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1 SIAM J. NUMER. ANAL. Vol. 5, No., pp c 014 Society for Inustrial an Applie Matematics DISCONTINUOUS GALERKIN METHODS FOR THE VLASOV MAXWELL EQUATIONS YINGDA CHENG, IRENE M. GAMBA, FENGYAN LI, AND PHILIP J. MORRISON Abstract. Discontinuous Galerkin metos are evelope for solving te Vlasov Mawell system, metos tat are esigne to be systematically as accurate as one wants wit provable conservation of mass an possibly total energy. Suc properties in general are ar to acieve witin oter numerical meto frameworks for simulating te Vlasov Mawell system. Te propose sceme employs iscontinuous Galerkin iscretizations for bot te Vlasov an te Mawell equations, resulting in a consistent escription of te istribution function an electromagnetic fiels. It is proven, up to some bounary effects, tat carge is conserve an te total energy can be preserve wit suitable coices of te numerical flu for te Mawell equations an te unerlying approimation spaces. Error estimates are establise for several flu coices. Te sceme is teste on te streaming Weibel instability: te orer of accuracy an conservation properties of te propose meto are verifie. Key wors. Vlasov Mawell system, iscontinuous Galerkin metos, energy conservation, error estimates, Weibel instability AMS subject classifications. 65M60, 74S05 DOI / Introuction. In tis paper, we consier te Vlasov Mawell VM system, te most important equation for te moeling of collisionless magnetize plasmas. In particular, we stuy te evolution of a single species of nonrelativistic electrons uner te self-consistent electromagnetic fiel wile te ions are treate as uniform fie backgroun. Uner te scaling of te caracteristic time by te inverse of te plasma, lengt by te Debye lengt λ D, an electric an magnetic fiels by mcω p /e wit m te electron mass, c te spee of ligt, an e te electron carge, te imensionless form of te VM system is frequency ω 1 p 1.1a 1.1b 1.1c t f + ξ f +E + ξ B ξ f =0, E t = B B J, t = E, E = ρ ρ i, B =0, wit ρ,t= f,ξ,tξ, Ω ξ J,t= f,ξ,tξξ, Ω ξ Receive by te eitors April 1, 013; accepte for publication in revise form January 13, 014; publise electronically April 9, 014. ttp:// Department of Matematics, Micigan State University, East Lansing, MI 4884 yceng@ mat.msu.eu. Tis autor was supporte by grant NSF DMS Department of Matematics an ICES, University of Teas at Austin, Austin, TX 7871 gamba@mat.uteas.eu. Tis autor was supporte by grant NSF DMS Department of Matematical Sciences, Rensselaer Polytecnic Institute, Troy, NY 1180 lif@rpi.eu. Tis autor was partially supporte by NSF CAREER awar DMS an an Alfre P. Sloan Researc Fellowsip. Department of Pysics an Institute for Fusion Stuies, University of Teas at Austin, Austin, TX 7871 morrison@pysics.uteas.eu. Tis autor was supporte by te U.S. Department of Energy, grant DE-FG0-04ER

2 1018 Y. CHENG, I. M. GAMBA, F. LI, AND P. J. MORRISON were te equations are efine on Ω = Ω Ω ξ, Ω enotes position in pysical space, an ξ Ω ξ in velocity space. Here f,ξ,t 0 is te istribution function of electrons at position wit velocity ξ at time t, E,t is te electric fiel, B,t is te magnetic fiel, ρ,t is te electron carge ensity, an J,t is te current ensity. Te carge ensity of backgroun ions is enote by ρ i, wic is cosen to satisfy total carge neutrality, Ω ρ,t ρ i = 0. In tis paper, we assume perioic bounary conitions in -space an Ω, Ω ξ to be bo omains. Te initial conitions are enote by f 0 = f,ξ,0, E 0 = E, 0, an B 0 = B, 0. We also assume tat te initial istribution function f 0,ξ H m Ω L 1 Ω ξ, i.e., is in a Sobolev space of orer m an is integrable wit finite energy in ξ-space, were L p m Ω ξ {ψ : Ω ξ ψ p 1 + ξ m/ ξ < }. Te initial fiels E 0 anb 0 are also assume to be in H m Ω. Te VM system as wie importance in plasma pysics for escribing space an laboratory plasmas, wit application to fusion evices, ig-power microwave generators, an large scale particle accelerators. Te computation of te initial bounary value problem associate to te VM system is quite callenging, ue to te igimensionality 6 + time of te Vlasov equation, multiple temporal an spatial scales associate wit various pysical penomena, nonlinearity, an te conservation of pysical quantities ue to te Hamiltonian structure [45, 46] of te system. Particle-in-cell PIC metos [6, 37] ave long been very popular numerical tools, in wic te particles are avance in a Lagrangian framework, wile te fiel equations are solve on a mes. Tis remains an active area of researc [4]. In recent years, tere as been growing interest in computing te Vlasov equation in a eterministic framework. In te contet of te Vlasov Poisson system, semi-lagrangian metos [1, 57], finite volume flu balance metos [7, 5, 6], Fourier Fourier spectral metos [41, 4], an continuous finite element metos [61, 6] ave been propose, among many oters. In te contet of VM simulations, Califano et al. [10] ave use a semi-lagrangian approac to compute te streaming Weibel SW instability, current filamentation instability [44], magnetic vortices [9], an magnetic reconnection [8]. Also, various metos ave been propose for te relativistic VM system [56, 5, 58, 38]. In tis paper, we propose te use of iscontinuous Galerkin DG metos for solving te VM system. Wat motivates us to coose DG metos, besies teir many wiely recognize esirable properties, is tat tey can be esigne systematically to be as accurate as one wants, meanwile wit provable conservation of mass an possibly also te total energy. Tis is, in general, ar to acieve witin oter numerical meto frameworks for simulating te VM system. Te propose sceme employs DG iscretizations for bot te Vlasov an te Mawell equations, resulting in a consistent escription of te istribution function an electromagnetic fiels. We will sow tat up to some bounary effects, epening on te size of te computational omain, te total carge mass is conserve an te total energy can be preserve wit a suitable coice of te numerical flu for te Mawell equations an unerlying approimation spaces. Error estimates are furter establise for several flu coices. Te DG sceme can be implemente on bot structure an unstructure meses wit provable accuracy an stability for many linear an nonlinear problems, it is avantageous in long time wave-like simulations because it as low ispersive an issipative errors [1], an it is very suitable for aaptive an parallel implementations. Te original DG meto was introuce by Ree an Hill [53] for a neutron transport equation. Lesaint an Raviart [43] performe te first error

