DISCONTINUOUS GALERKIN METHODS FOR THE ONE-DIMENSIONAL VLASOV-POISSON SYSTEM

Transcription

1 DSCONTNUOUS GALERKN METHODS FOR THE ONE-DMENSONAL VLASOV-POSSON SYSTEM BLANCA AYUSO, J. A. CARRLLO, AND CH-WANG SHU Abstract. We construct a new famly of sem-dscrete numercal scemes for te approxmaton of te one-dmensonal perodc Vlasov-Posson system. Te metods are based on te couplng of dscontnuous Galerkn approxmaton to te Vlasov equaton and several fnte element (conformng, non-conformng and mxed approxmatons for te Posson problem. We sow optmal error estmates for all te proposed metods n te case of smoot compactly supported ntal data. Te ssue of energy conservaton s also analyzed for some of te metods. Key words. conservaton Vlasov-Posson system; Dscontnuous Galerkn; mxed-fnte elements; energy AMS subject classfcatons. 65N3, 65M12, 65M15, 82D1. 1. ntroducton. Te Vlasov-Posson system s one of te basc and smplest models n te mesoscopc descrpton of large ensembles of nteractng partcles. n one-space dmenson and n dmensonless varables, te Vlasov equaton reads f t + v f x Φ f x v = (x, v, t Ω x R [,, (1.1 were te electrostatc feld, Φ x (x, t, s derved from a potental Φ(x, t tat satsfes: Φ xx = ρ(x, t 1 (x, t Ω x [,, (1.2 wt ρ(x, t beng te carge densty wc s defned by ρ(x, t = f(x, v, t dv for all (x, t Ω x [,. (1.3 R Te above system descrbes te evoluton of a collsonless plasma of carged partcles (electrons and ons n te case were te only nteracton (between partcles consdered relevant s te mean-feld force created troug electrostatc effects, ence neglectng te electromagnetc effects. f(x, v, t s te electron dstrbuton, wc s a non-negatve functon dependng on te poston: x Ω x R; te velocty: v R, and te tme: t R, wt Ω x denotng te spatal doman were te plasma s confned. As ons are muc eaver tan electrons, t s assumed tat ter dstrbuton s unform and snce te plasma sould be neutral, one as ρ(x, t dx = f(x, v, t dv dx = 1 for all t [,. (1.4 Ω x Ω x R We refer to te surveys [42, 12, 34 for good account on te state of te art n te matematcal analyss and propertes of te solutons of te Caucy problem for te Vlasov-Posson system. Departamento de Matemátcas, Unversdad Autónoma de Madrd, Madrd 2849, Span. E-mal: blanca.ayuso@uam.es. CREA and Departament de Matemàtques, Unverstat Autònoma de Barcelona, E-8193 Bellaterra, Span. E-mal: carrllo@mat.uab.es. Dvson of Appled Matematcs, Brown Unversty, Provdence, R 2912, USA. E-mal: su@dam.brown.edu. 1

2 2 B. AYUSO, J. A. CARRLLO AND C.-W. SHU Many efforts ave been dedcated to te numercal approxmaton of te Vlasov- Posson system wt eter probablstc or determnstc solvers. Snce te begnnngs of numercal plasma smulatons n te 6 s, partcle metods [11 ave been often preferred because of ter lower computatonal complexty. For tese metods, te moton of te plasma s approxmated by a fnte number of macro-partcles n te pyscal space tat follow backward te caracterstcs of te Vlasov equaton. Several works ave also analyzed ter convergence n one [3, 59 and ger dmensons [41. However, a well known drawback of tese metods s ter nerent numercal nose wc makes t dffcult to obtan an accurate descrpton of te dstrbuton functon n te pase space for many applcatons. To overcome ts lack of precson, Euleran solvers, metods dscretzng te Vlasov equaton on a mes of te pase space, ave been also consdered. Ter desgn as been explored by many autors and wt many dfferent tecnques: fnte volumes [36, 37; Fourer-Fourer transform [45; fnte elements [6, 61, splttng scemes [33, 17; and sem-lagrangan metods [39, 31, 16. All tese metods present dfferent pros and cons and we refer to [38 and te references teren for a dscusson. Fnte volumes are a smple and nexpensve opton, but n general, are low order. Fourer-Fourer transform scemes suffer from Gbbs penomena f oter tan perodc boundary condtons are mposed. Sem-lagrangan scemes can aceve g order allowng also for tme ntegraton wt larger tme steps. However, tey requre g order nterpolaton to compute te orgn of te caracterstcs, wc n turn destroys te local caracter of te reconstructon. Standard fnte element metods suffer from numercal oscllatons wen approxmatng te Vlasov equaton. n contrast, dscontnuous Galerkn (DG fnte elements are partcularly well suted for yperbolc problems and ter applcaton to non-lnear conservaton laws as already sown ter usefulness [26, 25, 28. Based on a totally dscontnuous fnte element spaces, DG metods are extremely versatle and ave numerous attractve features: local conservaton propertes; can easly andle rregularly refned meses and varable approxmaton degrees (p-adaptvty, weak approxmaton of boundary condtons and bult-n parallelsm wc permts coarse-gran parallelzaton. n addton, DG mass matrces are blockdagonal and can be nverted at a very low computatonal cost, gvng rse to very effcent tme-steppng algortms n te context of tme-dependent problems, as t s te case ere. Poneerng researc on dscontnuous Galerkn metods was pursued n [52, 48, 35, 57, 3. We refer to [24, 4 for a detaled storcal overvew and for more recent developments to [55, 5, 14, 5 and references teren. However, altoug DG metods can deal robustly wt partal dfferental equatons of almost any knd, ter applcaton n te realm of numercal approxmaton of knetc models as been consdered only very recently. n [2 and [9 te autors study, respectvely, te use of DG for te Boltzmann-Posson system n semconductors and for water-bag approxmatons of te Vlasov-Posson system. n [4, an L 1 -analyss s carred out for a smplfed lnear Vlasov-Boltzmann equaton wt a gven confnng force feld. Despte te fact tat te numercal performance of all tese Euleran solvers as been extensvely studed, to our knowledge, te ssue of ter convergence and error analyss for te Vlasov-Posson system, as not been tackled tll very recently, and only for te one-dmensonal perodc case. Te convergence and error analyss for a low order fnte volume sceme s contaned n [37. More recently, sem-lagrangan scemes ave been analyzed; of frst order n [7 and g order s consdered n [8, 1. n tese works te autors ave also proved a-pror error bounds n dfferent norms for bot te dstrbuton functon and te electrostatc feld. We also menton tat for

