A conservative and entropy scheme for a simplified model of granular media

Transcription

1 A conservatve and entropy scheme for a smplfed model of granular meda C. Buet 1, S. Corder 2, V. Dos Santos 3 1: CEA,91680 Bruyères le Chatel, France. emal: chrstophe.buet@cea.fr 2: Laboratore MAPMO,UMR 6628 Unversté Orléans, Orleans, France emal: stephane.corder@unv-orleans.fr 3: Laboratore Vson et Robotque, IUT de Bourges, 18 Bourges, France emal: Valere.Dossantos@bourges.unv-orleans.fr Abstract In ths paper, we present a numercal scheme for a non lnear Fokker- Planck equaton of one-dmensonal granular medum. We consder a knetc descrpton of a system of partcles undergong nearly elastc partcles and nteractng wth a thermal bath. We construct a numercal method whch preserve all the propertes of the contnuous model, conservaton laws and decay of the entropy. Moreover the dscretzaton s such that, on a fxed grd, we deal wth arbtrary small temperatures for the bath. Explct and mplct tme dscretzaton are analyzed 1

2 1 Introducton The model we consder has been proposed by Mac Namara and Young [23], and has been studed from mathematcal pont of vew ([5], [4], [3]). We also refer to ([1, 17]) for knetc modelsatons of nelastc collsons. The model s derved from a system of partcles movng n one dmenson and that undergo nelastc collson. The unknown of the knetc model s the dstrbuton functon that represents the number of partcles wth velocty v R, at tme t. The collsons are modeled by a Boltzmann type collson operator, for nelastc collson wth an hard sphere cross secton. The collsons beng nelastc, they lead to a decay of energy and the dstrbuton functon concentrates for large tme on zero velocty. When there s a lot (N 1) of such collson, weakly nelastc, the Boltzmann equaton reduces to a Fokker-Planck type equaton, [24, 25]. In ths work, we consder such a system of partcles mmersed n a thermal bath at constant temperature σ, whch counterbalances the loss of energy due to nelastc collsons. When σ = 0, the dstrbuton functon converges to a Drac mass at orgn n the sense of weak convergence of measure [3]. In the case σ > 0, the equlbrum states behave lke exp( v 3 ) for large veloctes, [4]: the equlbrum state are not Maxwellan. Let us also menton that some works has been devoted to the hydrodynamcal lmt of ths system: one obtans a system of 2 conservaton laws for the densty ρ and the momentum ρu, that can be wrtten as the Euler system for sotropc gases, wth a pressure law P (ρ) = ρ γ, γ = 1/3 (the exponent 1/3 yelds to mathematcal dffcultes snce the obtaned system les n between classcal gas dynamcs, γ > 1, and pressure less gas, γ = 0, that have very dfferent behavor). The am of ths paper s to present a determnstc dscretzaton that s compatble wth the known propertes of the operator. Moreover, the method s such that we can treat any temperature of the bath wth the same grd, n other words, as σ = 0 goes to zero the scheme degenerate to a conservatve and entropy dscretzaton of the pure granular case. The method s close to the one developed for the Fokker-Planck-Landau equaton or the Kompaneets equaton (Cf. Buet and al, [7, 9, 10]). For generalzed dscrete models of knetc equatons, we refer to ([20]). Let us emphasze that the proposed method has a lnear computatonal cost. Explct and Implct tme dscretzaton are detaled. Implct schemes appear to be more effcent. Some numercal results are presented. In partcular, the non-maxwellan equlbrum states are obtaned. 2 The one dmensonal model of granular meda We consder a smplfed 1D model of granular meda descrbed by example n [4]. The partcles are descrbed by ther dstrbuton functon f(v, t) of velocty 2

3 v and tme t. Ths functon obeys t f = v (F f + σ v f), (2.1) where F f s the pure granular term, F (v) = v v (v v )f(v )dv, (2.2) R and, σ s an arbtrary postve constant related to the temperature of the bath. All the more, σ v 2 f represents the thermal reservor, where σ s lnked to the temperature. Let us defne ρ and u f as the mass and the mean velocty respectvely ρ = f(v )dv, u f = 1/ρ f(v )v dv. (2.3) R 2.1 Propertes of the contnuous model The propertes of ths model are the conservaton of mass and momentum, the decay of the energy for (σ = 0) and of the entropy: Defnton 2.1 Let us defne the temperature and the entropy by respectvely T (f) = v u f 2 f(v)dv, (2.4) v E = σ[ln(f(v))]f(v)dv + 1 v v 3 f(v )f(v)dv dv, (2.5) 6 v v and let be E α (f) and H α (v), defned by E α (f) = v v α f(v )f(v)dv dv, (2.6) R H f,α (v) = v v α f(v )dv. (2.7) v We shall omt the dependency of H wth respect to f when there are no ambguty. Propertes Then, we get f E 3 (f) = 2 H 3 (f, v)dv, (2.8) v F = 2 v v f(v )dv = 2H 1 (v), (2.9) 3 v H 3 = F. (2.10) 3

