Research Article A Multilevel Finite Difference Scheme for One-Dimensional Burgers Equation Derived from the Lattice Boltzmann Method

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1 Appled Mathematcs Volume 01, Artcle ID 9590, 13 pages do: /01/9590 Research Artcle A Multlevel Fnte Dfference Scheme for One-Dmensonal Burgers Equaton Derved from the Lattce Boltzmann Method Qaoe L, Zhoushun Zheng, Shuang Wang, and Jankang Lu School of Mathematcs and Statstcs, Central South Unversty, Changsha , Chna Correspondence should be addressed to Zhoushun Zheng, 009zhengzhoushun@163.com Receved 13 February 01; Revsed 8 March 01; Accepted 8 March 01 Academc Edtor: June We Copyrght q 01 Qaoe L et al. Ths s an open access artcle dstrbuted under the Creatve Commons Attrbuton Lcense, whch permts unrestrcted use, dstrbuton, and reproducton n any medum, provded the orgnal work s properly cted. An explct fnte dfference scheme for one-dmensonal Burgers equaton s derved from the lattce Boltzmann method. The system of the lattce Boltzmann equatons for the dstrbuton of the fcttous partcles s rewrtten as a three-level fnte dfference equaton. The scheme s monotonc and satsfes maxmum value prncple; therefore, the stablty s proved. Numercal solutons have been compared wth the exact solutons reported n prevous studes. The L, L and Root-Mean- Square RMS errors n the solutons show that the scheme s accurate and effectve. 1. Introducton The lattce Boltzmann method LBM has been ntroduced as a new computatonal tool for the study of flud dynamcs and systems governed by partal dfferental equatons. It has made a rapd development n theory and applcaton over the last couple of decades snce ts ncepton 1 4. Ths method can be ether regarded as an extenson of the lattce gas automaton 5 or as a specal dscrete form of the Boltzmann equaton for knetc theory 6. The lattce Boltzmann models can also be used as partal dfferental equaton PDE solvers. By choosng approprate collson operator or equlbrum dstrbuton, the lattce Boltzmann model s able to recover the PDE of nterest. Recently, t has been developed to smulate lnear and nonlnear PDE such as Laplace equaton 7, Posson equaton 8, 9, the shallow water equaton 10, Burgers equaton 11, Korteweg-de Vres equaton 1, Wave equaton 13, 14, reacton-dffuson equaton 15, 16, and convecton-dffuson equaton 17, 18. The numercal schemes based on the LBM are gven as a system of two-level explct dfference equatons composed of the dstrbuton functons of fcttous partcles for each drecton n whch the partcles move. For one-dmensonal advecton-dffuson problems,

2 Appled Mathematcs Ancona 19 showed that the LB schemes wth the velocty model D1Q whch ncludes two veloctes wth speed 1 n opposte drectons to each other can be rewrtten as the DuFort- Frankel scheme 0 whch s a second-order three-level dfference scheme. Ths shows that the accuracy of the LB schemes based on the model D1Q s dentcal to that of the DuFort- Frankel scheme. Suga 1 have proposed a four-level explct fnte dfference scheme for 1D dffuson equaton whch s derved from the lattce Boltzmann method wth rest partcles. The consstency analyss of the scheme shows that the two parameters whch appear n the scheme, the relaxaton parameter and the amount of rest partcles, can be determned such that the scheme has the truncaton error of fourth order. In spte of the vast and successful applcatons, the numercal stablty of the method has not been well understood. For certan specfc class of lattce Boltzmann methods, for example, solvng for lnear and nonlnear convectve-dffusve equaton, there are some convergence and stablty results gven by Elton et al.. Many works have been developed on lattce Boltzmann method to the Burgers equaton n one or hgher dmenson 3 5. In those papers, the standard lattce Boltzmann method was used and the macroscopc quanttes were computed by the dstrbuton functon. However, those models are suffered from the stablty. In ths paper, we derve a three-level dfference scheme for 1D Burgers equaton based on the model D1Q from the LB schemes. It s generally recognzed that LBM s a fnte dfference scheme of Boltzmann equaton that has hgher-order dscretzaton error. We develop ths method wth the pont of vew above, but, at the same tme, we also regard the LBM wth BGK model as fnte dfference method for macroscopc equaton. We fnd such LB scheme s a three-level fnte dfference one, whch s monotonc and satsfes maxmum value prncple; therefore, we complete the proof of stablty. The rest of the paper s organzed as follows. Secton descrbes the LB scheme wth the velocty model D1Q and derves the three-level fnte dfference scheme whch s equvalent to the LB scheme. A stablty analyss of the scheme s gven n Secton 3. In Secton 4, numercal solutons are compared wth exact solutons reported n prevous studes. And the conclusons are gven n the end.. The Three-Level Fnte Dfference Scheme for 1D Burgers Equaton Based on the LB Schemes The one-dmensonal Burgers equaton take the followng form: u t u u x ν u x,.1 wth the ntal condton u x, 0 u 0 x. Here, the vscous coeffcent ν 1/ Re, Re s the Reynolds number. Hstorcally,.1 was frst ntroduced by Bateman 6 who gave ts steady solutons. It was later treated by Burgers 7 as a mathematcal model for turbulence and after whom such an equaton s wdely referred to as Burgers equaton. For a small value of ν, Burgers equaton behaves merely as hyperbolc partal dfferental equaton and the problem becomes very dffcult to solve as a steep shock-lke wave fronts developed.