3 DG METHODS FOR ONE-DMENSONAL VLASOV-POSSON SYSTEM 3 oter knetc models, fnte dfferences [53 and spectral metods [47, 46 ave been also consdered and analyzed. Te present paper s concerned wt te desgn and analyss of dscontnuous Galerkn approxmaton for te one-dmensonal perodc Vlasov-Posson system. We ntroduce a wole famly of Euleran scemes, based on te combnaton of DG approxmaton to te Vlasov equaton wt varous dfferent fnte element (conformng and nonconformng approxmatons to te electrostatc feld. Te frst one s a drect conformng approxmaton obtaned by takng advantage of te explct ntegraton of te Posson equaton n one dmenson. Suc approxmaton s equvalent to wat most autors, f not all, ave usually consdered for ts system. However, n spte of ts smplcty, t mgt not be te most approprate sceme n vew of te possble extenson/adaptaton of te numercal sceme to ger dmensons and to more complex knetc models. For ts reason, n te present paper we also examne a dfferent approac: snce te couplng n te Vlasov-Posson system s troug te electrostatc feld, te man nterest n te Posson problem s te approxmaton to Φ x rater tan to Φ, and terefore mxed fnte element metods seem to be te rgt coce. We explore Ravart-Tomas and several mxed DG approxmatons for te Posson problem. We also deal wt te convergence and error analyss for te proposed DG metods for te case of smoot compactly supported solutons. We derve optmal error bounds n te L 2 -norm for bot te dstrbuton functon and te electrostatc feld, for g order metods, namely k 1, k beng te polynomal degree of te DG approxmaton for te dstrbuton functon. Te analyss for pecewse constant approxmaton (k = s dfferent and wll be carred out somewere else. Altoug Vlasov equaton mgt be seen as a smple transport equaton, ts couplng wt Posson, brngs nto play n suc equaton, a non-lnear (quadratc and non-local term. Ts generates some dffcultes n te error analyss, precludng a stragtforward extenson of oter works. A key ngredent s te constructon of some projecton operators, nspred n tose ntroduced n [48, 54, 23, 62, but specally desgned for te Vlasov-Posson system. Tese specal projectons allow for avodng te loss of alf order, typcal of te standard error analyses of fnte element metods for yperbolc problems. We ave focused on sem-dscrete scemes; dscusson on sutable tme ntegrators and desgn and analyss of fully dscretzed scemes s outsde te scope of ts paper and wll be te subject of future researc. Fnally, we ws to note tat wle developng te metods, we ave taken specal care n ensurng tat pyscal propertes of te contnuous system are preserved. Te DG approxmaton for te Vlasov equaton ensures n an easy way tat te total carge of te system s preserved (1.4. We also dscuss te conservaton of te total energy for te proposed scemes. n partcular, we propose a full DG metod (DG for te Vlasov equaton and a partcular local dscontnuous Galerkn (LDG for te Posson problem, tat preserves te total dscrete energy of te system. To te best of our knowledge ts s te frst sceme proposed n te lterature for wc energy conservaton can be sown. Our proof owever requres a tecncal assumpton on te polynomal degree for te DG metods, namely k 2. Weter ts restrcton s really necessary or not, wll be te subject of future researc. For many oter full DG scemes presented n te paper, we provde a bound on te energy dsspated by te system. Extenson to ger dmensons, numercal valdaton of te results presented ere and numercal performance of te presented numercal scemes n callengng

4 4 B. AYUSO, J. A. CARRLLO AND C.-W. SHU questons suc as te Landau dampng of Langmur waves [63 or te Raman scatterng nstablty [9 wll be carred out somewere else. Te outlne of te paper s as follows. n secton 2 we descrbe te man propertes of te contnuous problem, we ntroduce te notatons and revse some basc results we need for te descrpton and analyss of te numercal metods. n secton 3 we present te numercal metods proposed to approxmate te one dmensonal perodc Vlasov-Posson system. Te error analyss for te presented metods s detaled n secton 4. We dscuss te energy conservaton propertes of te scemes n secton 5. Te paper s completed wt two appendces, Appendx A and Appendx B, contanng some proofs of tecncal and auxlary lemmas used n te convergence analyss. 2. Prelmnares, notaton and auxlary results. Trougout ts paper, we use te standard notaton for Sobolev spaces [1. For a bounded doman B R 2, we denote by H m (B te L 2 -Sobolev space of order m and by m,b and m,b te usual Sobolev norm and semnorm, respectvely. For m =, we wrte L 2 (B nstead of H (B. We sall denote by H m (B/R te quotent space consstng of equvalence classes of elements of H m (B dfferng by constants; for m = t s denoted by L 2 (B/R. We sall ndcate by L 2 (B te space of L 2 (B functons avng zero average over B. Ts notaton wll also be used for perodc Sobolev spaces wtout any oter explct reference to perodcty to avod cumbersome notatons Contnuous problem: Te 1D perodc Vlasov-Posson system. n te rest of te paper we take Ω x = [, 1 n (1.1-(1.2-(1.3-(1.4. Let f denote a gven ntal dstrbuton f(x, v, = f (x, v n (x, v [, 1 R. We mpose perodc boundary condtons on x for te transport equaton (1.1, f(, v, t = f(1, v, t for all (v, t R [,. and also for te Posson equaton (1.2, Φ(, t = Φ(1, t and Φ x (, t = Φ x (1, t for all t [,. (2.1 Notce tat (1.4 s coerent wt te 1-perodcty of Φ x. Let us also empasze tat te correct way of ncludng perodc boundary condtons s to assume tat f and Φ are te restrcton to [, 1 of perodc functons defned n R n te rgt spaces. To guarantee te unqueness of te soluton Φ (oterwse t s determned only up to a constant, we fx te value of Φ at a pont. We set Φ(, t = for all t [,. (2.2 However, notce tat snce te Posson equaton (1.2 s one-dmensonal t could be drectly ntegrated. More precsely, by usng twce te Fundamental Teorem of Calculus, t follows tat Φ s defned for all t [, as Φ(x, t = D + C E x + x2 2 x s ρ(z, t dz ds x [, 1, (2.3 were D and C E are ntegraton constants determned from (2.2 and (2.1; D =, C E = 1 z ρ(s, t ds dz 1 2 t [, T. (2.4

5 DG METHODS FOR ONE-DMENSONAL VLASOV-POSSON SYSTEM 5 Denotng ten by E(x, t = Φ x (x, t, and dfferentatng (2.3 wt respect to x, we fnd E(x, t = Φ x (x, t = C E + x x ρ(s, t ds x [, 1, (2.5 wt C E defned n (2.4. Trougout te paper, E wll be referred to as te electrostatc feld. Altoug te pyscal one s ndeed E, we sall use ts abuse n te notaton to follow te standard notaton for te Posson solvers n te dscontnuous Galerkn communty. Observe tat (2.1 mples tat te electrostatc feld as zero average n agreement wt te carge neutralty. n order to perform our error analyss we restrct our attenton to smoot compactly supported solutons f n a fxed tme nterval [, T for all T >. Gven a dstrbuton functon f(x, v, t, we wll denote by Q(t = 1 + sup{ v : x [, 1 and τ [, t suc tat f(x, v, τ }, for all t [, as a measure of te support of te dstrbuton functon. followng result s essentally contaned n [29, 58, 42. Teorem 2.1 (Well-posedness of te contnuous 1DVP. Gven f C 1 (R x R x, 1-perodc n x and compactly supported n v, Q( Q wt Q >. Ten te perodc Vlasov-Posson system (1.1-(1.2 as a unque classcal soluton (f, E, f C 1 (, T ; C 1 (R x R v and E C 1 (, T ; C 1 (R x tat s 1-perodc n x for all tme t n [, T for all T >, suc tat: Regularty: f n addton f C m (R x R x, m 2, ten, te dstrbuton functon f belongs to C m (, T ; C m (R x R v and te force feld E C m (, T ; C m (R x. Control of Support: Tere exsts a constant C dependng on Q and f suc tat Q(T CT for all T >. n te rest of ts work, we wll assume tat te ntal data f satsfes te ypoteses n Teorem 2.1, and tus, te unque classcal soluton to te perodc Vlasov-Posson system (1.1-(1.2 satsfes tat tere exsts L > dependng on f, T and Q suc tat supp( f(t Ω for all t [, T, were we ave defned Ω = J, wt = [, 1 and J = [ L, L. Te Vlasov transport equaton (1.1 s regarded as a transport equaton n Ω T := Ω [, T. Takng nto account te boundary condtons, te weak formulaton of te contnuous problem (1.1 reads: fnd (f, E suc tat f t φ dx dv vfφ x dx dv + Ef φ v dx dv = φ C (Ω. (2.6 Ω Ω Ω t s well known [42, 12, 34 tat te contnuous soluton of (1.1-(1.2 satsfes four mportant propertes: Postvty: f(t, x, v, for all (x, v, t Ω T. Carge conservaton: as gven n (1.4. L p -conservaton: Te f(t Lp (Ω = f Lp (Ω 1 p, t [, T. (2.7 Conservaton of te total Energy: ( d v 2 f(x, v, t dx dv + dt Ω E(x, t 2 dx =. (2.8