4 These propertes can be easly checked, by splttng the ntegrals nto v > v and v > v. The exstence and unqueness of an equlbrum state are gven n [4]. The strct convexty of E, follows from the followng result that wll be also useful to prove the entropy decay for an mplct scheme: Lemme 2.2 For any functon f n = {f : (1 + v 4 )f L 1 (R), f = 0 and vf(v)dv = 0}, E 3 (f) 0. Proof. Let f C0, and ntegrable. We have: H 3 (f, v) = f(v ) v v 3 dv = f(v v ) v 3 dv, and, dfferentatng wth respect to v v H 3 (f, v) = 3 f(v ) v v (v v )dv = v f(v v ) v 3 dv, whch gves the followng dentty 2 vh 3 (f, v) = 6H 3 (f, v) = H 3 ( 2 vf, v), Then, E 1 (f) = f(v)h 1 (f, v)dv = 1 f(v) v 2 6 H 3(f, v)dv = 1 v f(v) v H 3 (h, v)dv = E 3( v f(v)). Moreover v 2H 1(f, v) = 2f(v), E 1 ( v f) = f 2 dv < 0, and then E 1 ( v f) = 1 v 2 6 fh 3( v 2 f)dv = 1 6 E 3( v 2 f), so we have E 3 ( 2 v f) = f 2 (v)dv > 0, and 2 v f. Snce for any g C, wth compact support and such that g = 0 and vg(v)dv = 0, we can fnd a functon f C, ntegrable and such that g = 2 vf thus, E 3 (g) = 6 f 2 (v)dv. By densty, the result hold for any functon of. For sake of smplcty, we omt the ndex and set E = E 3 and H = H 3 n the remnder. 4

5 2.2 Pure Granular We suppose σ = 0 n (2.1), and let f = f(t, v) be a functon so that: By the defnton of F we have We can wrte the weak formulaton: t f = v (F f). (2.11) v F fdv = 0. (2.12) Proposton 2.3 Let f be a soluton of (2.11). Then, f satsfes for all test functons φ: t fφ = v φf f = v φ v v (v v )f(v)f(v )dvdv = 1 ( v φ v φ ) v v (v v )f(v)f(v )dvdv (2.13) 2 by symmetry of v and v. We can obtan also the weak form t fφ = 1 ( v (φ) v (φ )) (F F )ff dvdv (2.14) 2ρ These equatons verfy mass and momentum conservaton, and entropy decay. Proof. Takng respectvely φ = 1, v, v 2 n (2.13), we get respectvely mass and momentum conservaton, and the temperature decay. For φ = v 2 and ρ = 1 t fφ = 1 2 (2v 2v ) v v (v v )f(v)f(v )dvdv = v v (v v ) 2 f(v)f(v )dvdv = E(f). So d t v 2 f 0 mples the temperature decay, and the equlbrum corresponds to E = 0. Moreover, usng 3 v H = F, and d f E = 2 H we obtan for φ = 3H(v): d t E = 3 t fh = (1/2) (F (v) F (v )) v v (v v )f(v)f(v )dvdv. Note that F s strctly ncreasng: v F = 2 v v f(v )dv > 0. (2.15) Therefore (F (v) F (v ))(v v ) 0, and d t E 0. Thus we prove the decay of entropy). If we suppose that there s conservaton,.e. d t E = 0, as F s strctly ncreasng, we get f(v) = 0 and by the conservaton of mass, f(v) s of the form ρ δ(v u). 5

6 2.3 Case of the granular mmersed n a thermal bath Let us now consder the case σ 0 n the equaton (2.1). Let M defne as follows v M = F M, σ (2.16).e. M(v) = C exp( 1 H(v)). 3σ (2.17) We rewrte the equaton (2.1) t f = v (σm v (f/m)) = σ ( ) ρ v fm v (f/m) fm v (f/m) dv, v where ρ = f(v)dv denotes the mass and because the second term s zero by ntegratng by part and usng 2.16 and 2.12: M v (f/m) dv = v M (f/m) dv = 1 F (v )f(v )dv = 0 6σ Then, we can wrte the followng weak symmetrzed form of (2.1): t fφdv = σ ( v φ v φ ) ff ( v log(f/m) v log(f/m) ) dvdv. 2ρ (2.18) The mass and momentum conservaton, as the decay of entropy, are obvous, by choosng φ = 1, v, log(f/m), respectvely. Note that the above formulaton has a structure close to the Fokker-Planck- Landau one s, n the so called log form, that has been studed by the authors [7, 2]. The only terms n the above expresson that can be modfed wthout changng the propertes of the operator (conservaton of mass, momentum and the decay of entropy) s the product ff. 3 Dscretzaton In ths secton, we ntroduce a dscrete verson of the equaton (2.1), that degenerates correctly to the pure granular equaton, when σ 0,.e. an asymptotc preservng scheme. Moreover, ths scheme must conserve the propertes of conservaton of the mass, and momentum, as the decay of entropy. We frst deal wth the pure granular case, whch brngs up the followng problem: fnd a scheme that preserve the decay of entropy. Intally desgned for the Fokker-Planck lnear equatons, the method of Chang- Cooper (cf. [16]) allowed us to buld such a dscretzaton. For all the dfferent schemes, we consder an unform grd n velocty: v = v, 6