3 Appled Mathematcs 3 c c Fgure 1: D1Q model wth two veloctes n one dmenson..1. The Lattce Boltzmann Scheme Accordng to the theory of the LBM, t conssts of two steps: 1 streamng, where each partcle moves to the nearest node n the drecton of ts velocty; colldng, whch occurs when partcles arrvng at a node nteract and possbly change ther velocty drectons accordng to scatterng rules. Fcttous partcles are ntroduced at each of the mesh ponts x Δx...,, 1, 0, 1,,..., and they move wth the velocty c determned by the D1Q model from x to the neghborng mesh pont whch was shown n Fgure 1. The lattce Boltzmann schemes are establshed on grds wth two drectons c 1,c 1 c, c,. where c Δx/Δt s the speed n the system. Let f x, t denote the dstrbuton functon of the partcles movng wth velocty c. So the tme evoluton of the dstrbuton functon f x, t s gven by the followng lattce Boltzmann equaton LBE based on the Bhatnagar- Gross-Krook BGK model: f x c Δt, t Δt f x, t 1 f x, t f eq x, t,.3 where f eq x, t s the local equlbrum dstrbuton functon of partcles and s the dmensonless relaxaton tme whch controls the rate of approach to equlbrum. The change n the dstrbuton functon produced by the collson of partcles s approxmated by the second term on the rght-hand sde of.3. The macroscopc velocty u x, t s defned n terms of the dstrbuton functon as u x, t f x, t f eq x, t..4 In ths paper, f eq x, t are determned as to satsfy.4 and the followng condtons: c f eq c c f eq x, t u x, t, x, t c u x, t..5 Solvng these equatons determnes the equlbrum dstrbuton functons f eq u x, t 1 x, t f eq u x, t 1 x, t u x, t Δt, 4Δx u x, t Δt. 4Δx.6

4 4 Appled Mathematcs Applyng the Chapman-Enskog expanson 4 yelds the above Burgers equaton.1 from the LBE and the equlbrum dstrbuton functons gven by.3 and.6, respectvely. The vscosty ν s defned by ν 1/ Δx /Δt... The Multlevel Fnte Dfference Scheme Now, we let f n, denote f Δx, nδt and let u n denote u Δx, nδt. We note that the subscrpt, combnes nformaton about the channel or drecton of propagaton 1, 1 and locaton denotes a grd node. Usng the equlbrum dstrbuton functon.6, the lattce Boltzmann equaton.3 can be rewrtten by classcal fnte dfferent notaton f n 1 1, f n 1, 1 un f n 1 1, f n 1, 1 un Δt, u n.7 4Δx Δt. u n.8 4Δx Accordng to.4, the macroscopc velocty can be computed by f n 1 1, f n 1 1, u n 1 un 1 Δt 4Δx H f n 1, 1,fn 1, 1,un 1,un 1. f n 1, 1 f n 1, 1 u n 1 u n 1.9 In addton, f n 1, 1 f n 1, 1 u n 1 f n 1, 1 u n 1 f n 1, 1 u n 1 un 1 f n 1, 1 f n 1, 1,.10 whle f n 1, 1 f n 1, f n 1 1, f n 1 1, f n 1 1, f n 1 1, 1 1 u n 1. un 1 un 1 u n 1 1 un 1 Δt 4Δx Δt 4Δx 1 un 1 u n 1 u n 1.11