6 6 B. AYUSO, J. A. CARRLLO AND C.-W. SHU n dervng numercal metods for (1.1-(1.2, we wll try to ensure tat te resultng scemes wll be able to produce approxmate solutons, enjoyng some of tese propertes. As usual wt g-order scemes for yperbolc problems, we cannot expect to preserve postvty of te sceme. However, we wll be able to conserve te total energy for a partcular metod, see secton Dscontnuous Galerkn approxmaton: Basc notatons. Let {T } be a famly of parttons of our computatonal/pyscal doman Ω = J = [, 1 [ L, L, wc we assume to be regular [21 and made of rectangles. Eac cartesan mes T s defned as T := {T j = J j, 1 N x, 1 j N v } were = [x 1/2, x +1/2 = 1,..., N x ; J j = [v j 1/2, v j+1/2 j = 1,..., N v, and te mes szes x and v relatve to te partton are defned as < x = max 1 N x x := x +1/2 x 1/2, < v = max 1 j N v v j := v +1/2 v 1/2, were x and v j are te cell lengts of and J j, respectvely. Te mes sze of te partton s defned as = max ( x, v. For smplcty n te exposton we also assume tat v = corresponds to a node, v j 1/2 = for some j, of te partton of [ L, L. Te set of all vertcal edges s denoted by Γ x, and respectvely, we wll refer to Γ v as te set of all orzontal edges; Γ x := {x 1/2 } J j, Γ v := {v j 1/2 }, Γ = Γ x Γ v.,j,j By { } we sall denote te famly of parttons of te nterval ; := { : 1 N x } γ x := {x 1/2 }. Next, for k, we defne te dscontnuous fnte element spaces V k and Zk and a conformng fnte element space, W k+1, V k = { ψ L 2 ( : ψ P k (, x = 1,... N x, }, Z k := { z L 2 (Ω : z Q k (T j, (x, v T j = J j,, j }, W k+1 = { χ C ( : χ P k+1 (, x = 1,... N x, } L 2 (/R, were P k ( s te space of polynomals (n one dmenson of degree up to k, and Q k (T j s te space of polynomals of degree at most k n eac varable. Trace Operators: We denote by (ϕ + +1/2,v and (ϕ +1/2,v te values of ϕ at (x +1/2, v from te rgt cell +1 J j and from te left cell J j, respectvely; (ϕ ± +1/2,v = lm ε ϕ (x +1/2 ± ε, v, for all (x, v J or n sort-and notaton (ϕ ± x,j+1/2 = lm ε ϕ (x, v j+1/2 ± ε, (ϕ ± +1/2,v = ϕ (x ± +1/2, v, (ϕ ± x,j+1/2 = ϕ (x, v ± j+1/2, (2.9 for all (x, v J j. Te jump [ and average { } trace operators of ϕ at (x +1/2, v, v J j are defned by [ ϕ +1/2,v := (ϕ + +1/2,v (ϕ +1/2,v ϕ Z k, {ϕ } +1/2,v := 1 [ (ϕ /2,v + (ϕ +1/2,v ϕ Z k. (2.1

7 DG METHODS FOR ONE-DMENSONAL VLASOV-POSSON SYSTEM Tecncal tools. We start by defnng te space H m (T := {ϕ L 2 (Ω : ϕ Tj H m (T j T j T } m. n our error analyss, snce we consder a non-conformng approxmaton, we sall employ te followng semnorm and norms, ϕ 2 1, =,j ϕ 2 1,T j ϕ 2 m,t :=,j ϕ 2 m,t j ϕ H m (T, m ϕ,,t = sup T j T ϕ,,tj ϕ p L p (T :=,j ϕ p L p (T j ϕ L p (T, for all 1 p <. We also ntroduce te followng norms over te skeleton of te fnte element partton, ϕ 2,Γ x := (ϕ +1/2,v 2 dv, ϕ 2,Γ v = (ϕ x,j+1/2 2 dx ϕ H 1 (T.,j J j,j Ten, we defne ϕ 2,Γ = ϕ 2,Γ x + ϕ 2,Γ v. We notce tat all te above defntons apply also for te partton wt te obvous canges n te notaton. Projecton operators: For k, we denote by P k : L 2 ( V k te standard L2 - projecton onto te fnte element space V k defned locally,.e., for eac 1 N x, ( P k (w w q dx = q P k (. (2.11 Ts projecton s stable n L p ( for all p [32,.e., P k (w L p ( C w L p ( w L p (, 1 p. (2.12 We next ntroduce two more refned projectons (see [54, wc we denote by π ±, tat can be defned only for more regular functons, say w H 1/2+ɛ ( for all. Te projectons π + (w and π (w are te unque polynomals of degree at most k 1, tat satsfy for eac 1 N x ( π ± (w w q dx = q P k 1 (, (2.13 togeter wt te matcng condtons; π + (w(x + 1/2 = w(x+ 1/2 ; π (w(x +1/2 = w(x +1/2. (2.14 Provded w enjoys enoug regularty, say w H k+1 (, te followng error estmates can be easly sown for all tese projectons: } w P k (w, C k+1 w w π ± k+1, w H k+1 (. (2.15 (w, were C s a constant dependng only on te sape-regularty of te mes and te polynomal degree [21, 54. For te standard L 2 -projecton we wll also need estmates n te L -norm [56, w P k (w,, C k+1 w k+1,,. (2.16

8 8 B. AYUSO, J. A. CARRLLO AND C.-W. SHU Let k and let P : L 2 (Ω Z k be te standard L2 -projecton (n te twodmensonal case defned by P (w = (Px k Pv k (w;.e., for all and j, (P (w(x, v w(x, v ϕ (x, v dv dx = ϕ P k ( P k (J j. (2.17 J j From ts defnton, ts L 2 -stablty follows mmedately, but t can be sown to be stable n L p for all p [32, P (w L p (T C w L p (Ω w L p (Ω, 1 p. ( Te suggested numercal metods. n ts secton we formulate te numercal scemes we propose to approxmate te Vlasov-Posson system. Te frst one s a sceme were te DG approxmaton for te transport equaton s coupled wt a smple conformng approxmaton of ger degree for te electrostatc feld. Te second sceme results by combnng mxed fnte element approxmaton for te Posson problem togeter wt DG approxmaton to te transport equaton. Te last approac s based on fully DG approxmaton for bot varables, te electron dstrbuton f and te electrostatc feld. Due to te specal structure of te transport equaton: v s ndependent of x and E s ndependent of v; for all metods te DG approxmaton for te electron dstrbuton functon s done exactly n te same way. Terefore we start by ntroducng te DG metod for te transport equaton (1.1, and n wat follows, we denote by E te restrcton to of te fnte element approxmaton E to be defned later on. Let f ( = P (f be te approxmaton to te ntal data. Te numercal metod reads: fnd (E, f : [, T (W, Z k suc tat N x N v Bj (E ; f, ϕ = ϕ Z k, (3.1 =1 j=1 were te blnear form B j (E ; f, ϕ s defned for eac, j and ϕ Z k as: B j (E ; f, ϕ = T j + f t ϕ ϕ dv dx vf dv dx + T j x [ ((vf ϕ +1/2,v ((vf ϕ + 1/2,v J j [ ( ( ( E f ϕ ( E f ϕ + x,j+1/2 E ϕ f dv dx T j v dv (3.2 x,j 1/2 dx, were we ave used te sort and notaton gven n (2.9. Notce tat te expresson B j (E ; f, ϕ s n fact a blnear form. E s used only to empasse te nonlnear dependence on t. Here, te boundary terms are te so-called numercal fluxes, wc are notng but te approxmaton of te functons vf and Ef at te vertcal and orzontal boundares Γ x and Γ v, respectvely. By specfyng tem, te DG metod s completely determned. Te desgn of tese numercal fluxes s te key ssue to ensure te stablty of te numercal sceme. We consder te followng upwnd coce: vf = { v f f v, v f + f v <, Ê f = { E f + f E, E f f E <. (3.3