7 for = 1, N. The macroscopc quanttes are defned usng standard quadrature formula: for the mass ρ = f v, the momentum ρu f = f v v, and the temperature T = f (v u f ) 2 v. 3.1 Case of pure granular: σ = 0 Ths secton s devoted to the lmtng case σ = 0. We present two dfferent methods Frst method Let us frst consder the more natural dscretzed verson of the granular equaton (2.11). The term F s dscretzed usng a standard quadrature formula F = j v v j (v v j )f(v j ) v. (3.1) Ths satsfes F f = 0, (3.2) by symmetry and F +1 F ( v) 2 ρ and thus t s strctly ncreasng. Indeed, as the grd s unform, one gets: v +1 v = v, v +1 v j > v v j v, thus F +1 F v( j v +1 v j f j v) v j (v v j )f j v v j ( v +1 v j (v v j ))f j v = v j K,j f j v 1. f v +1 > v j then K,j = v > 0, 2. f v +1 < v j then K,j = 2v j v +1 v = 2(v j v +1 ) + v > v. 7

8 So we get the nequalty F +1 F ( v) 2 ρ. We consder the followng upwnd scheme for ths transport equaton (n the velocty varable): t f φ = 1 v ((φ +1 φ )F + (φ φ 1 )F + )f. (3.3) Let us menton that the N terms F can be computed n O(N) operatons usng a splttng of the sum nto the j before and after v that reduce the complexty of ther evaluaton. Note also that there exsts some ndex 0 such that F < 0 for all < 0 and F > 0 for all > 0 (ths ndex can move n tme). Moreover, we have F 1 < 0 and F N > 0 and thus, no boundary condton are needed: we don t have to prescrbe the value of f 0 of f N+1. The mass and momentum are conserved takng φ = 1, v n (3.3) and usng the condton (3.2). The evoluton of the temperature s dt dt = 6E ( v) (F F + )f. The second term of the rght hand sde s postve and one cannot conclude about the decay of the temperature. For the dscrete entropy, E, the same concluson holds. However, dscrete steady state for ths scheme exsts: consder state such that 0 wth f = 0 for all > 0 +1 or < 0. The values of the dstrbuton functon on the two non vanshng ponts and the value of 0 are determned from mass and momentum conservaton: Second method f f 0 = ρ, v 0 +1f v 0 f 0 = ρu f. (3.4) We defne the dscrete analog of H(v) and the entropy E (2.5) by and H = 1 v v j 3 f j v, 3 j E = 1 H f v. 2 A second dscretzaton can be obtaned n the pure granular equaton. By ntegraton by parts, we get: φ v (F f)dv = v φ(v) v v (v v )f(v)f(v )dv dv. 8

9 The analogous dscrete weak formulaton s t f φ v = ( v) 2 Dφ j f f j v v j (v v j ), (3.5),j where the fnte dfference operator at pont s uncentered n a drecton dependng on the pont j as Dφ j = φ +1 φ v 1 <j + φ φ 1 1 >j. (3.6) v Ths partcular choce can be physcally nterpreted gong back to a Boltzmann quas-elastc monodmensonnal model. Frst, a symmetry can be output of (3.5), to obtan t f φ v = 1 2 ( v)2,j ( Dφ j Dφ j) f f j v v j (v v j ). (3.7) Let us consder two partcles wth velocty v and v j (wth < j for example). The only elastc collson between these 2 partcles consst n swappng the both, but ths does not change the dstrbuton functon. If one allows slghtly non elastc collsons but preservng the momentum,.e. the postcollsonal veloctes have the same average (the partcle have the same mass) as before the collson. Therefore, the less nelastc post-collsonal veloctes are (v +1, v j 1 ) and (v 1, v j+1 ). To decrease the energy of the system s equvalent to decrease the relatve velocty, so the veloctes must get closer. The latter corresponds to a ncreasng of energy and s not physcally relevant. The choce whch corresponds to a mnmal decrease of energy, s thus (v, v j ) (v +1, v j 1 ) (for < j). Proposton 3.1 The scheme (3.6)-(3.5) preserves mass, and momentum, and the decay of energy and of entropy E = 1 2 H f dv wth H = 1 3 j v v j 3 f j v. Proof. Standng (φ = 1) n (3.6), one gets mass conservaton. For momentum, let (φ = v ), (3.6) wrte as 1 <j + 1 >j, and (3.5) s null. Usng (3.7) wth (φ = v 2 ) one gves: 2 dt dt = ( v) 2 <j +( v) 2 >j (v v 2 j 1 v 2 v 2 j )f f j v v j (v v j ) (3.8) (v v 2 j+1 v 2 v 2 j )f f j v v j (v v j ) 0. For the entropy, one obtans: 2 d dt E = ( v)2 (H +1 H (H j H j 1 ))f f j v v j (v v j ) <j +( v) 2 (H 1 + H j+1 H H j )f f j v v j (v v j ). >j 9