3 DG METHODS FOR VM EQUATIONS 1019 estimates for te original DG meto, wile Cockburn an Su, in a series of papers [0, 19, 18, 17, 1], evelope te Runge Kutta DG RKDG metos for yperbolic equations. RKDG metos ave been use to simulate te Vlasov Poisson system in plasmas [36, 35, 15] an for a gravitational infinite omogeneous stellar system [13]. Some teoretical aspects about stability, accuracy, an conservation of tese metos in teir semiiscrete form are iscusse in [35, 3, ]. Recently semi-lagrangian DG metos [54, 5] were propose for te Vlasov Poisson system. In [39, 40], DG iscretizations for Mawell s equations were couple wit PIC metos to solve te VM system. In a recent work [60], error estimates of fully iscrete RKDG metos are stuie for te VM system. Te rest of te paper is organize as follows: in section, we escribe te numerical algoritm. In section 3, conservation an te stability are establise for te meto. In section 4, we provie te error estimates of te sceme. Section 5 is evote to iscussion of simulation results. We conclue wit a few remarks in section 6.. Numerical metos. In tis section, we will introuce te DG algoritm for te VM system. We consier an infinite, omogeneous plasma, were all bounary conitions in are perioic, an f,ξ,t is assume to be compactly supporte in ξ. Tis assumption is consistent wit te fact tat te solution of te VM system is epecte to ecay at infinity in ξ-space, preserving integrability an its kinetic energy. Witout loss of generality, we assume Ω = L,L ] an Ω ξ =[ L ξ,l ξ ] ξ, were te velocity space omain Ω ξ is cosen large enoug so tat f = 0 at an near te pase space bounaries. We take = ξ = 3 in te following sections, altoug te meto an its analysis can be etene irectly to te cases wen an ξ take any values from {1,, 3}. In our analysis, te assumption tat f,ξ,t remain compactly supporte in ξ, given tat it is initially so, is an open question in te general setting. Te answer to tis question is important for proving te eistence of a globally efine classical solution, an its failure coul inicate te formation of sock-like solutions of te VM system. Weter or not te tree-imensional VM system is globally well-pose as a Caucy problem is a major open problem. Te limite results of global eistence witout uniqueness of weak solutions an well-poseness an regularity of solutions assuming eiter some symmetry or near neutrality constitute te present etent of knowlege [31, 3, 7, 3, 8, 30, 9]..1. Notations. Trougout te paper, stanar notation will be use for te Sobolev spaces. Given a boune omain D R wit =, ξ,or + ξ an any nonnegative integer m, H m D enotes te L -Sobolev space of orer m wit te stanar Sobolev norm m,d,anw m, D enotes te L -Sobolev space of orer m wit te stanar Sobolev norm m,,d an te seminorm m,,d. Wen m =0,wealsouseH 0 D =L D anw 0, D =L D. Let T = {K } an T ξ = {K ξ} be partitions of Ω an Ω ξ, respectively, wit K an K ξ being rotate Cartesian elements or simplices, an anging noes also being allowe; ten T = {K : K = K K ξ K T, K ξ T ξ } efines a partition of Ω. Let E be te set of te eges of T, efine by E = {e : e = K K,K,K T }, similarly let E ξ be te set of te eges of T ξ ; ten te eges of T will be E = {K e ξ : K T, e ξ E ξ } {e K ξ : e E, K ξ T ξ }. Here we take into account te perioic bounary conition in te -irection wen efining E an E. Furtermore,E ξ = Eξ i Eb ξ wit Ei ξ an Eb ξ being te set of interior

4 100 Y. CHENG, I. M. GAMBA, F. LI, AND P. J. MORRISON an bounary eges of T ξ, respectively. In aition, we enote te mes size of T as = ma, ξ = ma K T K, were = ma K T K wit K = iamk, ξ =ma Kξ T ξ Kξ wit Kξ =iamk ξ, an K =ma K, Kξ for K = K K ξ. Wen te mes is refine, we assume ξ,min is uniformly boune from above by a positive constant σ 0.Here,min =min K T K. Tis assumption is require tecnically in our error analysis, an it stems from tat te electric an magnetic fiels epen only on wile our error estimate is establise in terms of see, for instance, te use of an inverse inequality in 4.5 in te proof of Teorem 4.3. It is furter assume tat {T } is sape-regular wit = or ξ. Tat is, if enotes te iameter of te largest spere inclue in K,tereis ρ K K σ K T ρ K for a positive constant σ inepenent of. Net we efine te iscrete spaces { }.1a G k = g L Ω : g K=K K ξ P k K K ξ K T, K ξ T ξ = { g L Ω : g K P k } K, K T, U r = { U [L Ω ] : U K [P r K ], K T }.1b, were P r D enotes te set of polynomials of total egree at most r on D, ank an r are nonnegative integers. Note te space G k, wic we use to approimate f, is calle P-type, an it can be replace by te tensor prouct of P-type spaces in an ξ, { }. g L Ω : g K=K K ξ P k K P k K ξ, K T, K ξ T ξ, or by te tensor prouct space in eac variable, wic is calle Q-type { }.3 g L Ω : g K=K K ξ Q k K Q k K ξ, K T, K ξ T ξ. Here Q r D enotes te set of polynomials of egree at most r in eac variable on D. Te numerical metos formulate in tis paper, as well as te conservation, stability, an error estimates, ol wen any of te spaces above are use to approimate f. In our simulations of section 5, we use te P-type of.1a as it is te smallest an, terefore, reners te most cost-efficient algoritm. In fact, te ratios of te imensions of tese tree spaces efine in.1a,., an.3 are k n+ 1 n=0 1 : k n+ 1 n=0 1 :k +1 wit = ξ =. For piecewise functions efine wit respect to T or T ξ, we furter introuce te jumps an averages as follows. For any ege e = {K + K } E, wit n ± as te outwar unit normal to K ±, g ± = g K ±,anu ± = U K ±,tejumpsacrosse are efine as [g] = g + n + + g n, [U] = U + n + + U n, [U] τ = U + n + + U n, an te averages are {g} = 1 g+ + g, {U} = 1 U+ + U.

5 DG METHODS FOR VM EQUATIONS 101 By replacing te subscript wit ξ, one can efine [g] ξ,[u] ξ, {g} ξ,an{u} ξ for an interior ege of T ξ in Ei ξ. For a bounary ege e Eb ξ wit n ξ being te outwar unit normal, we use.4 [g] ξ = gn ξ, {g} ξ = 1 g, {U} ξ = 1 U. Tis is consistent wit te fact tat te eact solution f is compactly supporte in ξ. For convenience, we introuce some sortan notation, Ω = T = K T K, Ω = T = K T K, E = e E e, were again is or ξ. In aition, g 0,E = g + g 0,E T ξ 0,T E 1/ wit g ξ 0,E T ξ = E g ξs T ξ 1/, g 0,T E ξ = T E ξ g s ξ 1/, an g 0,E = E g s 1/. Tere are several equalities tat will be use later, wic can be easily verifie using te efinitions of averages an jumps;.5a 1 [g ] = {g} [g] wit = or ξ,.5b [U V] + {V} [U] τ {U} [V] τ =0,.5c [U V] +V + [U] τ U [V] τ =0, [U V] +V [U] τ U + [V] τ =0. We en tis subsection by summarizing some stanar approimation properties of te above iscrete spaces, as well as some inverse inequalities [16]. For any nonnegative integer m, letπ m be te L projection onto G m, an let Πm be te L projection onto U m,ten Lemma.1 approimation properties. Tere eists a constant C>0, suc tat for any g H m+1 Ω an U [H m+1 Ω ], te following ol: g Π m g 0,K + 1/ K g Πm g 0, K C m+1 K g m+1,k K T, U Π m U 0,K + 1/ K U Π m U 0, K C m+1 K U m+1,k K T U Π m U 0,,K C m+1 K U m+1,,k K T were te constant C is inepenent of te mes sizes K an K,butepenson m an te sape regularity parameters σ an σ ξ of te mes. Lemma. inverse inequality. Tere eists a constant C>0, suc tat for any g P m K or P m K P m K ξ wit K =K K ξ T, an for any U [P m K ], te following ol: g 0,K C 1 K g 0,K, ξ g 0,K C 1 K ξ g 0,K, U 0,,K C / K U 0,K, U 0, K C 1/ K U 0,K, were te constant C is inepenent of te mes sizes K, Kξ,butepensonm an te sape regularity parameters σ an σ ξ of te mes.