9 DG METHODS FOR ONE-DMENSONAL VLASOV-POSSON SYSTEM 9 We defne te numercal fluxes at te boundary Ω by ( vf 1/2,v = ( vf Nx+1/2,v, (Ê f x,1/2 = (Ê f x,nv+1/2 =, (x, v J, so tat te perodcty n x and te compactness n v are reflected. Te dscrete densty, denoted by ρ (x, t, s gven by ρ (x, t = f (x, v, t dv j J j x, t [, T. (3.4 Note tat from te defntons (1.3 and (3.4 of ρ and ρ, respectvely, and usng Caucy-Scwartz s nequalty t s stragtforward to see tat ρ(t ρ (t 2, 2L f (t f(t 2,T t [, T. (3.5 One of te nce features of te DG approxmaton for te transport s tat carge conservaton s ensured by constructon, as te followng result sows: Lemma 3.1. Partcle or Mass Conservaton: Let k and let f C 1 ([, T ; Z k be te DG approxmaton to f, satsfyng (3.1-(3.2. Ten, f (t dv dx = f ( dv dx = f dv dx = 1 t [, T. (3.6,j T j,j T j,j T j Proof. Note tat snce f ( = P (f, from te defnton of te L 2 -projecton (2.17 (wt ϕ = 1 togeter wt (1.4 we ave f ( dv dx = P (f dv dx = f dv dx = 1. (3.7,j T j,j T j,j T j We now fx an arbtrary T j and take n (3.2 te test functon ϕ = 1 n T j ; ϕ = elsewere. Notng tat suc a test functon verfes (ϕ +1/2,v = (ϕ + 1/2,v = 1, we ave B j (E : f, 1 = d f dv dx + [(vf dt +1/2,v (vf 1/2,v dv T j J j [(E f x,j+1/2 (E f x,j 1/2 dx. Moreover, note tat snce te coce of T j was done arbtrarly, te dentty above olds true for all, j. By summng t over all and j, te flux terms telescope and tere s no boundary term left because of te perodc (for and compactly supported (for j boundary condtons. Hence, takng nto account (3.1 we ave, = B j (E ; f, 1 = d f dv dx =, dt,j,j T j and so ntegraton n tme togeter wt (3.7 lead to (3.6. We next deal wt te approxmaton to te electrostatc feld E(x, t = Φ x (x, t. Te dscrete Posson problem reads, (Φ xx = 1 ρ x [, 1, Φ (1, t = Φ (, t. (3.8

10 1 B. AYUSO, J. A. CARRLLO AND C.-W. SHU Te well posedness of te above dscrete problem s guaranteed by (3.6 from Lemma 3.1 wc n partcular mples (Φ x (1, t = (Φ x (, t. (3.9 To ensure te unqueness of te soluton we set Φ (, t =. To get te soluton of te dscrete Posson problem at least two possble approaces arse: Drect ntegraton of te dscrete Posson problem (3.8, approxmaton of (1.2 wt some mxed fnte element metod; possbly dscontnuous. We next consder n detal tese approaces Conformng approxmaton to te electrostatc potental. Reasonng as n secton 2.1, drect ntegraton of te dscrete Posson problem (3.8 togeter wt Φ (, t = gves Φ (x, t = C E x + x2 2 x s ρ (z, tdzds x [, 1, (3.1 were C E s determned from te boundary condtons n (3.8, C E = 1 z ρ (s, t ds dz 1 2 Ten, dfferentaton w.r.t x n (3.1 leads to t [, T. (3.11 x E (x, t = CE + x ρ (s, t ds x [, 1. (3.12 Observe tat snce ρ V k, E turns out to be a contnuous polynomal of degree k + 1; so E s conformng. ts restrcton to s gven by x (x, t = E 1(x 1/2, t + (x x 1/2 f (s, χ, tdχds x, (3.13 E x 1/2 and E (x, t = for all x [x 1/2, x +1/2. Te boundary condton (3.9 reads J E (x 1/2, t = E Nx (x N x+1/2, t t [, T. (3.14 To sow tat E ndeed belongs to W k+1 we ave to verfy tat t as zero average. From (3.11 t follows stragtforwardly E (x dx = E (x 1/2, t + x 2 +1/2 x2 1/2 2 x x 1/2 ρ (x dx =. Fnally, we state a Lemma tat relates te error commtted n te approxmaton to E, wt te error n accumulated n te approxmaton to f. Ts result wll be used n our subsequent analyss and ts proof s gven n Appendx A. Lemma 3.2. Let k and let (E, f C ([, T ; W k+1 C 1 ([, T ; Z k be te conformng-dg approxmaton to te soluton of Vlasov-Posson system (E, f, soluton of (3.1 (3.2 (3.12. Ten, E(t E (t, C 1 f(t f (t,t t [, T, (3.15

11 DG METHODS FOR ONE-DMENSONAL VLASOV-POSSON SYSTEM 11 were 2L =meas(j and C 1 = (4L(1 + x 1/2. Furtermore, f te force feld E C ([, T ; H 1 (, te followng estmates also old for all t [, T, and E(t E (t,, C 2 f(t f (t,t wt C 2 = ((2L 1/2 + C 1, (3.16 E (t,, C 2 f(t f (t,t + E(t 1,. ( Mxed fnte element approxmaton for te Posson problem. We rewrte problem (3.8 as a frst order system: E = Φ x x [, 1; E x = ρ 1 x [, 1 (3.18 wt boundary condton Φ (, t = Φ (1, t =. n ts secton, we consder a mxed approxmaton to (3.18, wt te one-dmensonal verson of Ravart-Tomas elements, RT k k [51, 15. n 1D te mxed fnte element spaces turn out to be te (W k+1, V k-fnte element spaces. Note tat n partcular, d k+1 dx (W = V k. For k te sceme reads: fnd (E, Φ W k+1 V k suc tat E z dx + Φ z x dx = z W k+1, (3.19 (E x p dx = (ρ 1p dx p V k. (3.2 We refer to [51, 15 for te stablty and error analyss of te metod for lnear second order problems (see also [6 for te 1D-verson of te sceme n te lowest order case k =. However, n our case, te Posson problem s non lnear snce te source term n (1.2 depends on te soluton troug ρ. Terefore n te error analyss a consstency error appears. We ave te followng result, wose proof can be found n Appendx A. Lemma 3.3. Let k and let (E, Φ C ([, T ; W k+1 V k be te RT k approxmaton to te Posson problem (3.18 and let E C ([, T ; H k+1 ( Ten, te followng estmates old for all t [, T : E(t E (t, + E(t E (t 1, C k+1 E(t k+1, + 2L f(t f (t,t, E(t E (t,, C k+1 E(t k+1, + (2L 1/2 f(t f (t,t, (3.21 E (t,, E(t 1, + (2L 1/2 f(t f (t,t + C E(t 1, DG approxmaton for te Posson problem. Consder te DG approxmaton to te frst order system (3.18: fnd (E, Φ V r Vr suc tat for all : E z dx = Φ z x dx + [( Φ z +1/2 ( Φ z + 1/2 z V, r (3.22 E p x dx [(Êp +1/2 (Êp + 1/2 = (ρ 1p dx p V r, (3.23

12 12 B. AYUSO, J. A. CARRLLO AND C.-W. SHU were ( Φ 1/2 and (Ê 1/2 are te numercal fluxes. n ts work we focus on te followng famly of DG-scemes (see owever remark 3.5: { ( Φ 1/2 = {Φ } 1/2 c 12 [ Φ 1/2 + c 22 [ E 1/2, (3.24 (Ê 1/2 = {E } 1/2 + c 12 [ E 1/2 + c 11 [ Φ 1/2, were te parameters c 11, c 12 and c 22 depend solely on x 1/2, and are stll at our dsposal. At te boundary nodes due to perodcty n x we mpose ( Φ 1/2 = ( Φ Nx+1/2, (Ê 1/2 = (Ê Nx+1/2. Followng [18 we defne a(e, z := E zdx + c 22 [ E 1/2 [ z 1/2, b(φ, z := Φ z x dx + ({Φ } c 12 [ Φ [ z 1/2, c(φ, p := c 11 [ Φ 1/2 [ p 1/2, and A((E, Φ ; (z, p = a(e, z + b(φ, z b(p, E + c(φ, p. Tus, problem (3.22-(3.23 can be rewrtten as: fnd (E, Φ V r V r suc tat A((E, Φ ; (z, p = (ρ 1p dx (z, p V r V r. (3.25 Note tat A(, nduces te followng sem-norm (z, p H 1 ( H 1 ( : (z, p 2 A := A((z, p; (z, p = z 2, + c 1/2 22 [ z 2,γ x + c 1/2 11 [ p 2,γ x. (3.26 We also defne te norm for all r (E, Φ 2 r+1, := E 2 r+1, + Φ 2 r+2, (E, Φ H r+1 ( H r+2 (. (3.27 We next descrbe te specfc coces of te metods we consder (by specfyng te parameters n (3.24. We restrct ourselves to k 1, k beng te order of approxmaton used for f. ( Local dscontnuous Galerkn (LDG metod: we take r = k + 1 so te spaces are V r = V k+1 and we set c 22 = and c 11 = c 1 wt c a strctly postve constant. Ts metod was frst ntroduced n [27 for a tme dependent convecton dffuson problem (wt c 11 = O(1. n ts paper we take c 11 = c 1 wt c a postve constant, and c 12 = 1/2; tat s: { (Ê 1/2 = {E } 1/2 c 12 [ E 1/2 + c 1 [ Φ 1/2, ( Φ 1/2 = {Φ } 1/2 + c 12 [ Φ 1/2 c 12 = 1 2. (3.28 For te approxmaton of lnear problems, t as been proved (see [27, 18 convergence of order r + 1 and r for Φ and E, respectvely.