10 Usng the convexty of w w a 3, one fnd that the sequence H +1 H s ncreasng n or equvalently n v and then d dt E 0. Lemme 3.2 The system (3.5) has a global postve soluton, for any postve ntal data (f 0 > 0, for all ). Proof. The exstence of a soluton for ths sem-dscretzed system can be easly obtaned usng a Cauchy-Lpschtz theorem for small tme. Ths soluton s global n tme usng a mnoraton of the soluton by retanng only the loss term and the mass conservaton (that provdes an upperbound). Indeed, f f s soluton, the correspondng dscrete veloctes system for each s t f = ( v)[f 1 <j f j(v 1 v j ) 2 + f +1 >j f j(v +1 v j ) 2 ] ( v)f j f j(v v j ) 2. Moreover, ( v) j f j (v v j ) 2 < K, where K s a constant whch depends of the length of the doman, and of the conserved quanttes (mass and momentum). Thus, wth f = ρ =constant, we get ρ f (t) f 0 exp( Kt). Therefore, the maxmal soluton s global n tme. Equlbrum state for ths scheme: Lemme 3.3 dt = 0 f and only f f dt = 0 for < 0 and > 0 + 1, 0 and f 0, f 0 +1 are determned usng mass and momentum conservaton. Proof. We start from (3.8). The converse s obvously true. For the drect mplcaton, let f be such that the rght hand sde of (3.8) s null, but wth a fxed mass and momentum. Snce the mass s not null, they necessary exst an ndex 0 such that f 0 > 0. Snce n the r.h.s. of (3.8), all the terms n factor of f f j are non postve except for j = 1 orj = + 1, ths leads to f = 0 for < 0 1 and > Now, snce the term n factor of f 0 1f 0 +1 s also negatve thus one of f 0 1 and f 0 +1 s null. Equaton (3.4) determnes 0 and the value of f 0 f If one compares the two schemes proposed here, the frst one s uncentered at a macroscopc level that depends only on the ntegrated value of F whereas, the second conssts n a mcroscopc uncentered scheme where only the physcally relevant collson are allowed. 10

11 3.2 Chang-Cooper method for a Fokker-Planck lnear equaton Before usng the Chang-Cooper method for the granular meda, we brefly recall ths method on a more smple lnear Fokker-Planck equaton: set F = v n (2.1), t f = v (vf + σ v f), (3.9) that can be put under the form t f = σ v (M v (f/m)), where M s a Maxwellan, and M(v) = exp( v 2 /2σ). Orgnally, the method of Chang-Cooper was desgned to preserve equlbrum state of Fokker-Planck equaton. Another nterestng feature of ths method s ts ablty to degenerate correctly to an upwnd scheme, for the equaton of convecton, when σ 0, wth velocty grd fxed. Moreover, t s an entropy decayng method and not just an equlbrum state preservng method. We set M = M(v ). Ths method can be wrtten as t f = F +1/2 F 1/2, v wth F +1/2 = σ v M +1/2 ( f +1 M +1 f M ), where thus M +1/2 = M M +1 M +1 M (ln M +1 ln M ), F +1/2 = σ v (f +1 f ) + σ v ( 1 + M +1/2 )f +1 + σ M +1 v (1 M +1/2 )f. M Analyze each term: σ v (1 M +1/2 M ) = σ v (1 M +1 M +1 M (ln M +1 ln M )) = σ v ln M +1 M ( 1 ln M +1 ln M M +1 M +1 M ) = σ v ln M +1 Settng w = ln(m /M +1 ), then we get: 1 M ( 1 ). ln M M 1 M +1 M +1 σ v (1 M +1/2 M ) = σ v w( 1 w 1 exp w 1 ) = σ v wh(w) = σ v wθ, 11

12 where θ = h(w) and h s the functon h(x) = 1 x 1 e x 1. Now, h s postve on R, decreasng and vares between 0 and 1. Smlarly, the second term reads: and h( w) = h(w) + 1, thus Then σ v ( 1 + M +1/2 ) = σ v wh( w), M +1 σ v ( 1 + M +1/2 ) = σ w(θ 1). M +1 v F +1/2 = σ v (f +1 f ) + σ v ( w(θ 1))f +1 + σ v (wθ)f = σ v (f +1 f ) + σw v (θf + (1 θ)f +1 ). We get an upwnd scheme, mxed wth a θ-scheme : ths s the Chang- Cooper method. It s easy to verfy that ths dscretzaton correspond to the dscretzaton of the weak formulaton of the FPL equaton t f φ = σ(dφ) +1/2 (M +1/2 D(f/M) +1/2 ), (3.10) where Dg stands for the centered fnte dfference.e. D(g) +1/2 = (g +1 g )/ v and the coeffcents M +1/2 s an average of the value between M and M +1 to be defned. Such type of scheme s by constructon entropy decayng and provde the good equlbrum state, see [2]. By takng M +1/2 as M +1/2 = M M +1 M +1 M log(m +1 /M ), (3.11) ths gves the Chang-Cooper scheme. Ths scheme degenerates toward an upwnd scheme for t f = v (vf) when σ 0. Indeed, the scheme (3.10) wth the choce (3.11) reads: usng t f φ = 2σ(D φ) +1/2 log(m +1 /M ) M +1 M (f +1 M M +1 f ), σ log(m +1 /M ) = v v = vv +1/2,