6 10 Y. CHENG, I. M. GAMBA, F. LI, AND P. J. MORRISON.. Te semiiscrete DG metos. On te PDE level, te two equations in 1.1c involving te ivergence of te magnetic an electric fiels can be erive from te remaining part of te VM system; terefore, te numerical metos propose in tis section are formulate for te VM system witout 1.1c. We want to stress tat even toug, in principle, te initial satisfaction of tese constraints is sufficient for teir satisfaction for all time, in certain circumstance one may nee to consier eplicitly suc ivergence conitions in orer to prouce pysically relevant numerical simulations [48, 4]. Given k, r 0, te semiiscrete DG metos for te VM system are efine by te following proceure: for any K = K K ξ T,lookforf G k, E, B U r, suc tat for any g G k, U, V Ur, t f gξ f ξ gξ f E + ξ B ξ gξ.6a K K + + K ξ K K K ξ K f ξ n gs ξ f E + ξ B n ξ gs ξ =0,.6b t E U = B U + K K K.6c t B V = E V K K K wit.7 J,t= f,ξ,tξξ. T ξ n B Us J U, K n E Vs, Here n an n ξ are outwar unit normals of K an K ξ, respectively. All at functions are numerical flues tat are etermine by upwining, i.e., f ξ n := f ξ n = {f ξ} + ξ n.8a [f ] n, f E + ξ B n ξ := f E + ξ B n ξ = {f E + ξ B } ξ + E + ξ B n ξ.8b [f ] ξ n ξ, n E :=n Ẽ = n {E } + 1.8c [B ] τ,.8 n B :=n B = n {B } 1 [E ] τ, were tese relations efine te meaning of tile. For te Mawell part, we also consier two oter numerical flues: central flu an alternating flu, wic are efine by.9a.9b Central flu: Ẽ = {E }, B = {B }, Alternating flu: Ẽ = E +, B = B, or Ẽ = E, B = B +.

7 DG METHODS FOR VM EQUATIONS 103 Upon summing up.6a wit respect to K T an similarly summing.6b an.6c wit respect to K T, te numerical meto becomes te following: look for f G k, E, B U r, suc tat.10a.10b a f, E, B ; g =0, b E, B ; U, V =l J ; U, for any g G k, U, V Ur,were a f, E, B ; g =a,1 f ; g+a, f, E, B ; g, l J ; U = b E, B ; U, V = + t E U t B V + B U an a,1 f ; g = t f gξ T f ξ gξ + T a, f, E, B ; g = f E + ξ B ξ gξ T + f E + ξ B [g] ξ s ξ. E ξ E J U, B [U] τ s E V + Ẽ [V] τ s, E T ξ E f ξ [g] s ξ, Note, a is linear wit respect to f an g, yet it is, in general, nonlinear wit respect to E an B ue to.8b. Recall, te eact solution f as compact support in ξ; terefore, te numerical flues of.8a.8 an.9a an.9b are consistent an, consequently, so is te propose numerical meto. Tat is, te eact solution f,e, B satisfies a f,e, B; g =0 g G, k b E, B; U, V =l J; U U, V U. r.3. Temporal iscretizations. We use total variation iminising TVD ig-orer Runge Kutta metos to solve te meto of lines ODE resulting from te semiiscrete DG sceme, t G = RG. Suc time stepping metos are conve combinations of te Euler forwar time iscretization. Te commonly use tirorer TVD Runge Kutta meto is given by G 1 G = Gn + trg n, = 3 4 Gn G trg1,.11 G n+1 = 1 3 Gn + 3 G + 3 trg, were G n represents a numerical approimation of te solution at iscrete time t n.a etaile escription of te TVD Runge Kutta meto can be foun in [55]; see also [33] an [34] for strong-stability-perserving metos.

8 104 Y. CHENG, I. M. GAMBA, F. LI, AND P. J. MORRISON 3. Conservation an stability. In tis section, we will establis conservation an stability properties of te semiiscrete DG metos. In particular, we prove tat, subject to bounary effects, te total carge mass is always conserve. As for te total energy of te system, conservation epens on te coice of numerical flues for te Mawell equations. We also sow tat f is L stable, wic facilitates te error analysis of section 4. Lemma 3.1 mass conservation. Te numerical solution f G k wit k 0 satisfies 3.1 f ξ +Θ,1 t =0, t T were Θ,1 t = f mae + ξ B n ξ, 0s ξ. E b ξ Equivalently, wit ρ,t= T f ξ,ξ,tξ, for any T>0, te following ols: T 3. ρ,t + Θ,1 tt = ρ, 0. T 0 T Proof. Let g,ξ = 1. Note tat g G k, for any k 0, is continuous an g = 0. Taking tis g as te test function in.10a, one as t f ξ + T E b ξ f E + ξ B [g] ξ s ξ =0. Wit te numerical flu of.8b an te average an jump across Eξ b of.4, te secon term above becomes 3.3 f E + ξ B n ξ s ξ 3.4 = E b ξ E b ξ f E + ξ B n ξ + E + ξ B n ξ s ξ =Θ,1 t, an tis gives 3.1. Integrating in time from 0 to T gives 3.. Lemma 3. energy conservation 1. For k, r 0, te numerical solution f G k, E, B U r wit te upwin numerical flues.8a.8 satisfies 3.5 f ξ ξ + E + B +Θ, t+θ,3 t =0, t T wit Θ, t = [E ] τ + [B ] τ s, E Θ,3 t = f ξ mae + ξ B n ξ, 0s ξ. T Eξ b

9 DG METHODS FOR VM EQUATIONS 105 Proof. Step 1. Let g,ξ= ξ.notetatg G k for k an it is continuous. In aition, g =0, ξ g =ξ, anξ U ξ g = 0 for any function U. Taking tis g as te test function in.10a, one as f ξ ξ = f E ξξ t T T = E f ξξ = = T ξ E b ξ f E + ξ B [ ξ ] ξ s ξ 1 T Eξ b E + ξ B f + E + ξ B n ξ f n ξ ξ n ξ s ξ f E J E + ξ B n ξ + E + ξ B n ξ ξ s ξ E J E b ξ E b ξ f ξ mae + ξ B n ξ, 0s ξ. Step. Wit U = E an V = B,.10b becomes T J E = 1 t + 1 t = 1 t = 1 t E B + B E E B + E E B [E ] τ s Ẽ [B ] τ s E + B [E B ] + B [E ] τ Ẽ [B ] τ s, E E + B + 1 [E ] τ + [B ] τ s. E Te last equality uses te formulas of te upwin flues.8c.8 as well as.5b. Combining te results from te previous two steps, one conclues 3.5. Corollary 3.3 energy conservation. For k, r 0, antenumerical solution f G k, E, B U r wit te upwin numerical flu.8a.8b for te Vlasov part, an wit eiter te central or alternating flu of.9a.9b for te Mawell part, te following ols: f ξ ξ + E + B +Θ,3 t =0. t T T Proof. Te proof procees te same way as for Lemma 3.. Te only ifference is tat ere te equalities.5b.5c give [E B ] + B [E ] τ Ẽ [B ] τ =0,

10 106 Y. CHENG, I. M. GAMBA, F. LI, AND P. J. MORRISON wic ols for Ẽ an B efine in te central or alternating flu in te Mawell solver. Wit eiter te central or alternating flu for te Mawell solver, te energy oes not cange ue to te tangential jump of te magnetic an electric fiels as in Lemma 3.. Tis, on te oter an, may ave some effect on te accuracy of te metos see sections 4 an 5 an also [1]. Remark 3.4. In Lemma 3., te conservation error term Θ, satisfies Θ, 0, were equality ols wen special numerical flues are taken in te Mawell solver. In aition, bot te term Θ,1 in Lemma 3.1 an te term Θ,3 in Lemmas 3. an Corollary 3.3 epen on te compute f on te outflow portion of te computational bounary in ξ-space wit perioic bounary conitions, wic is etermine by te magnitue of te numerical electric an magnetic fiels. Wen te computational omain in ξ irection is not cosen large enoug, te effects of Θ,1 an Θ,3 to te conservation properties may be visible; see, e.g., Figure S1 in te supplementary material [14] of te present paper. Remark 3.5. Te propose scemes o not guarantee tat f 0; terefore Θ,1 in Lemma 3.1 an Θ,3 in Lemmas 3. an Corollary 3.3 may not necessarily be nonnegative. It is not ar to esign a positivity preserving limiter to ensure tat f 0 by following te iea evelope in [63]; owever, it is nontrivial to esign a strategy to ensure bot f being nonnegative an te energy conservation. Remark 3.6. Energy conservation ols as long as ξ G k. Inee, for k<, te energy conservation results of Lemma 3. an Corollary 3.3 can be obtaine if one replaces G k wit G k = Gk { ξ } = {g + c ξ g G k, c R}. Finally, we can obtain te L -stability result for f, a result tat is inepenent of coice of numerical flu in te Mawell solver. Tis result will be use in te error analysis of section 4. Lemma 3.7 L -stability of f. satisfies For k 0, te numerical solution f G k 3.6 f ξ + t T + T ξ ξ n [f ] s ξ E E ξ E + ξ B n ξ [f ] ξ s ξ =0. Proof. Taking g = f in.10a, one gets f ξ + R 1 + R =0, t T wit R 1 = T f ξ f ξ+ T ξ E f ξ [f ] s ξ an R = a, f, E, B ; f.

