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Hindawi Publishing Corporation Boundary Value Problems Volume 2010, Article ID 781750, 16 pages doi:10.1155/2010/781750 Research Article A Linear Difference Scheme for Dissipative Symmetric Regularized Long Wave Equations with Damping Term Jinsong Hu, 1 Youcai Xu, 2 and Bing Hu 2 1 School of Mathematics and Computer Engineering, Xihua University, Chengdu 610039, China 2 School of Mathematics, Sichuan University, Chengdu 610064, China Correspondence should be addressed to Youcai Xu, xyc@scu.edu.cn Received 24 August 2010; Accepted 14 November 2010 Academic Editor: V. Shakhmurov Copyright q 2010 Jinsong Hu et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. We study the initial-boundary problem of dissipative symmetric regularized long wave equations with damping term by finite difference method. A linear three-level implicit finite difference scheme is designed. Existence and uniqueness of numerical solutions are derived. It is proved that the finite difference scheme is of second-order convergence and unconditionally stable by the discrete energy method. Numerical simulations verify that the method is accurate and efficient. 1. Introduction A symmetric version of regularized long wave equation SRLWE, u t ρ x uu x − u xxt 0, ρ t u x 0, 1.1 has been proposed to model the propagation of weakly nonlinear ion acoustic and space charge waves 1.Thesec 2 solitary wave solutions are u x, t 3 v 2 − 1 v sec 2 1 2 v 2 − 1 v 2 x − vt , ρ x, t 3 v 2 − 1 v 2 sec 2 1 2 v 2 − 1 v 2 x − vt . 1.2 2 Boundary Value Problems The four invariants and some numerical results have been obtained in 1, where v is the velocity, v 2 > 1. Obviously, eliminating ρ from 1.1, we get a class of SRLWE: u tt − u xx 1 2 u 2 xt − u xxtt 0. 1.3 Equation 1.3 is explicitly symmetric in the x and t derivatives and is very similar to the regularized long wave equation that describes shallow water waves and plasma drift waves 2, 3. The SRLW equation also arises in many other areas of mathematical physics 4–6. Numerical investigation indicates that interactions of solitary waves are inelastic 7;thus, the solitary wave of the SRLWE is not a solution. Research on the wellposedness for its solution and numerical methods has aroused more and more interest. In 8, Guo studied t he existence, uniqueness, and regularity of the numerical solutions for the periodic initial value problem of generalized SRLW by the spectral method. In 9, Zheng et al. presented a Fourier pseudospectral method with a restraint operator for the SRLWEs and proved its stability and obtained the optimum error estimates. There are other methods such as pseudospectral method, finite difference method for the initial-boundary value problem of SRLWEs see 9– 15. In applications, the viscous damping effect is inevitable, and it plays the same important role as the dispersive effect. Therefore, it is more significant to study the dissipative symmetric regularized long wave