Applied Mathematics, 2011, 2, 14791485 doi:10.4236/am.2011.212210 Published Online December 2011 (http://www.SciRP.org/journal/am) Copyright © 2011 SciRes. AM Numerical Solution of Nonlinear KleinGordon Equation Using Lattice Boltzmann Method Qiaojie Li1, Zong Ji2, Zhoushun Zheng1, Hongjuan Liu1 1School of Mathematical Science and Computing Technology, Central South University, Changsha, China 2Modern Service and Trade College, Yunnan University of Finance and Economics, Kunming, China Email: qiaojie_li@foxmail.com Received October 24, 2011; revised November 24, 2011; accepted December 1, 2011 Abstract In this paper, in order to extend the lattice Boltzmann method to deal with more nonlinear equations, a one dimensional (1D) lattice Boltzmann scheme with an amending function for the nonlinear KleinGordon equation is proposed. With the Taylor and ChapmanEnskog expansion, the nonlinear KleinGordon equa tion is recovered correctly from the lattice Boltzmann equation. The method is applied on some test exam ples, and the numerical results have been compared with the analytical solutions or the numerical solutions reported in previous studies. The L2, L∞ and RootMeanSquare (RMS) errors in the solutions show the effi ciency of the method computationally. Keywords: Lattice Boltzmann, ChapmanEnskog Expansion, Nonlinear KleinGordon Equation 1. Introduction Nonlinear phenomena modeled by partial differential equation appear in many areas of scientific fields such as solid state physics, plasma physics, fluid dynamics, ma thematical biology and chemical kinetics. The nonlinear KleinGordon equation has attracted much attention in studying solutions and condensed mater physics, in in vestigating the interaction of solitons in a collisionless plasma, the recurrence of initial states, and in examining the nonlinear wave equations [1,2]. In the last few de cades, many powerful methods, such as the inverse sca ttering method, Baklund transformation, the auxiliary equation method [3,4], the Wadati trace method, Hirota bilinear forms, the tanhsech method, the sinecosine me thod, Jacobi elliptic functions, and the Riccati equation expansion method were used to investigate these types of equations (see [5] and references therein). A variety of finite difference scheme have been presented (see [6] and references therein) and the alternative approaches using spectral and pseudospectral methods have recently been presented [7,8]. To avoid the mesh generation, meshless techniques have attracted the attention of researchers in recent years. The radial basis function (RBF) as a truly meshless method was used to solve nonlinear KleinGor don equation in [9]. Recently, unlike convectional numerical methods which search for the macroscopic equation, the lattice Boltzmann method (LBM) has achieved much success in studying nonlinear equations and the evolution of complex systems [10,11]. By choosing appropriate collision or equilibrium distribution, the lattice Boltzmann model is able to recover the PDE of interest. This method is a new technique based on a mesoscopic kinetic equation for the particle distri bution functions. Compared with the conventional nume rical methods, the LBM provides many of the advantages, including geometrical flexibility, clear physical pictures, ease in incorporating complex boundary conditions, simpli city of programming and numerical efficiency. Recently, it has been developed to simulate linear and nonlinear PDE such as Laplace equation [12], Poisson equation [13,14], the shallow water equation [15], Burgers equation [16], Kortewegde Vires equation [17], Wave equation [18,19], reactiondiffusion equation [20,21], convectiondiffusion equation [2224]. In this paper, the initialvalue problem of the onedimen sional nonlinear KleinGordon equations is given by the following equation, () (,) tt xx uugufxt (1) where (,)uuxt represents the wave displacement at position x and time t, is a known constant and () u is the nonlinear force. The present work is motivated by the desire to extend