11 DG METHODS FOR VM EQUATIONS 107 Observe f R 1 = ξ ξ + f ξ [f ] s ξ T ξ K T K T ξ E f = ξ n s ξ + f ξ [f ] s ξ T ξ K T K T ξ E 1 = T ξ E [ξf ] s ξ + f ξ [f ] s ξ T ξ E = 1 [ξf ] + {f ξ} [f ] + 1 ξ n [f ] [f ] s ξ T ξ E = 1 [f ] + {f } [f ] ξ + 1 T ξ E ξ n [f ] s ξ = 1 ξ n [f ] s ξ, T ξ E were te fourt equality uses te efinition of te numerical flu.8a an te last one is ue to.5a. Similarly, f R = E + ξ B ξ ξ T K ξ T ξ K ξ + f E + ξ B [f ] ξ s ξ T E ξ = 1 [E + ξ B f] ξ + {f E + ξ B } ξ E ξ [f ] ξ + 1 E + ξ B n ξ [f ] ξ [f ] ξ s ξ = 1 [f ] ξ + {f } ξ [f ] ξ E ξ E + ξ B + 1 E + ξ B n ξ [f ] ξ s ξ = 1 E + ξ B n ξ [f ] ξ s ξ, T E ξ were te secon equality is ue to ξ E + ξ B = 0 an te efinition of te numerical flu in.8b, an te tir equality uses.5a an E + ξ B being continuous in ξ. Wit 3.7, we conclue L stability Error estimates. In tis section, we establis error estimates at any given time T>0for our semiiscrete DG metos escribe in section.. It is assume tat te iscrete spaces ave te same egree, i.e., k = r, an tat te eact solution satisfies f C 1 [0,T]; H k+1 Ω W 1, Ω an E, B C 0 [0,T]; [H k+1 Ω ] [W 1, Ω ]. Also, perioic bounary conitions in an compact support for f in ξ are assume. To prevent te proliferation of constants, we use A B to represent te inequality A constantb, were te positive constant is inepenent of te mes size,,an ξ, but it can epen on te polynomial egree k, mes parameters σ 0,σ an σ ξ, an omain parameters L an L ξ.

12 108 Y. CHENG, I. M. GAMBA, F. LI, AND P. J. MORRISON Defining ζ =Π k f f an ε =Π k f f, it follows tat f f = ε ζ. Analogously, if ζ E = Π k E E, ζ B = Π k B B, ε E = Πk E E,anε B = Πk B B, ten E E = ε E ζe an B B = ε B ζb. Wit te approimation results of Lemma.1, we ave 4.1 ζ 0,Ω k+1 f k+1,ω, ζ B 0,Ω k+1 B k+1,ω, ζ E 0,Ω k+1 E k+1,ω ; terefore, we nee only estimate ε, ε E,anεB. Te remainer of tis section is organize as follows: We first state Lemmas 4.1 an 4., wit wic te main error estimate is establise in Teorem 4.3 for te propose semiiscrete DG meto wit te upwin numerical flues. Ten, te proofs of Lemmas 4.1 an 4. will be given in subsections 4.1 an 4.. Lastly, for te propose meto using te central or alternating flu of.9a.9b for te Mawell solver, error estimates are given in Teorem 4.6. Lemma 4.1 estimate of ε. Base on te semiiscrete DG iscretization for te Vlasov equation of.10a wit te upwin flu.8a.8b, we ave 4. wit t ε ξ + ξ n [ε ] s ξ T T ξ E + E + ξ B n ξ [ε ] ξ s ξ E ξ k+1 ˆΛ+ k f k+1,ω ε E 0,,Ω + ε B 0,,Ω + f 1,,Ω ε E 0,Ω + ε B 0,Ω ε 0,Ω + k+ 1 f k+1,ω ε B 1/ 0,,Ω + ε E 1/ 0,,Ω + B 1/ 0,,Ω + E 1/ 1/ E + ξ B n ξ [ε ] s ξ T E ξ 1/ ξ n [ε ] s ξ, E + k+ 1 f k+1,ω T ξ ˆΛ = t f k+1,ω +1+ E 1,,Ω + B 1,,Ω f k+1,ω + E k+1,ω + B k+1,ω f 1,,Ω. 0,,Ω Lemma 4. estimate of ε E an εb. Base on te semiiscrete DG iscretization for te Mawell equations of.10b wit te upwin flu.8c.8, we ave ε E t + ε B + [ε E 4.3 ] τ + [ε B ] τ s T E ε 0,Ω + k+1 f k+1,ω ε E 0,Ω 1/ + k+ 1 E k+1,ω + B k+1,ω [ε E ] τ + [ε B ] τ s. E

13 DG METHODS FOR VM EQUATIONS 109 Teorem 4.3 error estimate 1. For k, te semiiscrete DG meto of.10a.10b for te Vlasov Mawell equations, wit te upwin flues of.8a.8 in bot te Vlasolv an Mawell solvers, as te following error estimate: 4.4 f f t 0,Ω + E E t 0,Ω + B B t 0,Ω C k+1 t [0,T]. Here te constant C epens on te upper bouns of t f k+1,ω, f k+1,ω, f 1,,Ω, E 1,,Ω, B 1,,Ω, E k+1,ω, B k+1,ω over te time interval [0,T], anit also epens on te polynomial egree k, mes parameters σ 0,σ,anσ ξ, an omain parameters L an L ξ. Proof. Wit several applications of Caucy Scwarz inequality an Λ = 1/ ˆΛ+ f k+1,ω 1+ E 1/ 0,,Ω + B 1/ 0,,Ω, 4. becomes ε ξ t T c k+1 Λ + k f k+1,ω ε E 0,,Ω + ε B 0,,Ω + f 1,,Ω ε E 0,Ω + ε B 0,Ω + k+1 f k+1,ω εe 0,,Ω + ε B 0,,Ω + ε 0,Ω c k+1 Λ + k 1 + f k+1,ω ε E 0,,Ω + ε B 0,,Ω + f 1,,Ω ε E 0,Ω + ε B 0,Ω + ε 0,Ω. Here an below, te constant c > 0 only epens on k, mes parameters σ 0,σ, an σ ξ, an omain parameters L an L ξ. Moreover, wit te inverse inequality of Lemma., an ξ,min being uniformly boune by σ 0 wen te mes is refine, we ave 4.5 k ε E 0,,Ω + ε B 0,,Ω c k ε E 0,Ω + ε B 0,Ω an tis leas to 4.6 ε ξ t T c k+1 Λ + k 1 + f k+1,ω + f 1,,Ω εe 0,Ω + ε B 0,Ω + ε 0,Ω. Recall =3,tenfork, tere is k 0 an, terefore, k <. Similarly, wit te Caucy Scwarz inequality, 4.3 becomes 4.7 t ε E + ε B c ε 0,Ω + k+ f k+1,ω + k+1 E k+1,ω + B k+1,ω + ε E 0,Ω.