equations with the damping term u t ρ x − υu xx uu x − u xxt 0, 1.4 ρ t u x γρ 0, 1.5 where υ, γ are positive constants, υ>0 is the dissipative coefficient, and γ>0 is the damping coefficient. Equations 1.4-1.5 are a reasonable model to render essential phenomena of nonlinear ion acoustic wave motion when dissipation is considered. Existence, uniqueness, and wellposedness of global solutions to 1.4-1.5 are presented see 16–20.Butit is difficult to find the analytical solution to 1.4-1.5, which makes numerical solution important. To authors’ knowledge, the finite difference method to dissipative SRLWEs with damping term 1.4-1.5 has not been studied till now. In this paper, we propose linear three level implicit finite difference scheme for 1.4-1.5 with u x, 0 u 0 x ,ρ x, 0 ρ 0 x ,x∈ x L ,x R , 1.6 and the boundary conditions u x L ,t u x R ,t 0,ρ x L ,t ρ x R ,t 0,t∈ 0,T . 1.7 We show that this difference scheme is uniquely solvable, convergent, and stable in both theoretical and numerical senses. Lemma 1.1. Suppose that u 0 ∈ H 1 , ρ 0 ∈ L 2 , the solution of 1.4–1.7 satisfies u L 2 ≤ C, u x L 2 ≤ C, ρ L 2 ≤ C, and u L ∞ ≤ C,whereC is a generic positive constant that varies in the context. Boundary Value Problems 3 Proof. Let E t u 2 L 2 u x 2 L 2 ρ 2 L 2 x R x L u 2 dx x R x L u x 2 dx x R x L ρ 2 dx, t ∈ 0,T . 1.8 Multiplying 1.4 by u and integrating over x L ,x R , we have x R x L uu t uρ x − υuu xx u 2 u x − uu xxt dx 0. 1.9 According to x R x L uu t dx 1 2 d dt x R x L u 2 dx, x R x L uρ x dx uρ| x R x L − x R x L ρ du − x R x L u x ρ dx, − x R x L uu xx dx −uu x | x R x L x R x L u x du x R x L u x 2 dx, x R x L u 2 u x dx 1 3 u 3 | x R x L 0, − x R x L uu xxt dx −uu xt | x R x L x R x L u xt du 1 2 d dt x R x L u x 2 dx, 1.10 we get d dt x R x L u 2 u 2 x dx − 2 x R x L u x ρ dx 2υ x R x L u x 2 dx 0, 1.11 Then, multiplying 1.5 by ρ and integrating over x L ,x R , we have x R x L ρρ t ρu x γρ 2 dx 0. 1.12 By x R x L ρρ t dx 1 2 d dt x R x L ρ 2 dx, 1.13 we get d dt x R x L ρ 2 dx 2 x R x L u x ρ dx 2γ x R x L ρ 2 dx 0. 1.14 4 Boundary Value Problems Adding 1.14 to 1.11,weobtain d dt x R x L u 2 u 2 x ρ 2 dx −2υ x R x L u x 2 dx − 2γ x R x L ρ 2 dx ≤ 0. 1.15 So Et is decreasing with respect to t, which implies that Etu 2 L 2 u x 2 L 2 ρ 2 L 2 ≤ E0, t ∈ 0,T. Then, it indicates that u L 2 ≤ C, u x L 2 ≤ C,andρ L 2 ≤ C. It is followed from Sobolev inequality that u L ∞ ≤ C. 2. Finite Difference Scheme and Its Error Estimation Let h and τ be the uniform step size in the spatial and temporal direction, respectively. Denote x j x L jhj 0, 1, 2, ,J, t n nτn 0, 1, 2, ,N, N T/τ, u n j ≈ ux j ,t n , ρ n j ≈ ρx j ,t n ,andZ 0 h {u u j | u 0 u J 0,j 0, 1, 2, ,J}. We define the difference operators as follows: u n j x u n j1 − u n j h , u n j x u n j − u n j−1 h , u n j x u n j1 − u n j−1 2h , u n j t u n1 j − u n−1 j 2τ , u n j u n1 j u n−1 j 2 , u n ,v n h J−1 j0 u n j v n j , u n 2 u n ,u n , u n ∞ max 0≤j≤J−1 u n j . 