13 DG METHODS FOR ONE-DMENSONAL VLASOV-POSSON SYSTEM 13 ( Mnmal dsspaton LDG and DG metods (MD-LDG and MD-DG: we set r = k and te spaces are taken as V r = V k. For te MD-LDG metod, te numercal fluxes are defned by takng n (3.24 c 22 =, c 12 = 1/2 and c 11 = except at a boundary node, tat s, { (Φ 1/2 = (Φ 1/2 (E 1/2 = (E + 1/2 + c 11 [ Φ 1/2, c 11 = { Nx 1, cr 1 = N x. (3.29 For te MD-DG metod te same coce apples except for (Φ 1/2 = (Φ 1/2 + c 22 [ E 1/2 wt c 22 = c/r. For te approxmaton of lnear problems, te MD- LDG metod was frst ntroduced for te 2D case n [23 but wt c 11 = O(1 rater tan O( 1 at te boundary. Te analyss n te one-dmensonal case for bot te MD-LDG and te MD-DG can be found n [19, were te autors sow tat te approxmaton to E, wt bot metods, superconverges wt order r + 1. ( General DG & ybrdzed LDG metod: we set r = k so tat te spaces are taken as V r = V k, and we take te numercal fluxes as n (3.24 wt: c 11, c 22, > c 12 bounded c 11 1 c 22. Superconvergence results are proved n [22 (for dmenson d 2 for te approxmaton of lnear problems. Anoter opton wc also provdes superconvergence and could be effcently mplemented, s te ybrdzed LDG metod (see [22 n wc te numercal fluxes can be recast n te form (3.24 by settng: ( ( ( τ (Ê 1/2 = τ + +τ (E + 1/2 + τ + τ + +τ (E 1/2 + τ τ + τ + +τ [ Φ 1/2, ( ( ( τ ( Φ 1/2 = + (Φ + 1/2 + τ (Φ 1/2 + 1 [ E 1/2, τ + +τ τ + +τ τ + +τ were τ ± are non-negatve constants. To aceve superconvergence, t s enoug to take n eac nterval one τ at one end and at te oter end we set τ =. Superconvergence can be sown by followng te analyss n [22 but usng te specal projectons defned troug (2.13-(2.14. As t appened wt RT k approxmaton, our Posson problem s nonlnear and terefore te estmates sown n [18, [19 and [22 are not drectly applcable. However, we ave te followng result, wose proof can be found n Appendx A. Lemma 3.4. Let k 1 and let (E, Φ C ([, T ; V r V r be te DG approxmaton (3.22-(3.23-(3.24 to te Posson problem (3.18, wt any of te tree coces (, ( or (. Let (E, Φ C ([, T ; H r+1 ( H r+2 ( Ten, te followng estmates old for all t [, T, E(t E (t 2, C 2(k+1 (E(t, Φ(t 2 r+1, + 2L f(t f (t 2,T, (3.3 were r s te order of polynomals of V r also olds as gven n (, (, (. Furtermore, t (E(t E (t, Φ(t Φ (t 2 A C2(k+1 (E(t, Φ(t 2 r+1, +2L f(t f (t 2,T. were r = k + 1 for ( and r = k for ( and (.

14 14 B. AYUSO, J. A. CARRLLO AND C.-W. SHU Remark 3.5. Snce k 1, one mgt consder any of te (consstent and stable DG metods tat ft n te framework gven n [4 for approxmatng te Posson problem (1.2. Most of te results sown n ts paper for te general LDG dscretzaton (wt general c 12, old (wt mnor canges n te proofs for any of te resultng metods. For te sake of concseness, te detals are omtted. 4. Error Analyss. We start by sowng a cell-entropy nequalty [44 for te proposed DG scemes (3.1, wc guarantees ter L 2 -stablty. We ten derve te error equaton and gve some auxlary results tat are used n te proofs of te man results, wc are gven at te end of te secton Stablty. Next proposton sows tat te above selecton of te numercal fluxes s enoug to preserve te L 2 -stablty of numercal soluton of (3.1-(3.2, for all k. Proposton 4.1 (L 2 -stablty. Let k and let f Z k be te approxmaton (3.1-(3.2 of problem (1.1, wt te numercal fluxes as n (3.3. Ten f (t,t f (,T t [, T. (4.1 Proof. By settng ϕ = f n (3.2 we ave B,j(E ;f, f = J j E [ (Ê f f (f 2 dv dx 1 J j t 2 J j (f 2 dv dx + v J j x,j+1/2 (Ê f f + v (f 2 x dv dx [ ( vf f +1/2,v ( vf f + 1/2,v x,j 1/2 dx. Takng nto account tat E depends only on x (troug f wle v s ndependent of x, ntegraton of te second and trd volume terms leads to dv B,j (E ; f, f = 1 d [ 2 dt f 2,T j + F+1/2,j F 1/2,j + Θ F 1/2,j + [Ĝ,j+1/2 Ĝ,j 1/2 + Θ G,j 1/2, (4.2 were F +1/2,j, Ĝ,j+1/2 are defned for all, j, as and Θ F 1/2,j = Θ G,j 1/2 = J j F +1/2,j = Ĝ,j+1/2 = J j [ v 2 (f 2 vf f [ E 2 (f 2 Ê f f [ v 2 (f 2 vf f [ E 2 (f 2 Ê f f 1/2,v dv + dx x,j 1/2 J j +1/2,v dv dx, x,j+1/2 [ v 2 (f 2 + vf f + [ E 2 (f 2 + Ê f f + 1/2,v dv, dx. x,j 1/2

15 DG METHODS FOR ONE-DMENSONAL VLASOV-POSSON SYSTEM 15 We next sow tat te coce (3.3 ensures tat bot Θ F 1/2,j and ΘG,j 1/2, for all and j, are non-negatve. By rewrtng our coce of te numercal fluxes (3.3 as: (vf = v{f } v 2 [ f, [E f = E {f } + E 2 [ f, (4.3 and usng tat [ f 2 = 2{f }[ f, t can be easly seen tat Θ F 1/2,j and ΘG,j 1/2 become [ v Θ F 1/2,j = 2 [ f 2 vf v [ f dv = 2 [ f 2 1/2,vdv, (4.4 Θ G,j 1/2 = J j 1/2,v J j [ Ê f [ f E 2 [ f 2 E dx = x,j 1/2 2 [ f 2 x,j 1/2dx. (4.5 Terefore, Θ F 1/2,j and ΘG,j 1/2 for all and j and so substtuton n (4.2 leads to 1 d [ f 2 dv dx + F+1/2,j 2 dt F 1/2,j + [Ĝ,j+1/2 Ĝ,j 1/2, T,j By summng n te above nequalty over and j, te flux terms telescope and tere s no boundary term left because of te perodc (for and compactly supported (for j boundary condtons. Hence, 1 d f 2 2 dt dv dx = 1 d T,j 2 dt f 2.T, (4.6,j and terefore, ntegraton n tme of te above nequalty yelds to (4.1. Remark 4.2. By carefully revsng te proof one realse tat n fact nequalty (4.6 s replaced by te dentty ( 1 d 2 dt f 2,T + v 1/2 [ f 2,Γ x + E 1/2 [ f 2,Γ v =. (4.7 Terefore, by defnng te norm f (t 2 := f (t 2,T + t v 1/2 [ f (s 2,Γ x ds+ t te concluson of Proposton 4.1 can be reformulated as: E 1/2 [ f (s 2,Γ v ds, (4.8 f (t 2 = f ( 2,T f ( 2 for all t [, T. Fnally, we note tat for te convergence and error analyss of numercal scemes for non-lnear problems, one usually needs to assume/prove tat some a-pror estmate on te approxmate soluton olds for all tme. n fact, wat s generally done s to assume tat tere exsts some C κ > suc tat, f f,t C κ, t [, T,

