格子Boltzmann方法求解Burgers方程_英文_

格子Boltzmann方法求解Burgers方程_英文_ L ATTICE BOL TZMA NN M ETHOD FO R BURGERS EQUATIO N SH EN Zhi2jun , YU AN Guang2wei , SH EN Lo ng2jun ( L aboratory of Com p ut at ional Physics , )I nst i t ute of A p plied Physics a n d Com p ut at ional M at he m at ics , Bei ji n g 100088 , P R Chi na () Abstract It is well known t hat lat tice Boltzmann met hodsLBMmake great success in many co m2 p utatio nal p hysics fields , expecially in fluid mechanics. A lat tice Boltzmann met hod wit h B GK model is developed to solve Burgers equatio n. Detailed analysis shows t hat t he calculating scheme is a t hree level no nlinear finite difference o ne. The maximum value p rinciple has been p roved and t he existence ,unique2 ness and stability are also discussed. The co mp utatio nal result s agree wit h seco nd order finite difference solutio ns very well . Ke y word word文档格式规范word作业纸小票打印word模板word简历模板免费word简历 s lat tice Boltzmann ; Burgers equatio n ; stability Clcnumber O24 ;O351 Document code A 0 Introduct ion () The lat tice Boltzmann met ho d LBMis a numerical scheme based o n kinetic t heo ry fo r mo d2 1 ,2 eling vario us mat hematical2p hysical p ro blems. It s micro scopic evolutio n can be viewed as aspace2time2velocit y discretizatio n of t he Boltzmann equatio n . U sually it s behavio r is simple but it s aim is to recover p hysical macro dynamics via t he simple means. Since it s geo met rically flexible , t rivially parallel ,numerically efficient and easy to co de ,LBM has made great success in many co m2 p utatio nal p hysics fields and extensively been used to solve vario us p ro blems ,including multip hase 1 ,2 flow ,chemical reacting flow setc. It is regretf ul t hat until now t here are lit tle knowledge abo ut numerical analysis in t heo ry of above mentio ned general p ro blems ,fo r example , stabilit y and co nvergence excep t so me numerical experiment s. Fo r certain specific classes of lat tice Boltzmann met ho ds ,fo r example ,solving fo r lin2 ear and no nlinear co nvective2diff usive equatio n , t here are so me co nvergence and stabilit y result s 3 given by Elto n B Het al . Alt ho ugh Elto n et al . have perfo r med a LB calculatio n to Burgers equatio n , t heir met ho d is derived f ro m real micro scopic behavio r ,in ot her wo rd ,lat tice gas ,and t he bridge bet ween micro2 Received date 1999209206 ( ) Foundation item Subsided by t he Special Funds fo r Majo r State Basic Research Project G1999032801 , t he Natio nal ( ) Nat ural Science Fo undatio n of China 19932010and t he Fo undatio n of L CP. ) ( Bio gra phy SHEN Zhi2jun 1966,, male , Liao ning , Engineer , Research directio n is numerical solutio ns of particle t rans2 scopic and macro scopic is Hilbert expansio n . It is too co mplicated and difficult fo r co nst ructing scheme and perfo r ming analysis. We int ro duce a simpler lat tice Boltzmann met ho d w hich base o n B GK mo del and Chap man2 Enskog expansio n fo r co mp uting solutio ns of 1D and 2D no nlinerar co nvective2diff usive equatio ns. In t his paper we o nly co nsider 1D Burgers equatio n ,namely ,t he visco us Burgers equatio n , ( )ρρρρ1 + = v x xt x ρ( ) ρ( )( )x , 0= x 2 0 ρ( ) w here v is visco us coefficient and here we assume | x | ?1 . 