2.1 Then, the average three-implicit finite difference scheme for the solution of 1.4–1.7 is as follow: u n j t − u n j xx t ρ n j x − υ u n j xx 1 3 u n j u n j x u n j u n j x 0, 2.2 ρ n j t u n j x γρ n j 0, 2.3 u 0 j u 0 x j ,ρ 0 j ρ 0 x j , 0 ≤ j ≤ J, 2.4 u n 0 u n J 0,ρ n 0 ρ n J 0, 1 ≤ n ≤ N. 2.5 Lemma 2.1. Summation by parts follows [12, 21] that for any two discrete functions u, v ∈ Z 0 h u j x ,v j − u j , v j x , v j , u j xx − v j x , u j x . 2.6 Boundary Value Problems 5 Lemma 2.2 discrete Sobolev’s inequality 12, 21. There exist two constants C 1 and C 2 such that u n ∞ ≤ C 1 u n C 2 u n x . 2.7 Lemma 2.3 discrete Gronwall inequality 12, 21. Suppose that wk, ρk are nonnegative functions and ρk is nondecreasing. If C>0 and w k ≤ ρ k Cτ k−1 l0 w l . 2.8 Then wk ≤ ρke Cτk . Theorem 2.4. If u 0 ∈ H 1 , ρ 0 ∈ L 2 , then the solution of 2.2–2.5 satisfies u n ≤ C, u n x ≤ C, ρ n ≤ C, u n ∞ ≤ C n 1, 2, ,N . 2.9 Proof. Taking an inner product of 2.2 with 2 u n j i.e., u n1 j u n−1 j and considering the boundary condition 2.5 and Lemma 2.1,weobtain 1 2τ u n1 2 − u n−1 2 1 2τ u n1 x 2 − u n−1 x 2 ρ n j x , 2u n j − υ u n j xx , 2u n j P, 2 u n j 0, 2.10 where P 1/3u n j u n j x u n j u n j x . Since ρ n j x , 2u n j − ρ n j , 2 u n j x , u n j xx , 2u n j −2 u n x 2 , P, 2 u n j 2 3 h J−1 j0 u n j u n j x u n j u n j x u n j 1 12 J−1 j0 u n j u n1 j1 u n−1 j1 − u n1 j−1 − u n−1 j−1 u n j1 u n1 j1 u n−1 j1 − u n j−1 u n1 j−1 u n−1 j−1 × u n1 j u n−1 j 1 12 J−1 j0 u n j u n j1 u n1 j1 u n−1 j1 u n1 j u n−1 j − 1 12 J−1 j0 u n j u n j−1 u n1 j u n−1 j u n1 j−1 u n−1 j−1 0, 2.11 6 Boundary Value Problems we obtain 1 2τ u n1 2 − u n−1 2 1 2τ u n1 x 2 − u n−1 x 2 − ρ n j , 2 u n j x 2υ u n x 2 0. 2.12 Taking an inner product of 2.3 with 2 ρ n j i.e., ρ n1 j ρ n−1 j ,weobtain 1 2τ ρ n1 2 − ρ n−1 2 u n j x , 2ρ n j 2γ ρ n j 2 0. 2.13 Adding 2.12 to 2.13, we have u n1 2 − u n−1 2 u n1 x 2 − u n−1 x 2 ρ n1 2 − ρ n−1 2 2τ ρ n j , 2 u n j x − u n j x , 2ρ n j − 4υτ u n x 2 − 4γτ ρ n j 2 . 2.14 Since ρ n j , 2 u n j x ρ n j , u n1 j x u n−1 j x ≤ ρ n 2 1 2 u n1 x 2 u n−1 x 2 , − u n j x , 2ρ n j − u n j x ,ρ n1 j ρ n−1 j ≤ u n x 2 1 2 ρ n1 2 ρ n−1 2 . 2.15 Equation 2.14 can be changed to u n1 2 − u n−1 2 u n1 x 2 − u n−1 x 2 ρ n1 2 − ρ n−1 2 ≤ Cτ u n1 x 2 u n x 2 u n−1 x 2 ρ n1 2 ρ n 2 ρ n−1 2 . 2.16 Let A n u n1 2 u n 2 u n1 x 2 u n x 2 ρ n1 2 ρ n 2 ,and2.16 is changed to A n − A n−1 ≤ Cτ A n A n−1 . 2.17 If τ is sufficiently small which satisfies 1 − Cτ > 0, then A n − A n−1 ≤ CτA n−1 . 2.18 Summing up 2.18 from 1 to n, we have A n ≤ A 0 Cτ n−1 l0 A l . 2.19 Boundary Value Problems 7 From Lemma 2.3,weobtainA n ≤ C, which implies that, u n ≤C, u n x ≤C,andρ n ≤C. By Lemma 2.2,weobtainu n ∞ ≤ C. Theorem 2.5. Assume that u 0 ∈ H 2 , ρ 0 ∈ H 1 , the solution of di fference scheme 2.2–2.5 satisfies: ρ n x ≤ C, u n xx ≤ C, u n x ∞ ≤ C, ρ n ∞ ≤ C n 1, 2, ,N . 