16 16 B. AYUSO, J. A. CARRLLO AND C.-W. SHU were,t usually refer to a stronger norm tan te one for wc te error analyss s carred out. For nstance,t =,,T f te error analyss s carred out n te L 2 or energy norm, see [49. We ws to stress tat n te present work, due to te structure of te contnuous problem, suc type of assumpton s not requred. Te man reason s tat altoug our L 2 -error analyss requres a bound on E,,, suc an estmate would depend ultmately on ρ (zero order moment of f, wc n general s more regular tan f tself. n te end, ts fact allows for gettng a bound for E,, dependng on te L 2 -error f f,t, for wc we can easly guarantee tat tere exsts c κ > suc tat, f f,t c κ, t [, T. (4.9 Estmate (4.9 follows from te L 2 conservaton property of te contnuous soluton (2.7 and te L 2 -stablty of ts approxmaton f gven n Proposton 4.1, togeter wt trangle nequalty and te L 2 -stablty of te standard L 2 projecton, (2.18 wt p = 2, f(t f (t 2,T 2( f(t 2,T + f (t 2,T 2 f 2,Ω + 2 P k (f 2,T 2(1 + C f 2,T = c κ. Let us pont out tat ts result allows us to obtan error estmates tat old for all and not only n te asymptotc regme Error equaton and specal projecton. To derve te error equaton te weak formulaton (2.6 s of lttle use, snce we sould take te test functon n Z. Hence, by allowng te test functon to be dscontnuous we fnd tat te true soluton satsfes te varatonal formulaton: were B,j (E; f, ϕ = N x N v B,j (E; f, ϕ = ϕ Z k, (4.1 =1 j=1 f T,j t ϕ dv dx [ + (vfϕ +1/2,v (vfϕ + 1/2,v J j dv dx + vf ϕ T,j x [ (E dv fϕ E f ϕ T,j v dv dx (4.11 x,j+1/2 ( E fϕ + x,j 1/2 dx, E beng te restrcton of te electrostatc feld E to ;.e., E = E. Subtractng (3.1 from (4.1 we obtan te error equaton, =,j B,j (E; f, ϕ B,j (E ; f, ϕ (4.12 =,j a,j (f f, ϕ +,j N,j (E; f, ϕ N,j (E ; f, ϕ ϕ Z. were te blnear form a(, =,j a,j(, gaters all lnear terms: [ f a,j (f, ϕ = t ϕ ϕ vf dv dx x J j + J j [ ( vf ϕ +1/2,v ( vf ϕ + 1/2,v dv

17 DG METHODS FOR ONE-DMENSONAL VLASOV-POSSON SYSTEM 17 and N,j (E ;, (resp. N,j (E;, carres te nonlnear part; N,j (E ; f, ϕ = E f ϕ dv dx J j v [ (Ê f ϕ (Ê f ϕ + x,j+1/2 N,j (E ; f, ϕ = E f ϕ dv dx J j v [ (E fϕ x,j+1/2 ( E + fϕ x,j 1/2 x,j 1/2 dx. dx, Notce tat due to te nonlnearty, te true soluton f does not satsfy te equatons defnng te numercal sceme (3.1 (3.2. n fact we ave a consstency error: N (E ; f, ϕ N (E; f, ϕ for all ϕ Z, wc s dden n te nonlnear error N (E; f, ϕ N (E ; f, ϕ. Specal Projecton: We next ntroduce te 2-dmensonal projecton operator Π : C (Ω Z k wc s defned n te followng way. Let T,j = J j be an arbtrary element of T and let w C (T,j. Te restrcton of Π (w to T,j s defned by { πx π Π (w = v (w, f sgn(e = constant, P k π v (w, f sgn(e constant, (4.13 were Px k denotes te standard L 2 -projecton onto P k ( defned n (2.11 and π x, π v are defned by π x (w = { π + x (w f E >, πx (w f E <, π v (w = { π v (w f v >, π v + (w f v <, (4.14 wt π x ± : C ( V k and π± v : C (J j V k beng te specal projecton operators n te x and v drecton respectvely, defned as n (2.13-(2.14. Te defnton of projecton Π s nspred n tose consdered n [48, 23 and tat ntroduced n [62 for te analyss of Runge-Kutta DG metods for conservaton laws, see Remark 4.4. Note tat takng nto account (4.13-(4.14 togeter wt (2.13-(2.14, t s stragtforward to see tat Π (w s unquely defned. Te next lemma, altoug elementary, provdes several approxmaton results needed for our analyss. Lemma 4.3. Let w H s+2 (T,j, s and let Π be te projecton operator defned troug (4.13-(4.14. Ten, w Π (w,tj C mn (s+2,k+1 w s+1,tj, w Π (w,e C mn (s+ 3 2,k+ 1 2 w s+1,tj, e =, J j T j. (4.15 Proof. From te defnton (4.13 we dstngus two cases. f T j s an element suc tat sgn(e (x s constant x T j, te proof s te same as [18, Lemma 3.2. f on te contrary, T j s suc tat x T j for wc E (x =, we ave Π (w = P k π v (w. But stll, snce Π s a polynomal preservng and lnear operator, estmates (4.15 follow also n ts case from Bramble-Hlbert lemma, trace Teorem and standard scalng arguments. Detals are omtted for te sake of concseness.

18 18 B. AYUSO, J. A. CARRLLO AND C.-W. SHU Summng estmates (4.15 from Lemma 4.3, over elements of te partton T, w Π (w,ω + 1/2 w Π (w,γ C k+1 w k+1,ω w H k+1 (Ω. (4.16 Now, denotng by we can wrte ω = Π (f f, ω e = Π (f f, (4.17 f f = [Π (f f [Π (f f = ω ω e. (4.18 Ten, by takng as test functon ϕ = ω Z k, te error equaton (4.12 becomes [ a(ω ω e, ω + N,j (E ; f, ω N,j (E ; f, ω =. (4.19,j We next defne K 1 (v, f, ω =,j K 1,j (v, ωe, ω, K 2 (E, f, ω =,j K 2,j (E, f, ω, (4.2 were K,j 1 (v, f, ω = vω e ωx dv dx T,j [ ( vω e (ω +1/2,v ( vω e (ω + 1/2,v dv, (4.21 J j K,j(E 2, f, ω = E ω e ωv dv dx T,j [(Ê ω e (ω x,j+1/2 (Ê ω e (ω + x,j 1/2 dx. (4.22 Te next two lemmas provde estmates for te terms defned n (4.2. Bot lemmas extend and generalze [23, Lemma 3.6 to te case of varable coeffcents and nonlnear problems, respectvely. To keep te readablty flow of te paper, te proofs of tese tecncal lemmas are postponed tll Appendx B. Remark 4.4. We ws to note tat te defnton (4.13 of Π s done n terms of E (and v, wle te defnton of te numercal fluxes s done n terms of E (and v. Ts s due to te non-lnearty of te problem and t s nspred n te deas used n [62. By defnng Π n terms of E rater tan E and usng te regularty of te soluton, we wll be able to estmate optmally te expresson K 2 wtout any furter assumpton on te mes partton T. Lemma 4.5. Let T be a cartesan mes of Ω, k 1 and let f Z k be te approxmate dstrbuton functon satsfyng (3.1-(3.2. Let f C ([, T ; H k+2 (Ω and let K 1 be defned as n (4.2. Assume tat te partton T s constructed so tat v = corresponds to a node of te partton. Ten, te followng estmate olds true K 1 (v, f, ω C k+1 ( f k+1,ω + CL f k+2,ω ω,t. (4.23