14 1030 Y. CHENG, I. M. GAMBA, F. LI, AND P. J. MORRISON Now, summing up 4.6 an 4.7, we get ε ξ + ε E t + ε B T T Λ k+1 +Θ ε ξ + ε E + ε B. T Here Λ epens on f,e, B in teir Sobolev norms t f k+1,ω, f k+1,ω, f 1,,Ω, E 1,,Ω, B 1,,Ω, E k+1,ω, B k+1,ω at time t, an Θ epens on f k+1,ω an f 1,,Ω at time t. Bot Λ an Θ epen on te polynomial egree k, mes parameters σ 0,σ,anσ ξ, an omain parameters L an L ξ. Now wit a stanar application of Gronwall s inequality, a triangle inequality, an te approimation results of 4.1, we conclue te error estimate 4.4. Remark 4.4. Teorem 4.3 sows tat te propose metos are k + 1 t orer accurate, wic is stanar for upwin DG metos applie to yperbolic problems on general meses. Te assumption on te polynomial egree k is ue to te lack of te L error estimate for te DG solutions to te Mawell solver an te use of an inverse inequality in anling te nonlinear coupling see in te proof of Teorem 4.3. If te computational omain in is one- or two-imensional =1 or, ten Teorem 4.3 ols for k 1. If te upwin numerical flu for te Mawell solver.10b is replace by eiter te central or alternating flu.9a.9b, we will ave te estimates for ε E an εb in Lemma 4.5 instea, provie an aitional assumption is mae for te mes wen it is refine. Tat is, we nee to assume tere is a positive constant δ<1 suc tat for any K T, 4.8 δ K 1 K δ, were K is any element in T satisfying K K. Tis assumption is also important in te presence of anging noes in te mes. Lemma 4.5 estimate of ε E an εb wit te nonupwining flu. Base on te semiiscrete DG iscretization for te Mawell equations of.10b, wit eiter te central or alternating flu of.9a.9b, we ave 4.9 t ε E + ε B ε 0,Ω + k+1 f k+1,ω ε E 0,Ω + cδ k E k+1,ω + B k+1,ω ε E + ε B 1/. Te proof of tis Lemma is given in subsection 4.3. Wit Lemma 4.5 an a proof similar to tat of Teorem 4.3, te following error estimates can be establise, but te proof is omitte. Teorem 4.6 error estimate. For k, te semiiscrete DG meto of.10a.10b for Vlasov Mawell equations, wit te upwin numerical flu.8a.8b in te Vlasov solver an eiter te central or alternating flu of.9a.9b in te Mawell solver, as te following error estimate: 4.10 f f t 0,Ω + E E t 0,Ω + B B t 0,Ω C k t [0,T].

15 DG METHODS FOR VM EQUATIONS 1031 Besies te epenence as in Teorem 4.3, te constant C also epens on δ of 4.8. Teorem 4.6 inicates tat wit eiter te central or alternating numerical flu for te Mawell solver, te propose meto will be kt orer accurate. Also, one can easily see tat te accuracy can be improve to k + 1 t orer as in Teorem 4.3 if te iscrete space for Mawell solver is one egree iger tan tat for te Vlasov equation, namely, r = k + 1. Tis improvement will require iger regularity for te eact solution E an B. In [], optimal error estimates were establise for some DG metos solving te multiimensional Vlasov Poisson problem on Cartesian meses wit tensor-structure iscrete space, efine in.3, an k 1. Some of te tecniques in [] are use in our analysis. In te present work, we focus on te P-type space G k in.1a in te numerical section, as it reners better cost efficiency an can be use on more general meses. Our analysis is establise only for k ue to te lack of te L error estimate of te DG solver for te Mawell part wic is of yperbolic nature, as pointe out in Remark 4.4. In te net tree subsections, we will provie te proofs of Lemmas 4.1, 4., an Proof of Lemma 4.1. Since te propose meto is consistent, te error equation is relate to te Vlasov solver, 4.11 a f,e, B; g a f, E, B ; g =0 g G k. Note, ε G k ; by taking g = ε in 4.11, one as 4.1 a ε, E, B ; ε =a Π k f,e, B ; ε a f,e, B; ε. Following te same lines as in te proof of Lemma 3.7, we get a ε, E, B ; ε = 1 ε ξ ξ n [ε ] s ξ t T T ξ E + 1 E + ξ B n ξ [ε ] ξ s ξ. T E ξ Net we will estimate te remaining terms in 4.1. Note were a Π k f,e, B ; ε a f,e, B; ε =T 1 + T, T 1 = a,1 Π k f; ε a,1 f; ε =a,1 ζ ; ε, T = a, Π k f,e, B ; ε a, f,e, B; ε. Step 1. EstimateofT 1. We start wit T 1 = t ζ ε ξ ζ ξ ε ξ+ T T T ξ E ζ ξ [ε ] s ξ = T 11 +T 1 +T 13. It is easy to verify tat t Π k =Π k t, an, terefore, t ζ =Π k t f t f. Wit te approimation result of Lemma.1, we ave 4.14 T 11 = t ζ ε ξ tζ 0,Ω ε 0,Ω k+1 t f k+1,ω ε 0,Ω. T

16 103 Y. CHENG, I. M. GAMBA, F. LI, AND P. J. MORRISON Net, let ξ 0 be te L projection of te function ξ onto te piecewise constant space wit respect to T ξ,ten 4.15 T 1 = ζ ξ ξ 0 ε ξ ζ ξ 0 ε ξ. T T Since ξ 0 ε G k an ζ =Π k f f wit Π k being te L projection onto G k,te secon term in 4.15 vanises. Hence T 1 ζ ξ ξ 0 ε ξ T ξ ξ 0 0,,Ωξ 1 K ζ 0,K K ε 0,K K K ξ =K T ξ ξ 0 0,,Ωξ 1 K f k+1,k ε 0,K k+1 K K K ξ =K T ξ ξ 1,,Ωξ k f k+1,ω ε 0,Ω 4.16 k+1 f k+1,ω ε 0,Ω. Te tir inequality above uses te approimating result of Lemma.1 an te inverse inequality of Lemma.. Te fourt inequality uses an approimation result similar to te last one of Lemma.1, an ξ,min being uniformly boune by σ 0 wen te mes is refine. Net, T 13 = {ζ } ξ + ξ n [ζ ] [ε ] s ξ T ξ E = {ζ } ξ ˆn ˆn + ξ n [ζ ] [ε ] s ξ, T ξ E were ˆn is te unit normal vector of an ege in E wit eiter orientation; tat is, ˆn = n,or n. Ten, T 13 ξ n 4.17 T ξ E {ζ } + [ζ ] [ε ] s ξ 1/ {ζ } + [ζ ] ξ n s ξ T ξ E 1/ ξ n [ε ] s ξ T ξ E 1/ 1/ = ξ n {ζ} s ξ ξ n [ε ] s ξ T ξ E T ξ E 1/ ξ 1/ 0,,Ω ξ ζ s ξ ξ n [ε ] s ξ K E T ξ K T k+ 1 f k+1,ω T ξ T ξ 1/ ξ n [ε ] s ξ. E 1/