13 wth v +1/2 = (v +1 + v )/2. Moreover, for such that v > 0 M M +1 M 1, as σ 0 whereas t tends to 0 f v < 0. Thus the lmt of the scheme as σ 0 s t f = v + +1/2 (f +1 f ) + v 1/2 (f f 1 ) t f = 2v +1/2 (f +1 f )1 v >0 + 2v 1/2 (f f 1 )1 v 1 <0 denotes the postve/negatve part of x.e. x ± = (x ± x )/ Dscretzaton wth σ 0 Takng the granular equaton (2.1), we look for a dscretzaton smlar to the FPl one s, based on the Chang-Cooper method. Moreover, ths scheme must preserve the propertes of conservaton and of decay, and degenerate to the equaton of pure granular, when σ A dscretzaton based on the symmetrc form We defne the dscrete analog of H(v) and the entropy E (2.5) by and E = σ H = 1 v v j 3 f j v, 3 j f ln(f )dv + 1 H f v. 2 We defne also a dscrete verson of (2.17) by ( M = exp H ). σ From the H s we defne dscrete of F by F +1/2 = H +1 H, v and usng the convexty of the functon w w a 3 one can verfy easly that F +1/2 F 1/2 thus the sequence F +1/2 s ncreasng n. One can remark that by splttng the sum n two parts j and j than the H s can be evaluated n O(n) operatons as explaned by the authors n ([11, 12]) for a smlar equaton, the sotropc Fokker-Planck-Landau equaton. 13

14 In order to preserve mass momentum and the decay of the entropy we dscretze the weak-symmetrzed form (2.18), t fφdv = σ ( v φ v φ ) ff ( v log(f/m) v log(f/m) ) dvdv, 2ρ as follows σ (Dφ +1/2 Dφ j+1/2 )f +1/2 f j+1/2 (3.12) 2ρ j (D log(f/m) +1/2 D log(f/m) j+1/2 ) where Dg +1/2 = (g +1 g )/ v and the averaged value of the product f +1/2 f j+1/2 has to be defned. On ths form conservaton of mass and mean velocty and the decay of the dscrete entropy E are easly verfed choosng φ = 1, v, H respectvely. We shall now defne the product f +1/2 f j+1/2 n such a way that, we wll have the much smplest scheme as possble, that degenerate correctly when σ go to 0, and for whch the collson term can be evaluated at the lower cost as possble, that s n O(n). Moreover, approxmatng f +1/2 f j+1/2 by any postve formulae allows to nsure conservaton of the mass, the mean velocty and the decayng of the entropy. The frst dea s to use the so called entropc average, see ([8]), that permts to get rd of the log term ( ) (f/m)j D(f/M) +1/2 (f/m) D(f/M) j+1/2 f +1/2 f j+1/2 = M +1/2 M j+1/2 D log(f/m) +1/2 D log(f/m) j+1/2 (3.13) Usng ths choce, one get σ 2ρ (Dφ +1/2 Dφ j+1/2 )M +1/2 M j+1/2 j ((f/m) j D(f/M) +1/2 (f/m) D(f/M) j+1/2 ), (3.14) that can be related to the non-log form of the Fokker-Planck equaton: (f/m) j D(f/M) +1/2 (f/m) D(f/M) j+1/2 = (f/m) j (f/m) +1 (f/m) (f/m) j+1. We defne the product M +1/2 M j+1/2 n such way that the scheme reduces to a good scheme when σ 0, and usng the analyss done for the lnear Fokker-Planck equaton, we choose C j = M +1/2 M j+1/2 = M +1M j M M j+1 M +1 M j M M j+1 log(m +1 M j /M M j+1 ). (3.15) 14

15 The ODE system has the followng structure t f = σ ( f+1 f j C j f ) ( j+1f f 1 f j + C 1,j f ) j+1f M j +1 M j M j+1 M M 1 M j M j+1 M (3.16) wth C j gven by (3.15). On ths form, the cost to evaluate the coeffcents of the dfferental system s quadratc. In order to recover a lnear cost, let us smplfy the expresson (3.15) for M +1 M j > M j+1 M.e. wth z j = H j+1 H j + H H +1 > 0. One approxmates Then, 1 M j+1 M M +1 M j = 1 M j+1 M (1 exp( z j /σ)) 1 M j+1 M (z j /σ). C j M j M +1 (1 + σ z j ) z j σ, (3.17) and dong the same type of approxmaton for j one obtan the dscrete weak formulaton n t f φ = 1 n 1 (Dφ +1/2 Dφ j+1/2 )( f +1f j f j+1f ) 2ρ v M +1 M j M j+1 M =0,j=0 (M j+1 M (σ + z j ) 1 {j>} + M +1 M j (σ z j ) 1 {j<} ), (3.18) and as F j+1/2 F +1/2 := H j+1 H j v n t f φ = σ ρ v =0 n 1,j=0 ( f +1f j M +1 M j H +1 H v = z j, we deduce v Dφ +1/2 (M j+1 M 1 {j>} + M +1 M j 1 {j<} ) f j+1f M j+1 M ) 1 ρ n 1,j=0 Dφ +1/2 (F j+1/2 F +1/2 ) (M j+1 M 1 {j>} M +1 M j 1 {j<} )( f +1f j M +1 M j f j+1f M j+1 M ). (3.19) Usng the choce (3.17) for the coeffcents C j, one gets a trdagonal matrx: for = 1,..., n 1 and the boundary terms are, t f = a +1 f +1 (a + b )f + b 1 f 1, (3.20) t f 0 = a 1 f 1 b 0 f 0 (3.21) t f n = a n f n + b n 1 f n 1, (3.22) 15