19 DG METHODS FOR ONE-DMENSONAL VLASOV-POSSON SYSTEM 19 Lemma 4.6. Let T be a cartesan mes of Ω, k 1 and let (E, f W Z k be te soluton to (3.1-(3.2 wt W a fnte element space, conformng or non-conformng, of at least frst order (W = W k+1 or W = V r. Let (E, f C ([, T ; W 1, ( H k+2 (Ω and let K 2 be defned as n (4.2. Ten, te followng estmate olds K 2 (E, f, ω C k E E,, f k+1,ω ω,t ( C k+1 ( f k+2,ω E,, + f k+1,ω E 1,, ω,t Auxlary results. We next prove two lemmas tat are needed for te proofs of te man Teorems 4.9, 4.13, and Te frst one reduces te expresson for te lnear part of te error equaton (4.19: Lemma 4.7. Let f C (Ω and let f Z k equalty olds a(f f, ω = ( ω t ω e t ω dxdv +,j T,j,j wt k 1. Ten, te followng J j v 2 [ ω 2 1/2,v dv + K1 (v, f, ω. Proof. From (4.18 we get a(f f, ω = a(ω, ω a(ω e, ω. Argung as for (4.4 n te proof of Proposton 4.1 (note tat ω Z, we ave for te frst term a(ω, ω = ωt ω dxdv + v,j J j,j J j 2 [ ω 2 1/2,vdv. (4.25 Te defnton (4.21 of K 1, te contnuty of f and te numercal fluxes (3.3 mply a(ω e, ω = ωt e ω dxdv vω e ωx dxdv [ vω e [ ω 1/2,v dv,j J j J j,j J j = ωt e ω dxdv K 1 (v, f, ω.,j J j wc togeter wt (4.25 completes te proof. Te oter auxlary Lemma deals wt te error comng from te nonlnear term: Lemma 4.8. Let E C (, f C (Ω and f Z k wt k 1. Ten, te followng dentty olds [N,j (E; f; ω N,j (E ; f, ω = (4.26,j =,j E 2 [ ω 2 x,j 1/2 dx,j T,j [E E f v ω dv dx K 2 (E, f, ω. Proof. Subtractng te nonlnear terms n (4.11 and (3.2 we ave N,j (E; f;ω N,j(E ; f, ω = [E Jj f E f ω dv dx v [ ([E f Ê f ω x,j 1/2 ([E f Ê f ω + x,j 1/2 dx. (4.27

20 2 B. AYUSO, J. A. CARRLLO AND C.-W. SHU Notce tat te ntegrand of te volume part above, can be decomposed as [ E f E f ± E f = [E E f + E (f f, (4.28 and so substtutng nto (4.27 we fnd were T 1 = T 3 = N,j (E; f; ω N,j(E ; f, ω = T 1 + T 2 + T 3, (4.29 [E E fω v dv dx, T 2 = E [f f ωv dv dx, J j J j [([(E f + Ê f (ω + x,j 1/2 ([(E f Ê f (ω x,j+1/2 dx. Snce neter E nor E depend on v, ntegraton by parts of T 1 gves T 1 = T 1a + T 1b : T 1 = [E E f J j v ω dv dx + (E E [(fω x,j+1/2 (fω + x,j 1/2 dx. Summng now over j and takng nto account te contnuty of f we fnd for T 1b, T 1b = (E E (f [ ω x,j 1/2 dx. (4.3 j j We next deal wt T 2. From te splttng (4.18 we ave T 2 = E ω ωv dv dx E ω e ωv dv dx = T 2a + T 2b, J j J j and so, ntegratng te frst term and summng over j we easly get T 2a = 1 E (ω 2 j j Jj 2 dv dx = E v j 2 [ (ω 2 x,j 1/2 dx. (4.31 We fnally deal wt te boundary terms collected n T 3. Summaton over j and te contnuty of E and f gves T 3 = [E f Ê f x,j 1/2 [ ω x,j 1/2 dx. j j Ten, reasonng as n (4.28, we deduce for all tat ( ( E f Ê f ±Ef = (E Ef + Ef Ê f = (E Ef + E (ω E (ωe, were n te last step we ave used te contnuty of f togeter wt te consstency of te numercal flux Ê f. Tus, substtutng back nto T 3, we nfer T 3 = ( (E E f [ ω + Ê ω [ ω Ê ωe [ ω dx j j x,j 1/2 = j T 3a + j T 3b + j T 3c.

21 DG METHODS FOR ONE-DMENSONAL VLASOV-POSSON SYSTEM 21 Ten, for te frst term, T 3a, recallng te expresson (4.3, we get [T 1b + T 3a =. (4.32 j Next, summng T 3b and T 2a from (4.31 and argung as for (4.5 n te proof of Proposton 4.1, we fnd [T 2a + T 3b = j j E 2 [ ω 2 x,j 1/2dx. (4.33 Fnally, recallng te defnton (4.22 of K 2 and addng up T 3c wt T 2b we get [T 2b + T 3c = K 2 (E, f, ω. j Tus, substtutng te above dentty togeter (4.33 and te expresson for T 1a nto te equaton (4.29 we reac (4.26 and so te proof s complete Approxmaton. We next sow te man convergence results of ts work provng a-pror error estmates for te electron dstrbuton f, for all te proposed metods. n eac case, as a byproduct result, we also get te correspondng convergence results for te electrostatc feld E. Te secton s closed wt some remarks about te comparson wt te convergence of oter metods. We start wt te result for te conformng-dg metod: Teorem 4.9 (Conformng-DG metod. Let k 1 and consder te unque compactly supported soluton of te Vlasov-Posson system (1.1-(1.2 gven by Teorem 2.1 wt f C 1 ([, T ; H k+2 (Ω and E C ([, T ; W 1, (. Let (E, f C ([, T ; W k+1 C 1 ([, T ; Z k be te conformng-dg approxmaton,.e. soluton of (3.1, (3.2 and (3.12. Ten, f(t f (t,t C k+1 t [, T, were C depends on te tme t, te polynomal degree k, te sape regularty of te partton and depends also on f and on E troug te norms C = C ( f(t k+2,ω, f t (t k+1,ω, L, E(t 1,,. Proof. Recallng te error equaton (4.19 a(ω ω e, ω + N (E ; f, ω N (E ; f, ω =, and usng Lemmas 4.7 and 4.8, we ave ωt ω dv dx + v,j T,j,j J j 2 [ ω 2 +1/2,v dv +,j = ωt e ω dv dx +,j T,j,j E 2 [ ω 2 x,j+1/2 dx T,j [E E f v ω dv dx K 1 (v, f, ω + K 2 (E, f, ω = T 1 + T 2 K 1 + K 2. (4.34