Q. J. LI ET AL. 1480 the lattice Boltzmann method to deal with evolution models characterized by nonlinear wave dispersion. By using Taylor expansion and the ChapmanEnskog expan sion, the secondorder nonlinear KleinGordon equation can be recovered from the present model correctly. The local equilibrium distribution function and the amending function are obtained. To make a comparison between numerical solutions and analytical ones, four KleinGor don equations with quadratic or cubic nonlinearity are considered. From the simulations, we find that the nu merical results are in excellent agreement with the ana lytical solutions. This indicates that the present method is an efficient and flexible approach for practical applica tion. The organization of the paper as follows. In Section 2, the lattice Boltzmann model is described. Numerical examples are simulated in Section 3. Summary and con clusion are presented in Section 4. 2. The Lattice Boltzmann Model The lattice Boltzmann model used on this study is the threevelocity lattice BhatnagarGrossKrook (LBGK) model. The directions of the discrete velocity are defined as () i c0,1, 2i 012 [,, ][0,,]ccc cc. where c is a constant. The lattice Boltzmann equation with an amending function is given as follow ,(,) 1(,) (,)(,) ii i eq ii i fxcttt fxt xtfxttF xt (2) where (,) i xt and i(,) eq xt are defined as the distri bution and equilibrium distribution function, respectively. i(,) xt is an amending function and is the dimen sionless relaxation time. i ct and are the lattice spacing and time step, respectively. t Unlike for the normal LBM, the first derivative of the macroscopic variable meets the following con servation laws (,)uxt (,) (,) (,) i eq i ii uxt fxtf xtt (3) Then, through choosing appropriate local equilibrium distributions, we can retrieve the corresponding macro scopic equation correctly. Indeed, applying the Taylor expansion to lefthand side of Equation (2) and retaining terms up to 3 ()t , we get 2 2 3 2 1 () i iii eq ii t tcf cf xt xt tfft The macroscopic equation can be recovered in the multiscale analysis using a small expansion parameter which is proportional to the ration of the lattice spac ing to the characteristic macroscopic length. To do this, the ChapmanEnskog expansion in time and space is applied: (1)2(2)2(2) 2 12 1 , , eq iiii ii ff fFF tt txx (5) where ()k i and (2) i are the nonequilibrium distribu tion functions and nonequilibrium amending function, which satisfy the solvability conditions () (2) (2) 0( 1) k i i i i fk FF (6) Substituting Equation (5) into (4), we have 2(1)2(2) 11 2 2 2(1) 11 2 (1)2( 2 )2( 2 ) 2 1 eq iii eq iii ii i cfff xt t tcff xt t ff F 2(2) i i f (7) Comparing the two sides of Equation (7) and treating terms in order of and 2 gives (1) 11 1 ():ii eq i cf xt t f (8) 2( 211 2 (2) (2) 11 (): 1 2 iii ii q eq ii e fc f txt tcff xt t 1) F (9) Applying Equation (8) to the left side of Equation (9), we can rewrite Equation (9) as (1) 21 (2) (2) 1 12 1 ii q ii e 1 i cf tx fF t t (10) In order to recover Equation (1), we must give appro priate local equilibrium distribution function. We choose q i e such that, (,) i eq i uxt ft i F (4) 2 0,(, ) eq eq iii ii isi cfccf cuxt (11) Copyright © 2011 SciRes. AM