5 Appled Mathematcs 5 Then,.10 becomes f n 1, 1 f n 1, 1 un 1 un 1 un 1..1 Substtute.1 to.9, we fnally obtan the followng three-level explct fnte dfference scheme 1 1 u n 1 un 1 un 1 1 u n 1 un 1 Δt u n 1 u n Δx 3. Stablty Analyss In ths secton, assumed the ntal value u 0 x s bounded and smooth enough, we wll prove the multlevel fnte dfference scheme s stable n L 1 L space. Suppose u 0 x L 1, u 0 x It s not dffcult to see that, f u n 1and 1, Δt Δx 1, 3. then the scheme.9 s monotonc ncrease. 1 means νδt Δx Now, we wll pont out that the soluton of the scheme.13 satsfes the maxmum value prncple. Lemma 3.1 maxmum value prncple. If ntal value u 0 x 1 and the restrctons 3. hold, then, for all Z, there are mn u 0 l l max u 0 l, n l

6 6 Appled Mathematcs Proof. It s known that f we take f 0 1, u0 /, f0 1, u0 /, and un L max u n,un S mn u n Z, then, for all, k Z, f 1 1, f 1 1,k f H 0 1, 1,f0 1,k 1,u0 1,u0 k 1 H u0 1, u0 k 1,u0 1,u0 k 1 u 0 L H, u0 L,u0 L,u0 L 1 1 u 0 L u0 L u 0 L, 1 u 0 L u0 L Δt 4Δx u 0 L u 0 L 3.5 and smlarly f 1 1, f 1 1,k H u0 1, u0 k 1,u0 1,u0 k 1 u 0 S H, u0 S,u0 S,u0 S u 0 S. 3.6 If we suppose u 0 S f n 1, f n 1,k u0 L s also correct. Partcularly k, we have u0 S un u 0 L, then f n 1 1, f n 1 1,k f H n 1, 1,fn 1,k 1,un 1,un k 1 H f n 1, 1,fn 1,k 1,u0 L,u0 L 1 1 f n 1, 1 f n 1,k 1 1 u0 L u 0 L. 3.7 Smlarly, we get f n 1 1, f n 1 1,k u0 S. 3.8 Let k, we can get mn u 0 l l max u 0 l, n l

7 Appled Mathematcs 7 Assume that ũ x, t s another soluton of.1 wth subect to ntal condton ũ x, 0 ũ 0 x, and the ntal condton satsfes ũ 0 x 1. Usng the same scheme.13 and same restrcton condton 3., we have the followng. Lemma 3.. If the condtons of Lemma 3.1 are fulflled, there are nequaltes max mn, ũ n 1, ũ n 1 max u 0, ũ0, mn u 0, ũ Denote that u n Δx {un, Z} s the dscrete soluton of LBE.7.9 at tme nδt, and u n Δx L 1 un Δx s the L1 norm of dscrete functon u n Δx. Then, the soluton s stable n the meanng of L 1. Theorem 3.3. If u n Δx, ũn Δx are the solutons of.13, u0 Δx, ũ0 Δx L1 R wth subect to the correspondng ntal condtons 3.1 and restrctons 3., then there are u n Δx ũn Δx L 1 u n Δx L 1 u 0 Δx ũ0 Δx, L 1. L 1 u 0 Δx Proof. Consder ũ n 1 max, ũ n 1 mn, ũ n Summng the absolute value to all,bylemma 3., we have ũ n 1 max, ũ n 1 max u 0, ũ0 mn, ũ n 1 mn u 0, ũ0 u 0 ũ If we let ũ Δx x, t 0n 3.11, we can get 3.1. Remark 3.4. The restrcton 3. s suffcent but not necessary. 4. Numercal Experments Example 4.1. We nvestgate the accuracy of the scheme by solvng.1 on the doman t, x 0,T 0, 1. The ntal condton s u x, 0 sn πx, 0 x 1, and the homogenous

8 8 Appled Mathematcs boundary condton s u 0,t u 1,t 0. In ths case, the exact Fourer soluton s gven by 8 n 1 u x, t πν a n exp n π νt n sn nπx a 0 n 1 a n exp n π νt cos nπx, 4.1 where a n a 0 exp πν 1 1 cos πx dx, 0 exp πν 1 1 cos πx cos nπx dx, n 1,, In comparson wth the analytcal solutons, the effcency of proposed model s valdated. The followng error norms are used to measure the accuracy: 1 L -error e L n 1/ e, L -error e L Max e, 1 n, The root mean square RMS error e RMS n 1 e 1/. 4.5 n The numercal solutons of.1, whch are computed by usng dfferent step sze at tme T 0.1 forν 1, are gven n Table 1. The above error norms are gven n Table for dfferent mesh sze. From Table, we fnd that the accuracy measured n L,L and RMS norm errors ncreases as the step sze decrease. The numercal solutons are n the symmetrc pattern as the exact solutons are. Table 3 and Fgure 1 show a comparson between numercal and exact solutons at dfferent tmes for ν The curves for dstrbuton of absolute errors at dfferent tmes are also shown n Fgure. It s known that the Fourer solutons for ν fal to converge because of the slow convergence of the nfnte seres 8. The numercal soluton cures for ν at dfferent tme are drawn n Fgure 3, whch shows the correct physcal behavor.