16 where the coeffcents a, and b are all postves and defned by the followng relatons, wth, for = 1,..., n: a = 1 ρ v [ M 1 ( σ M v F n 1 1/2) j= 1 M j+1 f j + (F 1/2 + σ 1 M j v ) j=0 f j + M 1 M and for = 0,..., n 1: n 1 j= 1 M j+1 1 f j F j+1/2 f j F j+1/2 ], (3.23) M j j=0 b = 1 ρ v [( σ v F n 1 +1/2) n 1 + j= j= f j+1 F j+1/2 M +1 M f j+1 + M +1 M (F +1/2 + σ v ) j=0 j=0 M j M j+1 f j+1 M j M j+1 f j+1 F j+1/2 ]. (3.24) Note that the coeffcents a and b can be evaluated n O(n) steps. From the equaton (3.20)-(3.22), we note that the system can be wrtten as a dfference of flows: t f = G +1/2 G 1/2, where G +1/2 = a +1 f +1 b f, the conservaton of the mass s obvous. Propertes Ths scheme conserves mass and momentum (φ = v ) for the system (3.19), and decays of the dscrete entropy E. Proof. By constructon the approxmaton of f +1/2 f j+1/2 s postve. Thus startng from (3.12) f we set φ = 1 and v respectvely we obtan the conservaton of the mass ρ and the mean velocty u. For the entropy, as for the contnuous model we have d dt E = ( Thus by taken φ = ln df dt ) f M ( ln f + H ) σ = df dt ln ( f M ). n (3.12) we have evdently d dt E 0. If f s such that f = M thus we have d E = 0. For the exstence of such dt equlbrum, we refer to [4], ther proof remans vald f we have a dscrete measure nstead of the Lebesgue measure. 16

17 3.3.3 Lmt when σ 0: a thrd dscretzaton If one consders the lmt σ 0 n the scheme ( ), one gets the form based on the weak formulaton ff (F F ) nstead of v v (v v ). Ths scheme degenerates correctly when σ 0, ndeed, z j s postve, defned by So, when σ 0, exp z j σ n =0 t f φ v = 1 n 1 2ρ.,j=0 exp z j σ = M j+1m M j M and the scheme becomes ( Dφ+1/2 Dφ j+1/2 ) (F +1/2 F j+1/2 )(f j+1 f.1 {j>} + f +1 f j.1 {j<} )( v) 2, (3.25) whch s a dscrete form of the weak symmetrzed form (2.14) of the pure granular operator. Propertes The system lmt conserves mass and momentum, and we have the decay of energy and of the dscrete entropy. Equlbrum states are such that f = 0 for < 0 and > 0 + 1, and 0, f 0, f 0 +1 are determned from mass and momentum, see 3.4. Proof. By takng φ = 1, v n (3.25), mass and momentum conservaton are obtaned. The sequence F +1/2 s ncreasng n and takng φ = v 2 we have the decay of the temperature t T = v n 1 ρ.,j=0 ( v+1/2 v j+1/2 ) (F +1/2 F j+1/2 ) (fj+1 f.1 {j>} + f +1 f j.1 {j<} )( v) 2, (3.26) where v +1/2 = v +v The sequence v +1/2 s also ncreasng n and then all the terms ( v +1/2 v j+1/2 ) ( F+1/2 F j+1/2 ) are non negatve, whch gves the result. For the entropy we have t E = 1 n 1 ρ.,j=0 ( F+1/2 F j+1/2 ) 2 (f j+1 f.1 {j>} + f +1 f j.1 {j<} )( v) 2. (3.27) Concernng the equlbrum, we start from the entropy producton term (3.27). Frst let us verfy that the sequence F +1/2 s a strctly ncreasng sequence n. By the defnton of F +1/2 we have F +1/2 F 1/2 = 1 v (H +1 + H 1 2H ), = 1 ( f j v+1 v j 3 + v 1 v j 3 2 v v j 3). v j 17