0 It is generally recognized t hat LBM are finite difference schemes of Boltzmman equatio n t hat has higher o rder discretizatio n erro r . We develop t his met ho d wit h t he point of view above but at t he same time ,we also regard t he LBM wit h B GK mo del as finite difference met ho d fo r macro2 scopic equatio n and analyze it s characters. We find such LB scheme is a t hree level no nlinear finite difference o ne ,w hich satisfies Maximum value p rinciple ,t herefo re we co mplete t he p roof of existe2 ness ,uniqueness and stabilit y. This paper is o rganized as follow s : a LBM wit h B GK mo del is int ro duced to calculate t he Burgers equatio n next sectio n . A stability analysis to t he LBM is given in t he seco nd sectio n . Al2 t ho ugh t he scheme is not mo noto nic ,it is co nservative and satisfies maximum value p rinciple . Fi2 nally we co mpare t he numerical result s wit h t ho se co mp uted by a standard seco nd o rder explicit co nservative ,mo noto nic finite difference met ho d. U sing different parameters , we can see t hat all t he result s f ro m LBM agree wit h t ho se f ro m difference met ho d very well . 1 L BM Method No r mal lat tice Boltzmann met ho ds aim at fluid mechanics ,so it s time scale is same wit h spa2 tial scale . We must increase time scale to seco nd o rder w hen we mo del Burgers equatio n . A spatial lat tice do main X is divided into small segment grids wit h micro scopic space Δsteplengt h x . A unit vecto r is assigned fo r each directio n , e= + 1 , e= - 1 . 1 2 We define o ur lat tice Boltzmann met ho d eq ( ) ( )f x , t - f x , t i i 2 ( εε) ( ) ε( )( )f x + ce, t + T - f x , t = - + hx , t 3 i i i i τ eq τ w here f is particle dist ributio n , f is equilibrium dist ributio n ,is a co nstant to mo dif y visco us i i coefficient , T is t ypical time , 2 ρ( ) + dx , t , 1 i = ( ) ( )hx , t =4 i 2 ρ( ) - dx , t , i = 2 ρ () w here d is a co nstant and is same wit h w hich in Eq. 1. J ust as general LBM , eqρ( ) ( )x , t = f =5 f i i 6 6 i i eq When f and coefficient s above are cho sen , we can employ Chap man2Enskog expansio n of i () ( ) t he evolutio n equatio n 2to match t he Burgers equatio n 1. Here we deter mine equilibrium dis2 t ributio n eq ( ) ρ( ) x , t = x , t 2 ( )/ 6 f i By using Taylo r expansio n ,we have 0 5 f f - f i1 2 i i2 22 ε( ) εεε)( ) ( ? f = - T + ce ? f + ce+ h x , t 7 i i i i i τ 5 t 2 eq 1 εL et f = f +f + ?, i i i (ε) O 1 eq τ( )ce? h = - f / 8 f -i i i i 2 ε)(O eq 2 5 f f 1 i i 1 2 2 eq ( )) ( T 9 + ce? f + c e? f = -i i i i τ 5 t 2 () () i ,we have Adding 8and 9wit h respect to 2 2 τρ 5 c 1 2 2 eq eq ( )( ) = 0 e?ce?2 hce?- f - + f i i i i i i6 6 5 t T 2 T i = 1 i = 1 2 2 ρ5 τρc 1 2 dc 5 2 τ (ρ - -( )) += 0 10 T 5 t T 5 x 2 () To recover Eq. 1,just let 2 τ1 c2 dc 1 (τ )( )-= v ; = 11 2 2 T T Once t hese coefficient s are deter mined ,let 2 Δεt = T, Δε x = cwe can co mp ute t he result s immediately. n ( ) To recapit ulate in difference notatio n ,we denote f = f x , t , i , j i j n Δ 1 1t1 n +n n2 n ) ρ)( (ρ = 1 - f + f + j , j j 1 1 , j +1 τ Δττ 4x2 Δ n +1 1 1tn n( )2 n12 ) ρf f (ρ ( ) = 1 - - + 2 , j jj 2 , j - 1 τ Δττ 24x n +1 n +1 n +1 ρ+ f f = 1 , j 2 , jj w here Δ 1 1 vt0 0 0 τ ρf ( )f = , = + 13 = j 1 , j 2 , j22 2 Δx subject to initial co nditio n 0 ρρ( )= x ( )14 0 j j 2 Sta bil ity Anal ysis 1 ? ( ) L ?L space .In t his sectio n ,we will p rove lat tice Boltzmann schemes 12are stable in ρ( ) Fo r t he sake of simplificatio n , we assume | x | ?1 . 