22 22 B. AYUSO, J. A. CARRLLO AND C.-W. SHU Notce tat te left and sde of te above equaton, s exactly wat results after summaton over and j n (4.2 from Proposton (4.1, see also (4.7. Ten, t s enoug to estmate te terms on te rgt and sde of te above equaton. Te frst term s drectly estmated by usng Caucy-Scwartz and te artmetc-geometrc nequaltes togeter wt te nterpolaton property (4.16 T 1 C 2 ( ωe t 2,T + ω 2,T C 2k+2 f t H k+1 (Ω + C ω 2,T. (4.35 Te second term on te rs of (4.34, s readly estmated by usng Hölder nequalty togeter wt estmate (3.16 from Lemma 3.2, te splttng (4.18, te artmetcgeometrc nequalty and te nterpolaton estmate (4.16, T 2 C E E,, f v,ω ω,t CC 2 f f,t f v,ω ω,t CC 2 ( ω e,t + ω,t f v,ω ω,t CC 2 2k+2 f 2 k+1,ω f v,ω + C 2 f v,ω ω 2,Ω, (4.36 were C 2 L 1/2 s te constant n Lemma 3.2. Estmate (4.23 from Lemma 4.5 and te artmetc-geometrc nequalty gve for te trd term, K 1 C 2k+2 L 2 f 2 k+2,ω + C ω 2,T. (4.37 Last term s bounded by usng estmate (4.24 from Lemma 4.6 and argung smlarly as for T 2 ; usng estmate (3.16 from Lemma 3.2, te splttng (4.18, te artmetcgeometrc nequalty and te nterpolaton estmate (4.16, K 2 C k f k+1,ω ( ω e,t + ω,t ω,t + C k+1 f k+2,ω E 1,, ω,t C 2k+2 ( f 2 k+2,ω E 2 1,, + C 2 k f 3 k+1,ω + C(1 + k f k+1,ω ω 2,T. Ten, by substtutng te above estmate togeter wt (4.35, (4.36 and (4.37 nto te error equaton (4.34, we conclude d dt ω (t 2,T A(t ω (t 2,T + 2k+2 B(t wt A(t = (C + L 1/2 f v,ω + CL 1/2 k f k+1,ω and B(t= C f 2 k+2,ω(l 2 + E 2 1,,+ f t 2 k+1,ω+cl 1/2 f 2 k+1,ω( f v 2,Ω+ k f k+1,ω. Terefore, ntegraton n tme of te above nequalty and a standard applcaton of Gronwall s nequalty gves te error estmate, ω (t 2,T C 2 2k+2, (4.38 were C s as stated n te clam. Hence, Teorem 4.9 follows from te trangle nequalty and te nterpolaton property (4.16. As a drect consequence of Teorem 4.9 togeter wt estmates (3.15 and (3.16 of Lemma 3.2, we obtan te followng result on te error of te electrostatc feld. Corollary 4.1. estmates old Under te ypotess of Teorem 4.9, te followng error E(t E (t, C C 1 k+1 t [, T, E(t E (t, C C 2 k+1 t [, T,

23 DG METHODS FOR ONE-DMENSONAL VLASOV-POSSON SYSTEM 23 were C 1 and C 2 are gven n (3.15 and (3.16, respectvely and C n Teorem 4.9. Next result establses te convergence for te RT k -DG metod: Teorem 4.11 (RT k -DG metod. Let k 1 and consder te unque compactly supported soluton of te Vlasov-Posson system (1.1-(1.2 gven by Teorem 2.1 wt f C 1 ([, T ; H k+2 (Ω and E C ([, T ; H k+1 (. Let ((E, Φ, f C ([, T ; (W k+1 V k C1 ([, T ; Z k be te RT k-dg approxmaton soluton of (3.1, (3.2, (3.19, and (3.2. Ten, f(t f (t,t C 4 k+1 t [, T, were C 4 depends on te tme t, te polynomal degree k, te sape regularty of te partton and depends also on f and on E troug te norms C 4 = C 4 ( f(t k+2,ω, f t (t k+1,ω, L, E(t k+1,, Proof. Te proof follows exactly te same lnes as te proof of Teorem 4.9. n ts case, to bound te error E E,, tat appears n te estmates for T 2 and K 2 one as to use estmate (3.21 from Lemma 3.3. We omt te detals for te sake of concseness. Corollary estmates old Under te ypotess of Teorem 4.11, te followng error E(t E (t, + E(t E (t 1, 2C 4 L 1/2 k+1 + C k+1 E k+1, E(t E (t,, C 4 L 1/2 k+1 + C k+1 E k+1, for all t [, T, were C 4 s te constant of Teorem Fnally, we sow te convergence for te full DG approxmaton: Teorem 4.13 (DG-DG metod. Let r k 1 and consder te unque compactly supported soluton of te Vlasov-Posson system (1.1-(1.2 gven by Teorem 2.1 wt f C 1 ([, T ; H k+2 (Ω and E C ([, T ; H r+1 (. Let ((E, Φ, f C ([, T ; V r Vr C1 ([, T ; Z k be te DG-DG approxmaton tat satsfes (3.1, (3.2, (3.22, and (3.23 wt any of te tree coces (, ( or (. Ten, f(t f (t,t C 4 k+1 t [, T, were C 5 depends on tme t, te polynomals degrees k and r, te sape regularty of te partton and depends also on f and on (E, Φ troug te norms C 5 = C 5 ( f(t k+2,ω, f t (t k+1,ω, L, (E, Φ r+1,. Proof. Te proof follows essentally te same lnes as te proof of Teorems 4.9 and 4.11, but dealng wt T 2 we use estmate (3.3 from Lemma 3.4; T 2 C E E, f v,,ω ω,t [C k+1 (E, Φ r+1, + (2L 1/2 f f,t f v,,ω ω,t C 2k+2 ( (E, Φ 2 r+1, + 2L f 2 k+1,ω f v,,ω + (C + (2L 1/2 f v,,ω ω 2,Ω. (4.39

24 24 B. AYUSO, J. A. CARRLLO AND C.-W. SHU Also, to bound for K 2 we frst note tat E = P k+1 (E snce E V r (and r = k +1 or r = k, so tat nverse nequalty, estmate (2.16 and te L 2 -stablty of te L 2 - projecton gve E E,, E P k+1 (E,, + C 1/2 P k+1 (E E, C k+1 E k+1,, + C 1/2 E E,. (4.4 Ten, usng estmate (4.24 from Lemma 4.6 togeter wt te above estmate and te L 2 -bound for te error E E gven n Lemma 3.4, we get K 2 C(1 + L 1/2 k 1/2 f k+1,ω ω 2,T + C 2k+2 ( E 2 1,, f 2 k+2,ω + (E, Φ 2 r+1, f 2 k+1,ω + k 1/2 f 3 k+1,ω, were we ave neglected g order terms of order O( 4k 1/4. Notng tat k 1, te proof can now be completed by argung as n te proof of Teorem 4.9. We omt te detals for te sake of brevty. Remark Takng nto account te defnton (4.8 of te norm (see Remark 4.2, observe tat n te proof of Teorems 4.9, 4.11 and 4.13, smlarly as ow t s obtaned te error estmate (4.38, we also get ω (t 2 C 2 s 2k+2 s =, 4, 5. (4.41 As a drect consequence of Teorem 4.13 and Lemma 3.4 we ave te followng corollary wose proof s omtted. Corollary Under te ypotess of Teorem 4.13, te followng error estmates old for all t [, T E(t E (t 2, C 2k+2 (E(t, Φ(t 2 r+1, + C2 5 L2k+2 were C 5 s te constant of Teorem 4.13, and E(t E (t 2, + c 1/2 11 [ Φ (t 2,γ x + c 1/2 22 [ E (t 2,γ x C 6 2k+2, wt C 6 = C5 2L + C (E(t, Φ(t 2 r+1, were r = k + 1 for ( and r = k for ( and (. Remark 4.16 (Order of convergence attaned by oter metods. As noted n te ntroducton, tere are very few works dealng wt te convergence and error analyss of Euleran solvers for te (perodc Vlasov-Posson system. Hg order scemes ave been only analyzed n te context of sem-lagrangan metods [7, 8, 1. Altoug, t s dffcult to compare ter results wt ours, snce tese analyss deal wt fully dscrete scemes, we just menton brefly wat one can expect to aceve wt tese metods n te case of a constant Courant-Fredrcs-Levy CFL (ν = dt/ =constant and n te case were te tme step dt were taken te largest possble. n [7, error estmates n L of frst order (for CFL=constant and slgtly better tan frst order (at most of order 4/3 for te largest possble tme step, are sown assumng te ntal data s of class C 2. Hg order scemes, by usng polynomals of degree k n te reconstructon, are consdered n [8, 1. Tere, te autors prove error bounds for te dstrbuton functon and te electrostatc feld n L 2 and L, respectvely, of at most order k (f CFL=constant and of order 2(k + 1/3 f te largest possble tme step wants to be used. Tese works typcally requre te tecncal assumpton f W k+1, (Ω.