Q. J. LI ET AL.1481 where 22 3 s cc is called the lattice Boltzmann sound speed. Equation (11) leads to three linear equations for (,) i eq f t. Solving these equations determines the equi librium distribution functions 0 1 2 (,) (,) (,) 3 (,) (,) 6 (,) (,) 6 eq eq eq uxt uxt fxtt uxt fxt uxt fxt (12) Meanwhile, the amending function (,) i xt is taken as (,)(,)(,) () ii i xtF xtfxtgu (13) such that (,) (,) i i xtF xt . For simplicity, only one case is given here 0 1 2 2 (,)(,) () 3 1 (,)(,) () 6 1 (,)(,)() 6 xtf xtgu xtf xtgu xtf xtgu (14) Summing Equation (8) and Equation (10) over i, and using Equation (6) and (11), we obtain 1 0 u tt (15) (1) (2) 21 1 12i i i ucf F tt x (16) Using Equation (8) and (11), we get (1) 11 11 2 1 ()( () eq ii ii eq eq i ii ii ii i s cftccf xt t ccfcf xt tcu x ) i (17) Then substituting Equation (17) into Equation (16), we have 2 21 1 2 s uu ct F tt xx (2) 1 (18) When Equation 2 (15) (18) is applied, the fi nal nonlinear KleinGordon equation is recovered as 22 22 (,)(,) () uu xtf xtgu tx (19) where 21 2 s ct In the computational process, in order to obtain u(x,t), we can apply backward difference to the item (,)uxt t (,)(,)(,)uxtuxtuxtt tt (20) 3. Numerical Simulation Results In this section, we present the result of our LBM numeri cal experiments for the relevant equations. In comparison with the analytical solutions and results derived by exist ing literature, the efficiency of proposed model is vali dated. The distribution function (,) i xt is initialized with i(,) eq xt (,uxt for all nodes at . The macroscopic variable is initialized by the initial condition and the nonequilibrium extrapolation scheme proposed by Guo [25] is used for boundary treatment. The following error norms are used to measure the accuracy 0t ) 1) L2error 1 2 2 2 1 n i i L errore 2) L∞error Max ,1 i Lerrorei n 3) The root mean square (RMS) error 1 22 1 n i i e RMS errorn where (,) (,) ii i euxtuxt , and are the numerical solution and analytical one. (,) i ux t(,) i uxt Example 1. The KleinGordon equation with quadratic nonlinearity in the interval 11 22 cos cos tt xx uux txtu 2 The initial conditions are given by (,0),(,0) 0 t uxx u x The exact solution is given in [9] (,) cosuxt xt We extract the boundary condition from the exact so lution. In Table 1, the L∞, L 2 and RMS errors are ob tained for 1,3, 5,7,10t . The graph of analytical and LBM solution for 1t and are given in Figure 1 and the spacetime graph of the LBM solution is given in Figure 2. 10t Example 2. Consider the nonlinear KleinGordon equ ation with quadratic nonlinearity in interval 01 . Copyright © 2011 SciRes. AM
Q. J. LI ET AL. 1482 Table 1. L∞, L2 and RMS errors with dx = 0.02 and dt = 2 × 10–5. errors t L∞error L 2error RMS 1 1.9558e–03 1.1135e–03 1.1294e–04 3 1.3664e–03 7.6676e–03 7.6295e–04 5 1.5260e–03 8.5602e–03 8.5178e–04 7 1.6201e–03 9.5926e–03 9.5450e–04 10 1.0465e–03 6.9848e–03 6.9501e–04 Figure 1. Analytical and LBM solutions with dx = 0.02 and dt = 2 × 10–5 for different time. Figure 2. Spacetime graph of the LBM solutions up to t = 10 with dx = 0.02 and dt = 2 × 10–5. 22 66 6 tt xx uxtxtxut 2 u The initial conditions are given by (,0) 0,(,0)0 t uxu x The exact solution is given in [9] 33 (,)uxt xt The Boundary condition is determined by the ana lytical solution. In Table 2, the L∞, L2 and RMS errors are obtained for 1, 2,3, 4,5t . The graph of analytical and LBM solution for 5t and the spacetime graph of the LBM solution are given in Figure 3 and Figure 4, respectively. Example 3. The nonlinear KleinGordon equation with cubic nonlinearity in interval 11 . 3 tt xx uu uu We take 2.5,1, 1.5 as the same in [9]. The initial conditions are given by 2 ( ,0)tan(),( ,0)sec() t uxBKxu xBcKKx The exact solution is (,) tanuxtBKxct where B and 2 2 c . In Ta ble 3, the L∞, L2 and RMS errors are obtained for two values of c (0.5c and ) for 0.05c1,2,3,4t . The graph of analytical and LBM solution for 4t and the spacetime graph of the LBM solution for each value of c are given in Figure 5 and Figure 6, respec tively. Example 4. We consider the nonlinear KleinGordon equation with the form [9]. 3;[0,1.28] tt xx uu uux with initial data 2π ( ,0)1cos,( ,0)0 1.28 t x uxAu x The boundary conditions are given by (0, )0,(1.28,)0 xx ut ut Table 2. L∞, L2 and RMS errors with dx = 0.01 and dt = 5 × 10–5. errors t L∞error L 2error RMS 1 5.8742e–04 1.9270e–03 1.9174e–04 2 4.6618e–03 2.1643e–02 2.1535e–03 3 1.5139e–02 4.9465e–02 4.9219e–03 4 3.4225e–02 8.5102e–02 8.4679e–03 5 6.3219e–02 9.3035e–02 1.2970e–02 Copyright © 2011 SciRes. AM