9 Appled Mathematcs 9 Table 1: Comparson of the LB fnte dfference solutons wth exact soluton at T 0.1forν 1wth 1. x Numercal soluton Exact soluton N 10 N 0 N Table : Error norms for ν 1atT 0.1wthdfferent step sze. N e L e L e RMS E 0.789E E E E E E E E 05 Example 4.. Consder Burgers equaton wth the followng forms: 1 u u u u t x 1 u Re x, 1 x 3,t>0, u x, 0 1 x 1 x tan, Re x 3, [,t 1 1 Re tan Re t 4 Re t [ 1 3 3Re tan Re t 4 Re t 3 u,t ], t > 0, ], t > It possesses the exact soluton 3 u x, t [ ] 1 x Re x tan. 4.7 Re t Re t

10 10 Appled Mathematcs Table 3: Comparson of the LB fnte dfference solutons wth exact soluton for ν wth dx 0.005,dt 0.003, and 1.1 atdfferent tmes. t x Numercal Exact Numercal Exact Numercal Exact t = Ux, t t = 1.4 t = X t =.6 a Numercal solutons Absolute error t = 1.4 t =.0 t = X b Absolute errors Fgure : Numercal solutons a and dstrbuton of absolute errors b for ν at dfferent tmes wth dx 0.005, 1.1, and dt In the computaton, we compare the result wth the D1Q and D1Q3 lattce Boltzmann model whose equlbrum dstrbuton functons are taken as f eq u x, t 1 x, t f eq u x, t x, t u x, t, 4c u x, t, 4c f eq 0 x, t u x, t, 3 f eq u x, t 1 x, t 6 f eq u x, t x, t 6 u x, t, 4c u x, t. 4c 4.8

11 Appled Mathematcs 11 Ux, t t = 0 t = 0. t = 0.4 t = 0.8 t = 1.4 t = Fgure 3: Numercal solutons for ν 0.001, at dfferent tmes wth dx 0.001, 1anddt X Ux, t Exact soluton Our model Fgure 4: Comparson of the exact soluton and our model. Parameters are: Re 500, dx 0.01, dt 0.00, 1. X Let Re 500, we gve the results of our model, and exact soluton as Fgure 4 at t 0.4. Table 4 shows the results of the D1Q, D1Q3, our model and the exact soluton at dfferent lattce at tme t 0.4. The global relatve errors u E x,t u N x,t GRE un x,t, 4.9 whch are used to measure the accuracy are presented n Table 5. From Fgure 4 and Table 4, we fnd that the D1Q, D1Q3, and our model are all n excellent agreement wth the exact solutons. The accuracy of the multlevel fnte dfference model s even hgher than the D1Q and D1Q3 model. It should be ponted out that n order to

12 1 Appled Mathematcs Table 4: Comparson of the results wth D1Q, D1Q3, our model, and exact soluton. x D1Q model D1Q3 model Our model Exact soluton Table 5: Global relatve errors wth dfferent models. D1Q model D1Q3 model Our model GRE 3.383E E E 04 attan better accuracy, the LB model requres a relatvely small tme step Δt but the multlevel fnte dfference model does not have ths restrcton. 5. Concluson In the current study, a three-level explct fnte dfference scheme for 1D Burgers equaton s derved by rewrtng the LB scheme. Furthermore, t s proved that the scheme s condtonally stable. The effcency and accuracy of the proposed scheme are valdated through detal numercal smulaton. It can be found that the numercal solutons are n excellent agreement wth the analytcal solutons. In order to derve LB scheme n a hgher dmenson, the LBM wth the multspeed velocty model wll be useful, n whch dfferent free parameters wll be assgned for dfferent values of the speed. Applcaton of our method to D and 3D equatons s left for future work. Acknowledgments Ths work was supported by the Natonal Natural Scence Foundaton of Chna no and Natonal Basc Research Program of Chna 011CB References 1 U. Frsch, B. Hasslacher, and Y. Pomeau, Lattce-gas automata for the Naver-Stokes equaton, Physcal Revew Letters, vol. 56, no. 14, pp , S. Chen and G. D. Doolen, Lattce Boltzmann method for flud flows, n Annual Revew of Flud Mechancs, vol. 30 of Annual Revew of Flud Mechancs, pp , Annual Revews, Palo Alto, Calf, USA, R. Benz, S. Succ, and M. Vergassola, The lattce Boltzmann equaton: theory and applcatons, Physcs Report, vol., no. 3, pp , 199.

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