2.20 Proof. Differentiating backward 2.2–2.5 with respect to x,weobtain u n j x t − u n j xxx t ρ n j x x − υ u n j xxx 1 3 u n j u n j x u n j u n j x x 0, 2.21 ρ n j x t u n j x x γ ρ n j x 0, 2.22 u 0 j x u 0,x x j , ρ 0 j x ρ 0,x x j , 0 ≤ j ≤ J, 2.23 u n 0 x u n J x 0, ρ n 0 x ρ n J x 0, 0 ≤ n ≤ N. 2.24 Computing the inner product of 2.21 with 2 u n x i.e., u n1 x u n−1 x and considering 2.24 and Lemma 2.1,weobtain 1 2τ u n1 x 2 − u n−1 x 2 1 2τ u n1 xx 2 − u n−1 xx 2 ρ n x x , 2u n x − υ u n xx x , 2u n x R, 2 u n x 0, 2.25 where R 1/3u n j u n j x u n j u n j x x . It follows from Theorem 2.4 that u n j ≤ C j 0, 1, 2, ,J . 2.26 By the Schwarz inequality and Lemma 2.1,weget R, 2 u n x 2 3 u n j u n j x u n j u n j x x , u n x − 2 3 u n j u n j x u n j u n j x , u n x x − 2 3 h J−1 j0 u n j u n j x u n j u n j x u n j xx ≤ 2 3 Ch J−1 j0 u n j x · u n j xx ≤ C u n x 2 u n xx 2 ≤ C u n1 x 2 u n−1 x 2 u n1 xx 2 u n−1 xx 2 . 2.27 8 Boundary Value Problems Noting that ρ n x x , 2u n x − 2 u n x x ,ρ n x − ρ n x ,u n1 x x u n−1 x x ≤ ρ n x 2 1 2 u n1 xx 2 u n−1 xx 2 , u n xx x , 2u n x −2 u n xx 2 , 2.28 it follows from 2.25 that u n1 x 2 − u n−1 x 2 u n1 xx 2 − u n−1 xx 2 ≤−4υτ u n xx 2 Cτ u n1 x 2 u n−1 x 2 u n1 xx 2 u n−1 xx 2 ρ n x 2 . 2.29 Computing the inner product of 2.22 with 2 ρ n x i.e., ρ n1 x ρ n−1 x and considering 2.24 and Lemma 2.1,weobtain 1 2τ ρ n1 x 2 − ρ n−1 x 2 u n x x , ρ n x 2γ ρ n x 2 0. 2.30 Since u n x x , 2ρ n x u n x x ,ρ n1 x ρ n−1 x ≤ u n xx 2 1 2 ρ n1 x 2 ρ n−1 x 2 , 2.31 then 2.30 is changed to ρ n1 x − ρ n−1 x ≤−4γτ ρ n x 2 Cτ u n xx 2 ρ n1 x 2 ρ n−1 x 2 . 2.32 Adding 2.29 to 2.32, we have u n1 x 2 − u n−1 x 2 u n1 xx 2 − u n−1 xx 2 ρ n1 x 2 − ρ n−1 x 2 ≤−4υτ u n xx 2 − 4γτ ρ n x 2 Cτ u n1 x 2 u n−1 x 2 u n xx 2 u n1 xx 2 u n−1 xx 2 ρ n1 x 2 ρ n x 2 ρ n−1 x 2 ≤ Cτ u n1 x 2 u n−1 x 2 u n xx 2 u n1 xx 2 u n−1 xx 2 ρ n1 x 2 ρ n x 2 ρ n−1 x 2 . 2.33 Leting B n u n1 x 2 u n x 2 u n1 xx 2 u n xx 2 ρ n1 x 2 ρ n x 2 ,weobtainB n −B n−1 ≤ CτB n B n−1 . Choosing suitable τ which is small enough to satisfy 1 − Cτ > 0, we get B n − B n−1 ≤ CτB n−1 . 2.34 Boundary Value Problems 9 Summing up 2.34 from 1 to n, we have B n ≤ B 0 Cτ n−1 l0 B l . 2.35 By Lemma 2.3,wegetB n ≤ C, which implies that ρ n x ≤C, u n xx ≤C. It follows from Theorem 2.4 and Lemma 2.2 that u n x ∞ ≤ C, ρ n ∞ ≤ C. 3. Solvability Theorem 3.1. The solution u n of 2.2–2.5 is unique. Proof. Using the mathematical induction, clearly, u 0 , ρ 0 are uniquely determined by initial conditions 2.4. then select appropriate second-order methods such as the C-N Schemes and calculate u 1 and ρ 1 i.e. u 0 , ρ 0 ,andu 1 , ρ 1 are uniquely determined. Assume that u 0 ,u 1 , ,u n and ρ 0 ,ρ 1 , ,ρ n are the only solution, now consider u n1 and ρ n1 in 2.2 and 2.3: 1 2τ u n1 j − 1 2τ u n1 j xx − υ 2 u n1 j xx 1 6 u n j u n1 j x u n j u n1 j x 0, 3.1 1 2τ ρ n1 j γ 2 ρ n1 j 0. 3.2 Taking an inner product of 3.1 with u n1 , we have 1 2τ u n1 2 1 2τ u n1 x 2 υ 2 u n1 x 2 1 6 h J−1 j0 u n j u n1 j x u n j u n1 j x u n1 j 0. 