0 j At first , n +1 n +1 n +1 ρ= f + f j 1 , j 2 , j Δ 1 1 tnnn 2 nnn 2 ( ) ( ) (ρ ρ) (ρ ) (ρ) ) ( = 1 - f + f + + - + 1 , j - 1 2 , j +1 j - 1 j +1 j - 1 j +1 τΔττ 24x nnn n( ρρ) ( )?H f , f ,, 15 1 , j - 1 2 , j - 1 j - 1 j +1 τTo t he end of po sitive scheme ,we must let ?1 . That means 2 ΔΔ( )2 vt / x ?1 16 In additio n ,we need ΔΔ( )t / x ?1 17 ( ) ( ) Lemma 1 . When parameters satisf y 16 , 17 and - 1 ?u , u?1 , t hen ( )H u , u, u, u 3 4 1 2 3 4 mo noto nically no ndecrease wit h respect to it s argument s. Proof . Δ5 H 1 t= + u 3 τ Δτ5 u22x 3 Δ5 H 1 t= - u 4τ Δτ5 u22x 4 U nder given co nditio ns ,co nclusio ns are o bvio us. () () Remark 1 . In f act ,co nditio ns 16and 17just guarantee L emman 1 to hold. ) () ( ρ( ) Lemma 2 . maximum value p rincipleIf | x | ?1 and steplengt h rest rict co nditio ns 16and 0 () 17hold ,t hen 0 0 n +1 ρρρ( )min?| | ?max, n ?0 18 l j l l l Proof . fo r Π j , k belo ng to integer set 1 1 n +1 n +1 nnn n ( ) ( )f (ρρ)= 1 - f + f + f + + 1 , j - 1 2 , k +1 j - 1 k +1 1 , j 2 , kτ τ2 Δ tn 2 n 2 ρ) )(ρ) ( (+ - k +1 j - 1 Δτ4x Because 0 0 0 ρf = / 2 = f 1 , j 2 , j j we have Δ t1 2 1 1 0 000 2 ρ)(ρ( (ρ ) ρ) ) (= + - f + + f j - 1 k +1 j - 1 k +1 1 , j 2 , k Δτ24x Δ 1 t0 000ρρρ(ρ ) + + | - ? | j - 1 k +1 j - 1 k +1 Δτ 2 2x so 0 0 0 1 1 0 0 0ρ(ρρ(ρρρmin?min ,) ? f + f ?max ,) ?max j - 1 k +1 1 , j 2 , k j - 1 k +1 l l l l and 0 1 0ρρρmin??max l j l l l () Suppo se t his relatio ns 18is also co rrect fo r n ,by L emma 1 , n +1 n +1 n n 0 0 ( ρρ)+ f ? H f , f , max , max f 1 , j 2 , k 1 , j - 1 2 , k +1 l l l l 1 1 nn0 ( ) ( = 1 - f ) ρ+ f + max 1 , j - 1 2 , k +1 l τ τ l 0ρ?max l l and n +1 n +1 0 f ρ ?min+ f l1 , j 2 , kl This co mpletes t he p roof of L emma 2 . Summary all above ,we have () () Theorem 1 . U nder t he steplengt h rest rictio n co nditio ns of 16and 17and initial co nditio n 0 ρ( ) ( ) || ?1 , t he solutio n of t he schemes 12 , 14 exist s and is unique , f urt her mo re , t here is j n ρ| | ?1 . j Now we t ur n to p roof of stabilit y. ρ( ρ( ) ρ( ) ) Suppo se `x , t is anot her solutio n w hich satisfies initial co ndito n ` x , 0= `x , and t 0 ρ( ) () he initial co nditio n satisfies | `x | ?1 . We empoly same schemes 12and same steplengt h re2 st 0 () () rictio n co nditio ns 16, 17to co mp ute it . L et ( ) ( ) u ? v = min u , v . u ? v = max u , v , () () Lemma 3 . U nder co nditio ns 16and 17, 0 0ρρρ( )?`? ?` 19 j jj j 6 j j 0 0ρ( )ρρ? ?`20 ?` jj j j 6 j j Proof . We just p rove , Δ t 1 1n +1 n +1 n n 2 n nnn ? f ( ) ρ(ρ )ρρ? 1 - f f ? `f + + ?? ` `1 , j +1 1 , j +11 , j 1 , j jjj j τ τ Δτ24x , Δ 1 1tnn +1 n +1 n2 n n n n ) ρ( (ρ )ρρ? f ? 1 - f ? `f + ?`- ?` f 2 , j - 1 2 , j +12 , j j2 , j j j j τ τ Δτ2 4x t hen n +1 n +1 n +1 n +1 ρ( )?`? f ? `f + f ? `f j j 1 , j 1 , j 2 , j 2 , j 6 j j 1 1 n nn nnn ( ρρ( )) f ? `f + f ? `f ?` ?1 - + 1 , j 1 , j 2 , j 2 , j j j 6 6 τ τ j j Since 0 0 0 0 00ρ( ) ρf ? `f + f ? `f = ?` 1 , j 1 , j 2 , j 2 , j j j 6 6 j j () It is know n t hat t he co nclusio n is right fo r n = 0 ,using inductio n ,we can p rove 19. () Similarly ,we can also p rove 20. In additio n n +1 n +1 n +1 n +1 n +1 n +1 ρρρρρρ| - | = ?