18 On can verfy easly that we have the lowerbound ( v+1 v j 3 + v 1 v j 3 2 v v j 3) ( v) 2 so that F +1/2 F 1/2 ( v) 2 ρ. (3.28) Consder f such that the rght hand sde of (3.27) s null. Snce the mass ρ of f s not null there exst 0 such that f 0 > 0. All the terms nvolvng n the rght hand sde of (3.27) are null snce they have all the same sgn. Usng then (3.28) we obtan that f j = 0 for j > and j < 0 1. And = 0 1 and j = 0 gves f 0 +1f 0 1 = 0. In other words, one of the both terms s null. That shows that f s as we have clamed t: the values of 0, f 0 and f 0 +1 are unquely determned by the values of the mass and the momentum. 3.4 Tme dscretzaton We shall present the tme dscretzaton of system n the form (3.20)-(3.22) Explct scheme In the explct case, the system can be wrtten: f n+1 f n = t ( ) a n +1f+1 n c n f n + b n 1f 1 n.e. f n+1 = Id + t f n+1 f n = tq n f n c n 0 a n b n 0 c n 1 a n b n n 2 c n n 1 a n n b n n 1 c n n f n, where c n = a n + bn. A suffcent condton to have a postve matrx s to take t satsfyng 1 c t > 0 for all. Therefore, we choose t max = (sup c ) 1. ( ) Notce that t max behave lke O. v 2 σ+( v)sup F +1/2 The propertes of conservaton of mass and momentum, of the system can be checked easly (φ = 1, v ). 18

19 3.4.2 Implct scheme The mplct scheme reads:.e. f n+1 f n = ta n+1 +1 f n+1 +1 tcn+1 f n+1 + tb n+1 1 f n+1 1, f n = (Id tq(f n+1 ))f n+1, where Q(f n+1 ) s the prevous matrx, at range n + 1, nstead of n. We use an teratve method: f n = (Id tq(f n ))f n+1 and we defne the process as: (Id tq(g p ))g p+1 = f n g 0 = f n to resolve t, where g p s the p eme terate. At each teraton, mass and postvty are preserved. The conservaton of mass follows from structure of the matrx Q, whch corresponds to a conservatve scheme. The last pont follow from a classcal algebra result [6]: Let N be a matrx R n R n, trdagonal, whose dagonal coeffcents are postves, the others negatves. Then N 1 s postve, f and only f, there exsts a dagonal postve matrx D, such that D 1 ND s dagonal domnant. N = (Id tq(g p )), take G Ker(Q) postve whch s always possble, then N G = G, where G s the dagonal postve matrx formed wth the coeffcents of G. Thus G 1 N G = Id whch s a dagonal domnant matrx. Ths concludes the proof. One can also notce that f f n s an equlbrum then g p = f n. Lemme 3.4 For the mplct scheme the entropy s decayng. Proof. The entropy E can be wrtten as: E(f) = σ (f ln f + 1 v v 3 f(v)f(v )dv )dv = σ 6 (ln f ln N)fdv, where N(v) = exp( 1 6σ v v f(v )dv ) = exp( 3H(v)/2σ). We get E n+1 E n = σ ( f n+1 ln( f N )n+1 dv f n ln( f N )n dv ) = σ ( f n (ln( f N )n+1 ln( f N )n )dv + t Q n+1 f n+1 ln( f N )n+1 dv ), wth Q n+1 = Q(f n+1 ), ( E n+1 E n σ f n ln( f n+1 f ) n f n ln( N n+1 N n )dv + t ) Q n+1 f n+1 ln(n) n+1 dv. 19

20 Usng the entropy decay on the weak form (2.18) wth φ = ln(f/m) (M = N 2 ), one gets E n+1 E n σ ( v f n ln(n) n dv + v f n+1 ln(n) n+1 dv 2 v f n ln(n) n+1 dv ) σ ( v f n (ln(n) n ln(n) n+1 )dv + v (f n+1 f n ) ln(n) n+1 dv ). By defnton of N, we ve: E n+1 E n 1 v v 3 [f n (f n+1 f n ) f n+1 (f n+1 f n )]dvdv, 2 v v 1 v v 3 [(f n+1 f n )(f n+1 f n )]dvdv. 2 v v Snce from lemma (2.2) the applcaton g g(v)g(v ) v v 3 dvdv s postve for functons g such that g = vg = 0, we have the decay of entropy. The proof for the dscrete formulaton follow the same lnes: ) ( E n+1 E n σ f n (ln(n ) n ln(n ) n+1 ) v + The defnton of N gves: E n+1 E n 1 2 (f n+1 f n ) ln(n ) n+1 v v v j 3 [(f n+1 j fj n )(f n+1 f n )]( v) 2. (3.29) j By a dscrete verson of the lemma (2.2), whch can be obtaned by regularzng the sum of Drac mass, we conclude to the decay of entropy.. 4 Numercal results We llustrate the method on two tests, a Drac and a Gaussan for ntal data. For the two tests the velocty domans s [0, 12.5], the total mass s one and the two ntal data are centered at v = 4. For the gaussan the varance s 1. For all, the run of the tme step s taken equals to 0.1. We use two unform grds of 50 and 200 ponts. For the explct scheme, we use sub cyclng technc to respect the CFL constrant that guarantee postvty of the scheme. For the mplct scheme the teratve process s stopped when the error on the relatve velocty s smaller than