17 DG METHODS FOR VM EQUATIONS 1033 Te approimation results of Lemma.1 are use for te last inequality. Step. EstimateofT. Note, T = a, Π k f,e, B ; ε a, f,e, B; ε = a, ζ, E, B ; ε +a, f,e, B ; ε a, f,e, B; ε =T 1 + T + T 3, wit T 1 = ζ E + ξ B ξ ε ξ, T T = E ξ ζ E + ξ B [ε ] ξ s ξ, T 3 = a, f,e, B ; ε a, f,e, B; ε. For T 1, we procee as for te estimate of T 1. Let E 0 = Π 0 E, letb 0 = Π 0 B be te L projection of E, B, respectively, onto te piecewise constant vector space wit respect to T,ten ζ E + ξ B ξ ε ξ = ζ E E 0 + ξ B B 0 ξ ε ξ T T + ζ E 0 + ξ B 0 ξ ε ξ, T an te secon term above vanises ue to E 0 + ξ B 0 ξ ε G k, an terefore ζ E + ξ B ξ ε ξ ζ E E 0 + ξ B B 0 ξ ε ξ, T T E E 0 + ξ B B 0 0,,Ω E E 0 0,,Ω + B B 0 0,,Ω k f k+1,ω ε E 0,,Ω + ε B 0,,Ω 1 K ξ ζ 0,K Kξ ξ ε 0,K, K K ξ =K T k+1 K K K ξ =K T + Π k E E 0 0,,Ω + Π k B B 0 0,,Ω ε 0,Ω. 1 K ξ f k+1,k ε 0,K, Note tat Π k E E 0 = Π k E E 0, an Π k is boune in any Lp -norm 1 p [, ], ten Π k E E 0 0,,Ω E E 0 0,,Ω E 1,,Ω, an similarly Π k B B 0 0,,Ω B 1,,Ω. Hence 4.18 ζ E + ξ B ξ ε ξ T k f k+1,ω ε E 0,,Ω + ε B 0,,Ω + E 1,,Ω + B 1,,Ω ε 0,Ω. For T, we follow te estimate of T 13.NotetatE an B only epens on, an ξ is continuous,

18 1034 Y. CHENG, I. M. GAMBA, F. LI, AND P. J. MORRISON ζ E + ξ B [ε ] ξ s ξ T E ξ = {ζ E + ξ B } ξ + E + ξ B n ξ [ζ ] ξ [ε ] ξ s ξ T E ξ = {ζ } ξ E + ξ B ˆn ξ ˆn ξ + E + ξ B n ξ [ζ ] ξ T E ξ [ε ] ξ s ξ ˆn ξ = n ξ or n ξ E + ξ B n ξ {ζ } ξ + [ζ ] ξ T E ξ [ε ] ξ s ξ, 1/ E + ξ B n ξ {ζ} s ξ T E ξ 1/ E + ξ B n ξ [ε ] s ξ. E ξ In aition, E ξ E + ξ B n ξ {ζ } s ξ E + ξ B 1/ 0,,Ω E 1/ 0,,Ω + B 1/ 0,,Ω 1/ 1/ {ζ } s ξ E ξ ζs ξ K T ξ K T k+ 1 f k+1,ω ε E 1/ 0,,Ω + ε B 1/ 0,,Ω + E 1/ 0,,Ω + B 1/ 0,,Ω, an terefore 4.19 T k+ 1 f k+1,ω ε E 1/ 0,,Ω + ε B 1/ 0,,Ω + E 1/ 0,,Ω + B 1/ 1/ E + ξ B n ξ [ε ] s ξ. E ξ 1/ 0,,Ω Finally, we estimate T 3.Sincef is continuous in ξ, an ξ E E + ξ B B = 0, T 3 = a, f,e, B ; ε a, f,e, B; ε = fe E + ξ B B ξ ε ξ T + fe E + ξ B B [ε ] ξ s ξ T E ξ = ξ f E E + ξ B Bε ξ; T

19 DG METHODS FOR VM EQUATIONS 1035 terefore, T 3 E E + ξ B B 0,Ω f 1,,Ω ε 0,Ω E E 0,Ω + B B 0,Ω f 1,,Ω ε 0,Ω 4.0 ε E 0,Ω + ε B 0,Ω + ζ E 0,Ω + ζ B 0,Ω f 1,,Ω ε 0,Ω ε E 0,Ω + ε B 0,Ω + k+1 E k+1,ω + B k+1,ω f 1,,Ω ε 0,Ω. Now we combine te estimates of 4.14 an , an get te result of Lemma Proof of Lemma 4.. Since te propose meto is consistent, te error equation is relate to te Mawell solver, 4.1 b E E, B B ; U, V =l J J, U U, V U k. Taking te test functions in 4.1 to be U = ε E an V = εb gives 4. b ε E, εb ; εe, εb =b ζ E, ζb ; εe, εb +l J J, ε E. Following te same lines of step in te proof of Lemma 3., 4.3 b ε E, ε B ; ε E, ε B = 1 ε E t + ε B + 1 [ε E ] τ + [ε B ] τ s. E It remains to estimate te two terms on te rigt sie of 4., 4.4 b ζ E, ζb ; εe, εb = t ζ E εe ζ B εe ζb [ε T T E E ] τ s + t ζ B εb + ζ E εb + ζe [ε T T E B ] τ s = ζb [ε E ] τ s + ζe [ε B ] τ s E E ζ 1/ 1/ B + ζ E s [ε E ] τ + [ε B ] τ s E E 1/ ζ E 0, K + ζ B 0, K [ε E ] τ + [ε B ] τ s K T E 1/ k+ 1 E k+1,ω + B k+1,ω [ε E ] τ + [ε B ] τ s. E

20 1036 Y. CHENG, I. M. GAMBA, F. LI, AND P. J. MORRISON All of te volume integrals of 4.4 vanis ue to t Π k = Πk t an ε E, εb, ε E, εb Uk. An, for te last two inequalities, te efinition of te numerical flues are use togeter wit te approimation results of Lemma.1. Finally, l J J ; ε E = J J ε E J J 0,Ω ε E 0,Ω = f f ξξ ε E T ξ 0,Ω 0,Ω f f 0,Ω ξ 0,Ωξ ε E 0,Ω 4.5 ε 0,Ω + ζ 0,Ω ε E 0,Ω ε 0,Ω + k+1 f k+1,ω ε E 0,Ω. Combining , we conclue Lemma Proof of Lemma 4.5. Te proof procees in a manner similar to tat of Lemma 4. of subsection 4.. Base on te error equation 4.1, relate to te Mawell solver wit some specific test functions, we get 4.. Wit eiter te central or alternating flu of.9a.9b, we ave b ε E, εb ; εe, εb = 1 t ε E + ε B. Te same estimate as tat of 4.5 can be obtaine for te secon term on te rigt of 4.. To estimate te first one, b ζ E, ζb ; εe, εb = t ζ E εe t ζ B εb + ζ B εe E ζb [ε E ] τ s ζ E εb + E ζe [ε B ] τ s = ζb [ε E ] τ s + ζe [ε B ] τ s E E 4.7 1/ 1 K ζ 1/ B + ζ E s K [ε E ] τ + [ε B ] τ s e E 4.8 e E e e