- ?``` j j j j j j n +1 n +1n +1 n +1ρρρρρ?- `| = - `?` j j j j j j 66j j j 000 0 ρρρρ? ?-? ``j j j j 6 6 j j 0 0 ρρ= | - `| j j 6 j 1 () () () ( ) Theorem 2 . L at tice Boltzmann schemes 12, 13and 14is stable in t he meaning L R un2 () () der steplengt h rest rictio n co nditio ns 16, 17. In additio n ,f ro m n n n n n n ρρ)( + f =+ - f + f f 1 , j - 1 2 , j +1 j - 1 j +1 1 , j +1 2 , j - 1 n nn - 1 ρρρ=+ - j - 1 j +1 j () We rew rite schemes 15as 1 1 n +1 n n - 1 n nn ρρρ)ρ ) (ρ) (ρ( + - + = 1 - + j - 1 j +1 jj - 1 j +1 j τ τ2 Δ tn 2 n 2 (ρ) )ρ) ( (+ -( ) 21 j - 1 j +1 τΔ4x Obvio usly it is t hree level centered difference scheme . It is easy to show t hat it s t runcate erro r 2 Δ t2 2 (ΔΔ) is O x + t + O . Δ x 3 Computat ions ( ) ( ( ) We co mp ute solutio ns of Eq. 1o n t he do main t , x ?0 , T ] ×[ 0 , L . The bo undary ρ( ) ( π) co nditoins are perio dic ,initial co nditio n is x , 0= - co s 2,and t he diff usio n coefficient is x - 5 v = 2 . () () Since t he number of coefficient s in scheme 2are greater t han t he number defined in 11, t here are so me f reedo ms fo r t he cho sen of t he coefficient s. Fo r example ,we can let τ ( ) T = 8 , c = 1/ 4 , d = 8 22 = 1 , τ ( )= 3/ 2 , T = 16 , c = 1/ 4 , d = 32/ 3 23 In o rder to co mpare o ur result s wit h ot hers ,we co nsider a mo noto ne explicit finite difference scheme Δ tm +1 m 2 m 2 m ρ(ρ)ρρ) (=- -[ ] i i +1 i - 1 i Δ4x ( )24 Δ vt mm m ρρρ+ [-2 + ] i +1 i - 1 i2(Δ) x This is a 2 nd o rder co nservative difference scheme ,t he 2 Δ(Δ) stabilit y co nditio n is t ?x / 2 v . ( ) ( )In f act , Case 1 22 is just t he same wit h 24 2 Δ(Δ) w hen t = x / 2 v , so we just co mp ute Case 2 . Fig11 Initial value Fig. 1 show s it s initial value , Fig. 2 is in t = 1/ 4 . We can see t hese t wo result s agree wit h rat her well . ( ) Remark 2 . The stabilit y co nditio n 16 is reaso nable . U sually t he step sizes of difference Δvt () ( ) schemes satisf y = O 1w hen solving Burgers equatio n 1,co mp utatio n will be stable p rovid2Δ x Δed t hat x is small eno ugh . Fig12 LBM and Difference 13 LBM and Difference Fig ττsolutio ns at t = 1/ 4 ,= 110 solutio ns at t = 1/ 4 ,= 019 () ()Remark 3 . Fro m t he following result s of numerical test s ,we know co nditio ns 16and 17 are o nly sufficient . but necessary co nditio ns are undeter mined since t hey depend o n initial values. L et ( )τ 8 , c = 5 / 8 , d = 32 5/ 9 25 T = = 019 , The numerical result s are show n in Fig. 3 . Ref erences 1 Grunall D , Chen S , Eggert K. A lat rice Boltzmann model for multip hase fluid flows J . Phys Fl ui ds , 1993 ,5 :2557 . Dawso n S P , Chen S , Doolen G D. L at tice Boltzmann Co mp utatio ns for reactio n2diff usin equatio n J . J 2 Chem Phys , 1993 ,98 :1514 . Elto n B H ,L evermore C D , Rodrigure G H. Co nvergence of co nvective2Diff usio n lat tice Boltzmann met hods 3 J . S IA M J N u mer A n al , 1995 ,32 :1327 . 格子 Boltzmann 方法求解 Burgers 方程 沈智军 , 袁光伟 , 沈隆钧 ( )北京应用物理与计算数学研究所计算物理实验室 ,北京 100088 () [ 摘 要 ] 众所周知 ,格子方法 包括格子气和格子 Boltzmann 方法在计算物理领域取得巨大进 展 。与之形成鲜明对比 ,格子方法的数学理论始终处于停滞不前的状况 。为求解 Burgers 方程 ,一 类带有 B GK 模型格子方法被构造出来 ,经过变量替换 ,发现他们属于三层非线性差分方法 。使用 极值原理 ,给出此类格式稳定性的严格证明 。最后 ,从数值实验中可以看出 ,使用 LBM 得到的结 果 ,与经典二阶守恒差分方法的结果符合得非常好 。 [ 关键词 ] 格子 Boltzmann ;Burgers 方程 ;稳定性