntation of t DG mtod in tim-discrtization lads to an ultra-convrgnc of ordr 2r+ in tim stp for t numrical fluxs w.r.t. t at t grid points. Numrical rsults in Tabls 4 and 5 sow tis pnomnon from two diffrnt ways. On t on and, w st r=k and coos = t 2 in Tabl 4. Tn t ultra-convrgnc rat ofo t 3 ando t 5 for k= and k=2 ar obsrvd numrically. On t otr and, w lt t= but coos r= and k=2, a convrgnc rat of ordro t 3 of t L 2 -rror is obsrvd in Tabl D numrical xampl W considr t following 3D Maxwll s quation

20 Z. Q. Xi, B. Wang and Z. Zang / Commun. Comput. Pys., 4 203, pp Tabl 6: T convrgnc rat of L 2 rror for r=k= 3-D cas. tim tim stp ms E,T E,T 0 ordr H,T H,T 0 ordr T= N= N= N= T=0 N= N= N= T=00 N= N= N= Tabl 7: T convrgnc rat of L 2 rror for r=k= 2 3-D cas. tim tim stp ms E,T E,T 0 ordr H,T H,T 0 ordr T= N= N= N= T=0 N= N= N= T=00 N= N= N= H t + E=R, E t H=R 2, in Ω=[0,] 3. Hr R, R 2 ar cosn suc tat t xact solution is y y 2 z z 2 E= x x 2 z z 2 tcost+x+y+z, y z H = z x tcost+x+y+z. x x 2 y y 2 x y Lik t 2-D cas w firstly coos t tim stp siz t qual to t spatial ms siz and t polynomials of t sam dgr for bot tmporal and spatial variabls, i.. r=k. T L 2 -rrors and tir corrsponding convrgnc ordr ar sown in Tabls 6 and 7 for r=k= and r=k=2 rspctivly. It is obsrvd tat t convrgnc rat of bot E and H in L 2 -norm is O k+, wic is bttr tan t tortical prdiction also. To sow t unconditional stability, w tak t tim stp siz t largr tan t spatial ms siz and r=k=. T L 2 -rrors ar listd in Tabl 8. It is noticd tat t rlativ rrors do not incras wn T= 20,30 vn wn t tim stp siz t=0.2,0.3 is largr tan t spatial ms siz =0.25.

21 936 Z. Q. Xi, B. Wang and Z. Zang / Commun. Comput. Pys., 4 203, pp Tabl 8: T unconditional stability of t spac-tim DG mtod for r=k=, ms 3-D cas. t E,T E,T 0 Rlativ rror H,T H,T 0 Rlativ rror in prcntag in prcntag 0.2 T= T= T= T= T= T= Tabl 9: T ultra-convrgnc of ordr 2r+ in t, = t 2, k= r, T= 3-D cas. r t ms E,T E,T 0 ordr H,T H,T 0 ordr Tabl 0: T ultra-convrgnc of ordr 2r+ in t, = t, r=, k= 2 3-D cas. tim tim stp ms E,T E,T 0 ordr H,T H,T 0 ordr T= N= N= N= T=0 N= N= N= T=00 N= N= N= Numrical rsults in Tabls 9 and 0 sow t ultra-convrgnc of our mtod numrically. In Tabl 9 w st r=k and coos = t 2. Tn t ultra-convrgnc rat of O t 3 and O t 5 for k= and k=2 ar obsrvd numrically. In Tabl 0, w coos t tim stp siz t qual to t spatial ms siz, and r=, k=2. An ultra-convrgnc of ordro t 3 is obsrvd numrically. Rmark 4.. Altoug t tortical rror bound is O k+/2 + t r+, our numrical tsts indicat tat t actual rror sms to bo k+ + t 2r+. Trfor, w propos t following two stratgis in practic to optimiz t scm: Us t sam polynomial spac P k 3 P k and diffrnt lmnt scal = t 2 k+ ; 2 Us t sam lmnt scal t= and diffrnt polynomial spacsp k 3 P r wit 2r=k. T first stratgy suggsts to us t largr tim stps wil t scond on rcommnds to us igr ordr polynomial spacs for spatial discrtization.

22 Z. Q. Xi, B. Wang and Z. Zang / Commun. Comput. Pys., 4 203, pp Concluding rmarks A spac-tim DG mtod is proposd to solv tim-dpndnt Maxwll s quation in omognous mdia. T L 2 -stability is provd. Basd on a tcnical oprator dcomposition, t convrgnc rat of O k+ 2+ t r+ in t L 2 -norm is stablisd undr t standard Galrkin finit lmnt framwork. T proposd spac-tim DG mtod is ssntially an implicit scm. T main advantags of it ovr t traditional xplicit tim stp approacs ar its unconditionally stabl proprty, and 2 its ultra-convrgnc in tim stps. Ts favorabl proprtis mak it possibl to comput long tim bavior of tim-dpndnt Maxwll s quations and offst t disadvantag of t computational cost of t implicit mtod. As xplaind in Rmark 4., it is advisd to us t 2 in our scm instad of t=o in most xplicit mtods. A systmatic study of comparison of t proposd spac-tim DG mtod wit t xplicit tim stp approacs spatial smi-discrtization by DG would b a sparat work. Otr futur works includ t rigorous justification of t ultra-convrgnc in tim stps, t -vrsion and p-vrsion spac-tim DG mtods for Maxwll s quations in disprsiv mdia and mta-matrials, and tir corrsponding tortical analysis and applications. Acknowldgmnts T first autor is partially supportd by t NSFC 704 and and t Scinc and Tcnology Grant of Guizou Provinc LS[200]05. T scond autor is partially supportd by t NSFC 704 and and Hunan Provincial Innovation Foundation for Postgraduat #CX200B2. T tird autor is partially supportd by t US National Scinc Foundation troug grant DMS-5530, t Ministry of Education of Cina troug t Cangjiang Scolars program, t Guangdong Provincial Govrnmnt of Cina troug t Computational Scinc Innovativ Rsarc Tam program, and Guangdong Provinc y Laboratory of Computational Scinc at t Sun Yat-sn Univrsity. W also tank Profssor Cuanmiao Cn for is lpful discussions. Rfrncs [] J. H. Argyris and D. W. Scarpf, Finit lmnts in tim and spac, Nucl. Engrg. Ds., 0 969, [2] D. Boffi, P. Frnands, L. Gastaldi, and I. Prugia, Computational modls of lctromagntic rsonators: analysis of dg lmnt approximation, SIAM J. Numr. Anal., , [3] R. Bonnrot and P. Jamt, Numrical computation of t fr boundary for t twodimnsional Stfan problm by spac-tim finit lmnts, J. Comput. Pys., , 63-8.

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