21 4.9 cδ K DG METHODS FOR VM EQUATIONS 1037 K 1 K ζ B + ζ E s 1/ 1/ K ε E + ε B s K T K cδ k K E k+1,k + B k+1,k K ε E 0,K + ε B 0,K 1/ 1/ K cδ k E k+1,ω + B k+1,ω ε E + ε B 1/. As before, all volume integrals of 4.6 vanis ue to t Π k = Π k t an ε E, εb, ε E, εb Uk. In 4.7, K is any element containing an ege e. To get 4.9, we use te efinitions of te numerical flues, jumps, as well as te assumption 4.8 on te ratio of te neigboring mes elements. Here cδ is a positive constant epening on δ. We obtain 5.1 by applying an approimation result of Lemma.1 an an inverse inequality of Lemma.. From all te above, we conclue Lemma Numerical results. In tis section, we perform a etaile numerical stuy of te propose sceme in te contet of te streaming Weibel SW instability first analyze in [50]. Te SW instability is closely relate to te Weibel instability of [59], but erives its free energy from transverse counterstreaming as oppose to temperature anisotropy. Te SW instability an its Weibel counterpart ave been consiere bot analytically an numerically in several papers e.g., [50, 10, 9, 8, 49] ere we focus on comparison wit te numerical results of Califano et al. in [10]. We consier a reuce version of te VM equations wit one spatial variable,, an two velocity variables, ξ 1 an ξ. Te epenent variables uner consieration are te istribution function f,ξ 1,ξ,t, a two-imensional D electric fiel E = E 1,t,E,t, 0, an a one-imensional 1D magnetic fiel B =0, 0,B 3,t, an te reuce VM system is were 5.3 j 1 = B 3 t f t + ξ f +E 1 + ξ B 3 f ξ1 +E ξ 1 B 3 f ξ =0, = E 1, E 1 t = B 3 j 1, f,ξ 1,ξ,tξ 1 ξ 1 ξ, j = Te initial conitions are given by E t = j, f,ξ 1,ξ,tξ ξ 1 ξ. f,ξ 1,ξ, 0 = 1 πβ e ξ /β [δe ξ1 v0,1 /β +1 δe ξ1+v0, /β ], E 1,ξ 1,ξ, 0 = E,ξ 1,ξ, 0 = 0, B 3,ξ 1,ξ, 0 = b sink 0,

22 1038 Y. CHENG, I. M. GAMBA, F. LI, AND P. J. MORRISON wic for b = 0 is an equilibrium state compose of counterstreaming beams propagating perpenicular to te irection of inomogeneity. Following [10], we trigger te instability by taking β = 0.01, b = te amplitue of te initial perturbation to te magnetic fiel. Here, Ω =[0,L y ], were L y =π/k 0,anweset Ω ξ =[ 1., 1.]. Two ifferent sets of parameters will be consiere, coice1: δ =0.5,v 0,1 = v 0, =0.3,k 0 =0., coice: δ =1/6,v 0,1 =0.5,v 0, =0.1,k 0 =0.. For comparison, tese are cosen to correspon to runs of [10]. Table 5.1 Upwin flu for Mawell s equations, L errors, an orers wit N uniform cells in eac of te, ξ 1, ξ irections. Run to T =5an back to T =10. Space G 1, U 1 G, U G 3, U 3 N=0 N=40 N=80 Error Error Orer Error Orer f 0.18E E E B 3 0.6E E E E 1 0.1E E E E 0.10E-05 0.E E f 0.56E E E-0.9 B 3 0.3E E E E E E E E 0.16E-06 0.E E f 0.1E E E B E E E E E E E E 0.14E E E Accuracy test. Te VM system is time reversible, an tis provies a way to test te accuracy of our sceme. In particular, let f,ξ,0, E, 0, B, 0 enote te initial conitions for te VM system an f,ξ,t, E,T, B,Ttesolution at t = T.Ifwecoosef, ξ,t, E,T, B,T as te initial conition at t =0, ten at t = T we teoretically must recover f, ξ,0, E, 0, B, 0. In Tables 5.1 an 5., we sow te L errors an orers of te numerical solutions wit tree flu coices for te Mawell s equations: te upwin flu, te central flu, an one of te alternating flues Ẽ = E + an B = B. Te parameters are tose of coice 1, wit symmetric counterstreaming. In te numerical simulations, uniform meses are use, wit N cells in eac of, ξ 1,anξ irections. In aition, te tir orer TVD Runge Kutta meto is applie in time, wit te CFL number C cfl =0.19 for te upwin an central flues, an C cfl =0.1 for te alternating flu in P 1 an P cases. For P 3,wetake t = O 4/3 to ensure tat te spatial an temporal accuracy is of te same orer. From Tables 5.1 an 5., we observe tat te scemes wit te upwin an alternating flues acieve optimal k + 1t orer accuracy in approimating te solution compare to k + 1 t orer an kt orer accuracy establise in te previous section, wile for o k, te central flu gives suboptimal approimation of some of te solution components. Conservation properties. Te purpose ere is to valiate our teoretical result about conservation troug two numerical eamples, te symmetric case an te nonsymmetric case. We first use parameter coice 1 as in te Califano et al. [10], te symmetric case wit tree ifferent flues for Mawell s equations. Te results are illustrate in Figure 5.1. In all te plots, we ave rescale te macroscopic quantities

23 DG METHODS FOR VM EQUATIONS 1039 Table 5. Central an alternating flues for Mawell s equations, L errors, an orers wit N uniform cells in eac of te, ξ 1, ξ irections. Run to T =5an back to T =10. G 1 U 1 G U G 3 U 3 G 1 U 1 G U G 3 U 3 Central N=0 N=40 N=80 Error Error Orer Error Orer f 0.18E E E B E E E E E E E E 0.9E E E f 0.56E E E-0.9 B 3 0.8E E E E E E E E 0.16E-06 0.E E f 0.1E E E B E E E E E E E E 0.14E E E Alternating N=0 N=40 N=80 Error Error Orer Error Orer f 0.18E E E B 3 0.9E E E E 1 0.4E E E E 0.10E-05 0.E E f 0.56E E E-0.9 B 3 0.8E-06 0.E E E 1 0.3E E E E 0.16E-06 0.E E f 0.1E E E B E E E E E E E E 0.14E E E by te pysical omain size. For all tree flues, te mass carge is well conserve. Te largest relative error for te carge for all tree flues is smaller tan As for te total energy, we coul observe relatively larger ecay in te total energy from te simulation wit te upwin flu compare to te one wit te oter two flues. Tis is epecte from te analysis in section 3. In fact, te largest relative error for te total energy is boune by for te upwin flu, an boune by for central an alternating flues. As for momentum conservation, it is well known tat te two species VM system conserves te following epression for te total linear momentum: 5.6 P = ξf ξ + E B, were te first term represents te momentum in te particles wile te secon tat of te electromagnetic fiel. In fact, tis is true for te full energy-momentum an angular momentum tensors [51]. Eac component of te spatial integran of 5.6, te components of te momentum ensity, satisfies a conservation law, a result tat relies on bot species being ynamic an one tat relies on te constraint equations 1.1c being satisfie. However, in tis paper we ave fie te constant ion backgroun by carge neutrality an, consequently, momentum is not conserve in general. Tis lack of conservation oes not appear to be wiely known, but it is known tat te enforcement of constraints may or may not result in te loss of conservation [47]. For

24 1040 Y. CHENG, I. M. GAMBA, F. LI, AND P. J. MORRISON Upwin Alternating Central Upwin Alternating Central 0.05 m t a Mass t b Total energy 6E-10 4E-10 Upwin Alternating Central 4E-05 Upwin Alternating Central E-10 E-05 P1 0 P 0 -E-10 -E-05-4E-10-4E-05-6E t t c Momentum P 1 Momentum P Fig Streaming Weibel instability wit parameter coice 1 as in Califano et al. [10] δ = 0.5,v 0,1 = v 0, =0.3,k 0 =0., te symmetric case. Te mes is wit piecewise quaratic polynomials. Time evolution of mass, total energy, an momentum for te tree numerical flues for te Mawell s equations. eample, te single species Vlasov Poisson system wit a fie constant ion backgroun oes inee conserve momentum. However, for te SW application, it is not ifficult to sow tat te following component is conserve: 5.7 P 1 = ξ 1 fξ 1 ξ + E B 3, wile te component P is not. Since conservation of P 1 relies on te constraint equations an since our computational algoritm oes not enforce tese constraints, conservation of P 1 serves as a measure of te gooness of our meto in maintaining te initial satisfaction of te constraints. From Figure 5.1, we see tat all tree flu formulations conserve P 1 relatively well, but, as epecte, tere is a large accumulating error in P, particularly for te alternating flu case. Similarly, for a general VM system witout constraints, te following epression