Q. J. LI ET AL.1483 Table 3. L∞, L2 and RMS errors with dx = 0.01 and dt = 5 × 10–5. errors t L∞error L 2error RMS c = 0.5 1 1.4189e–04 6.6508e–04 6.6171e–05 3 4.6601e–04 1.5438e–03 1.5362e–04 3 1.9445e–03 4.9588e–03 4.9342e–04 4 2.8219e–02 7.1870e–02 7.1513e–03 c = 0.05 1 5.6970e–05 2.9718e–04 2.9570e–05 2 7.4878e–05 3.8699e–04 3.8507e–05 3 1.1972e–04 5.2203e–04 5.1944e–05 4 1.4008e–04 4.4143e–04 4.3924e–05 Figure 3. Analytical and LBM solutions at t = 5 with dx = 0.01 and dt = 5 × 10–5. Figure 4. Spacetime graph of the LBM solutions up to t = 5 –5 Figure 5. Analytical and LBM solutions at t = 4 withx = d 0.01 and dt = 5 × 10–5 for different c. Figure 6. Spacetime graph of the LBM solutions up to t = 4 with dx = 0.01 and dt = 5 × 10–5 for different c. with dx = 0.01 and dt = 5 × 10. Copyright © 2011 SciRes. AM
Q. J. LI ET AL. 1484 boundary co 7 . Conclusions the current study, a new lattice Boltzmann model is his work was supported by the National Natural Sci For the above problem due to the periodic nditions, the continuous solutions remain always symmetric with respect to the center of the spatial in terval. Authors of [26] also studied this problem and found undesirable characteristics in some of the nu merical schemes, in particular a loss of spatial symme try and the onset of instability for larger values of the parameter A (amplitude) in the initial condition of the equation. We solved the above problem using lattice Boltzmann method for several values of A. In Figure 7, we show the approximate solutions for A = 1 with 0.0128x and 5 1.8286 10t . Figure 8 pre roxima 100 with 0.0128x and 5 1.8286 10t Figure , we cpatial symmetry is kept for different amplitude A. It indicates that the present lattice Boltzmann method is comparable with other numerical schemes. sents the app and Figure 8 te solutions for A an find that the s . From 4 In proposed to solve 1D nonlinear KleinGordon equation. The efficiency and accuracy of the proposed model are validated through detail numerical simulation with quadratic and cubic nonlinearity. It can be found that the LBGK results are in excellent agreement with the ana lytical solution. It should be point out that in order to attain better accuracy the lattice Boltzmann model re quires a relatively small time step tand the proper range is form 10–4 to 10–6. Detailed stability analysis of present model is needed in further study. 5. Acknowledgements T Figure 7. LBM solution at t = 0, 3, 50 with A = 1, dx = 0.0128 and dt = 1.8286 × 10–5. Figure 8. LBM solution with A = 100 at different time t nce Foundation of China (50874123, 51174236) and . References ] P. J. Caudrey, I. C. Eilbeck and J. D. Gibbon, “The Sine rris, uxiliary Equation Method and New Solu . e National Basic Research Program of China (2011CB60 6306). 6 [1 Gordon Equation as a Model Classical Field Theory,” Nuovo Cimento, Vol. 25, No. 2, 1975, pp. 497511. [2] R. K. Dodd, I. C. Eilbeck, J. D. Gibbon and H. C. Mo “Solitons and Nonlinear Wave Equations,” Academic, London, 1982. [3] Sirendaoreji, “A tions of KleinGordon Equations,” Chaos, Solitons & Fractals, Vol. 31, No. 4, 2007, pp. 943950. doi:10.1016/j.chaos.2005.10.048 [4] Sirendaoreji, “A New Auxiliary Equation and Exact Tra velling Wave Solutions of Nonlinear Equation,” Physics Letters A, Vol. 356, No. 2, 2006, pp. 124130. doi:10.1016/j.physleta.2006.03.034 [5] A. M. Wazwaz, “New Travelling Wave Solutions to the Boussinesq and KleinGordon Equations,” Communica tions in Nonlinear Science and Numerical Simulation, Copyright © 2011 SciRes. AM
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