21 4.1 Computatonal cost Let us frst compare the computatonal cost between explct and mplct. The costs correspond to σ = 1. For the mplct scheme the number of terates vares from 12, at the frst tme step, to 1 after few tme steps when startng from the gaussan, and from 5 to 1 when startng from the Drac, and ths wth the two grds. The number of terates seems to does not depend of the number of ponts. For the explct scheme, the number of sub cycles s nearly constant durng the computaton. Startng from the gaussan or the Drac we need 60 subcycles wth 50 ponts and 283 subcycles wth 200 ponts. The cost of the teratons s the same for the mplct and the explct. More the grd s fne, larger s the cost rato explct/mplct. For very coarse grd (ten ponts) the cost of the explct scheme when startng from the gaussan s smaller than the cost of the mplct but only for the frst tme steps of the relaxaton. One nterestng feature of the mplct scheme concerns t s ablty to reach an equlbrum state, despte the fact that the teratve process s stopped, when the error on the relatve velocty s smaller than O(10 n ). Ths mplementaton of the mplct scheme, strctly speakng, s not conservatve for the mean velocty, but we observe that for any value of the allowed error, n long tme, the scheme converge toward an equlbrum state for whch, the mean velocty s nearly those of the ntal condton up to an error of O(10 n ). We plot the dfference between the average velocty after total relaxaton and the ntal one, n functon of the error, for the two types of ntal condton, a Drac mass and a Gaussan, see Fgure (1) ntal condton gaussan gaussan Drac 1e-005 1e-005 1e-006 1e-006 1e-007 1e-007 1e-008 1e-008 1e-009 1e-009 1e-010 1e-010 1e-011 1e-011 1e-012 1e e Fgure 1: Gaussan and Drac ntal condton: dfference between the average velocty after total relaxaton and the ntal one n functon of the error, n log scale 21

22 4.2 Graphcs Fgure (2) shows the evoluton n tme of the temperature startng from the Drac mass, wth 50 meshes, σ = 1 and explct and mplct scheme. Fgure (3) shows the evoluton of the temperature startng from the Gaussan, wth 50 and 200 grd ponts and agan wth σ = 1 and usng explct and mplct verson of the scheme. Startng from the Gaussan, we can compare for σ = 1 and σ = the evoluton of the dstrbuton functon on the fgure (4) and the relaxaton of the temperature (6). Implct scheme s used wth 50 grd ponts. We can also show on fgure (5), the relaxaton of the temperature for the two ntal condton when σ = 1, usng 50 grd ponts and mplct scheme. 0,6 T 1,5 entropy 0,5 0,4 0,3 explct mplct 1 0,5 explct mplct 0,2 0 0,1 0-0,5-0, temps temps Fgure 2: Intal condton: Drac mass. Evoluton of the temperature and the entropy, 50 ponts, σ = ,9 0,8 0,7 0,6 0, temps T explct 50 explct 200 mplct 50 mplct 200-0,66-0,68-0,7-0,72-0,74-0,76-0,78-0,8 entropy explct 50 explct 200 mplct 50 mlct 200-0, temps Fgure 3: Intal condton: Gaussan. Evoluton of the temperature and the entropy, 50 and 200 ponts, σ = 1. To fnsh let us show the behavor of the equlbrum state f : we compare the gaussan exp( v 2 /2) wth the equlbrum state obtaned wth 22

23 0,5 0,4 gaussan_mp_f50_s3 t= e+00 2,5 2 dstrbuton functon, sgma= t= e+00 0,3 t= e-01 1,5 t= e-01 0,2 t= e+00 1 t= e+00 0,1 t= e+01 0,5 t= e , v -0, v Fgure 4: Intal condton: Gaussan. Evoluton of the dstrbuton functon, 50 ponts, σ = 1 and σ = T 0,8 0,6 0,4 0,2 0 gaussan 50 Drac 50-0, temps Fgure 5: Evoluton of the temperature startng from Drac mass and Gaussan, σ = 1, 50 ponts. ths ntal data and correspondng to σ = 1 and whch behave for large v as exp( a v 3 ). We plot log f n functon of v, n logarthmc scale, see Fgure 7. It s s easly seen on Fgure (7 that for large velocty, the dstrbuton functon has the expected behavor, the slope of log f(v) tends to 3 for the steady states of granular meda, and to to for maxwellan. 5 Concluson In 1D, wthout frcton term, the Chang-Cooper method allows us to construct a dscretzaton for the granular equaton. Moreover, the dscretzaton s performed at a lnear cost n the number of grd ponts. An mplct scheme has been used. It converges n large tme to the equlbrum state descrbed n [4], and t preserves the propertes of the equa- 23

24 1,4 T 1,2 1 0,8 sgma= sgma=1. sgma=3. 0,6 0,4 0, temps Fgure 6: Evoluton of the temperature startng from Gaussan wth dfferent values of the temperature of the thermal bath, σ = 1 and σ = 10 5, 50 ponts gaussan L=2000, 4000 meshes t=0 totally relaxed x*x/2 x*x*x/ Fgure 7: The dstrbuton functon n log scale for a gaussan and the equlbrum state assocated for σ = 1. ton. Moreover, the cost of such mplct scheme s less than the explct one. Ths mplct method could also be used for the numercal methods descrbed n ([11, 12]) for the sotropc Fokker-Planck-Landau equaton. In ths study, the frcton term was gnored. Snce the dscretzaton s process s appled to a symmetrc form of the equaton, the frcton term cannot be ncluded wthn the presented framework. However, t should be possble to take t nto account usng splttng techncs. 24

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