25 DG METHODS FOR VM EQUATIONS E1 E E1 E a Parameter Coice 1. flu Electric fiel, upwin b Parameter Coice. flu Electric fiel, upwin 0.1 B B c Parameter Coice 1. Magnetic fiel, upwin flu Parameter Coice. Magnetic fiel, upwin flu Fig. 5.. Streaming Weibel instability. Te mes is wit piecewise quaratic polynomials. Te electric an magnetic fiels at T = 00. for te total angular momentum is conserve: 5.8 L = ξfξ + E B. However, because te SW application breaks symmetry, tere is no relevant component of te angular momentum tat is conserve for tis problem, but for a more general application one may want to track its conservation. Comparison an interpretation. In Figure 5.3, we plot te time evolution of te kinetic, electric, an magnetic energies. In particular, we plot te separate components efine by K 1 = 1 fξ 1 ξ 1 ξ, K = 1 fξ ξ 1 ξ,e 1 = 1 E 1, an E = 1 E. Figure 5.3a sows for coice 1 te transference of kinetic energy from one component to te oter wit a eficit converte into fiel energy. Tis eficit is consistent wit energy conservation, as evience by Figure 5.1. Observe te magnetic an inuctive electric fiels grow initially at a linear growt rate com-

26 104 Y. CHENG, I. M. GAMBA, F. LI, AND P. J. MORRISON Kinetic energy K1 energy K energy Electric energy Magnetic energy E1 energy E energy KE γ= t a Parameter Coice 1. Kinetic energies γ= t b Parameter Coice 1. Fiel energies Kinetic energy K1 energy K energy 10 - Electric energy Magnetic energy E1 energy E energy KE t t c Parameter Coice. Kinetic energies Parameter Coice. Fiel energies Fig Streaming Weibel instability. Te mes is wit piecewise quaratic polynomials. Time evolution of kinetic, electric, an magnetic energies by alternating flu for te Mawell s equations. parable to tat of Table I of [10]. Saturation occurs wen te electric an magnetic energies simultaneously peak at aroun t = 70 in agreement wit [10]; owever, in our case we acieve equipartition at te peak, wic may be ue to better resolution. Here we ave also sown te longituinal component E, not sown in [10], wic in Figure 5.3b is seen to grow at twice te growt rate. Tis beavior was anticipate in [11] in te contet of a two-flui moel an seen in kinetic VM computations of te usual Weibel instability [49]. It is ue to wave coupling an a moulation of te electron ensity inuce by te spatial moulation of B3. Te growt at twice te growt rate of te magnetic fiel B 3 is seen in Figure 5.3b, an te ensity moulation, incluing te epecte spikes, is seen in Figure 5.4. We ave also calculate te first four Log Fourier moes of te fiels E 1, E, B 3, an tese are sown in Figure

27 DG METHODS FOR VM EQUATIONS ρ 1 ρ a Parameter Coice 1. t =55 b Parameter Coice. t = ρ 1 ρ c Parameter Coice 1. t =8 Parameter Coice. t = ρ 1 ρ e Parameter Coice 1. t = 15 f Parameter Coice. t = 15 Fig Plots of te compute ensity function ρ for te streaming Weibel instability at selecte time t. Temesis100 3 wit piecewise quaratic polynomials. Te upwin flu is applie.

28 1044 Y. CHENG, I. M. GAMBA, F. LI, AND P. J. MORRISON logfm LogFME1 LogFME1 logfm3-10 logfm1 logfm logfm3 logfm4-10 logfm logfm t t a Parameter Coice 1. Log Fourier moes of b Parameter Coice. Log Fourier moes E1 of E1 logfm -5-5 LogFME LogFME logfm4-10 logfm3-15 logfm1 logfm logfm3 logfm logfm t t c Parameter Coice 1. Log Fourier moes of Parameter Coice. Log Fourier moes E of E logfm1 logfm3-5 LogFMB3 LogFMB logfm logfm 0 logfm1 logfm logfm3 logfm4 100 t t e Parameter Coice 1. Log Fourier moes of f Parameter Coice. Log Fourier moes of B3 B3 Fig Streaming Weibel instability. Te mes is 1003 wit piecewise quaratic polynomials. Te first four Log Fourier moes of E1, E, B3 compute by te alternating flu for te Mawell s equations.

29 DG METHODS FOR VM EQUATIONS 1045 a Parameter Coice 1. =0.05π, t =55 b Parameter Coice. =0.05π, t =55 c Parameter Coice 1. =0.05π, t =8 Parameter Coice. =0.05π, t =8 e Parameter Coice 1. =0.05π, t = 15 f Parameter Coice. =0.05π, t = 15 Fig D contour plots of te compute istribution function f for te streaming Weibel instability. Te mes is wit piecewise quaratic polynomials. Te upwin flu is applie.

30 1046 Y. CHENG, I. M. GAMBA, F. LI, AND P. J. MORRISON 5.5. Here, te nt Log Fourier moe for a function W, t [36] is efine as logfm n t = log 10 1 L L 0 W, t sinkn + L 0 W, tcoskn. In Figure 5.6 we plot te D contours of f at selecte locations an time t, wen te upwin flu is use in te Mawell solver. Te times cosen correspon to tose for te ensity of Figure 5.4, an we see tat at late times consierable fine structure is present, wic is consistent wit te Log Fourier plots. For completeness, we also inclue in Figure 5. plots of te electric an magnetic fiels at te final time obtaine by te upwin flu. For coice, wit te nonsymmetric parameter set, te results are inclue in Figures 5., 5.3, 5.4, 5.5, an 5.6, jutapose wit tose for parameter coice 1. Insofar as we can make comparison wit [10], our results are in reasonable agreement. Similar energy transfers take place, but te equipartition of te magnetic an electric energies at te peak is not acieve. All moes saturate now at nearly te same values, eviently resulting from te broken symmetry. Also, at long times, contours of te istribution function are isplaye. Here te wrapping of te istribution function as two intertwine istorte cyliners is observe as in [10], altoug for late times tere is a loss of localization. 6. Concluing remarks. In summary, we ave evelope iscontinuous Galerkin metos for solving te Vlasov Mawell system. We ave proven tat te meto is arbitrarily accurate, conserves carge, can conserve energy, an is stable. Error estimates were establise for several flu coices. Te sceme was teste on te streaming Weibel instability, were te orer of accuracy an conservation properties were verifie. In te future, we will eplore oter time stepping metos to improve te efficiency of te overall algoritm. In our evelopment, te constraint equations of 1.1c were not consiere; in te future, we plan to investigate tem togeter wit some correction tecniques for te continuity equation. Te propose meto as been clearly establise as sufficient for investigating te streaming Weibel instability, an te long time nonlinear pysics of tis system can be furter investigate an moele. In te future, we will also apply te meto to stuy oter important plasma pysics problems, especially tose of iger imension. Acknowlegments. We woul like to tank F. Pegoraro for elpful corresponence. Also, support from te Department of Matematics at Micigan State University an te Institute of Computational Engineering an Sciences at te University of Teas, Austin are gratefully acknowlege. REFERENCES [1] M. Ainswort, Dispersive an issipative beavior of ig orer iscontinuous Galerkin finite element metos, J. Comput. Pys., , pp [] B. Ayuso, J. A. Carrillo, an C.-W. Su, Discontinuous Galerkin metos for te multiimensional Vlasov-Poisson problems, Mat. Moels Metos Appl. Sci., to appear. [3] B. Ayuso, J. A. Carrillo, an C.-W. Su, Discontinuous Galerkin metos for te oneimensional Vlasov-Poisson system, Kinet. Relat. Moels, 4 011, pp [4] T. Bart, On te role of involutions in te iscontinuous Galerkin iscretization of Mawell an magnetoyroynamic systems, in IMA Vol. 4, Mat. Appl., Springer, New York, 006, pp