3.3 Since 1 6 h J−1 j0 u n j u n1 j x u n j u n1 j x u n1 j 1 12 J−1 j0 u n j u n1 j1 − u n1 j−1 u n j1 u n1 j1 − u n j−1 u n1 j−1 u n1 j 1 12 J−1 j0 u n j u n1 j u n1 j1 u n j1 u n1 j u n1 j1 − 1 12 J−1 j0 u n j−1 u n1 j−1 u n1 j u n j u n1 j−1 u n1 j 0, 3.4 then it holds 1 2τ u n1 2 1 2τ υ 2 u n1 x 2 0. 3.5 10 Boundary Value Problems Taking an inner product of 3.2 with ρ n1 and adding to 3.5, we have 1 2τ u n1 2 1 2τ υ 2 u n1 x 2 1 2τ γ 2 ρ n1 2 0, 3.6 which implies that 3.1-3.2 have only zero solution. So the solution u n1 j and ρ n1 j of 2.2– 2.5 is unique. 4. Convergence and Stability Let vx, t and ∅x, t be the solution of problem 1.4–1.7;thatis,v n j ux j ,t n , ∅ n j ρx j ,t n , then the truncation of the difference scheme 2.2–2.5 is r n j v n j t − v n j xx t ∅ n j x − υ v n j xx 1 3 v n j v n j x v n j v n j x , 4.1 s n j ∅ n j t v n j x γ∅ n j . 4.2 Making use of Taylor expansion, it holds |r n j | |s n j | Oτ 2 h 2 if h, τ → 0. Theorem 4.1. Assume that u 0 ∈ H 1 , ρ 0 ∈ L 2 , then the solution u n and ρ n in the senses of norms · ∞ and · L 2 , respectively, to the difference scheme 2.2–2.5 converges to the solution of problem 1.4–1.7 and the order of convergence is Oτ 2 h 2 . Proof. Subtracting 2.2 from 4.1 subtracting 2.3 from 4.2, and letting e n j v n j − u n j , η n j ∅ n j − ρ n j , we have r n j e n j t − e n j xx t η n j x − υ e n j xx Q, 4.3 s n j η n j t e n j x γη n j , 4.4 where Q 1 3 v n j v n j x − u n j u n j x 1 3 v n j v n j x − u n j u n j x . 4.5 Computing the inner product of 4.3 with 2 e n ,weget e n1 2 − e n−1 2 e n1 x 2 − e n−1 x 2 −4υτ e n x 2 2τ r n j , 2e n j − η n j x , 2e n j − Q, 2 e n j . 4.6 [...]... 2002 17 Y D Shang and B Guo, “Global attractors for a periodic initial value problem for dissipative generalized symmetric regularized long wave equations, ” Acta Mathematica Scientia Series A, vol 23, no 6, pp 745–757, 2003 18 B Guo and Y Shang, “Approximate inertial manifolds to the generalized symmetric regularized long wave equations with damping term,” Acta Mathematicae Applicatae Sinica, vol 19,... 64–72, 1989 16 Boundary Value Problems 11 Y D Shang and B Guo, “Legendre and Chebyshev pseudospectral methods for the generalized symmetric regularized long wave equations, ” Acta Mathematicae Applicatae Sinica, vol 26, no 4, pp 590–604, 2003 12 Y Bai and L M Zhang, A conservative finite difference scheme for symmetric regularized long wave equations, ” Acta Mathematicae Applicatae Sinica, vol 30, no 2,... 2003 19 Y Shang and B Guo, “Exponential attractor for the generalized symmetric regularized long wave equation with damping term,” Applied Mathematics and Mechanics, vol 26, no 3, pp 259–266, 2005 20 F Shaomei, G Boling, and Q Hua, “The existence of global attractors for a system of multidimensional symmetric regularized wave equations, ” Communications in Nonlinear Science and Numerical Simulation, vol... 13 T Wang, L Zhang, and F Chen, “Conservative schemes for the symmetric regularized long wave equations, ” Applied Mathematics and Computation, vol 190, no 2, pp 1063–1080, 2007 14 T C Wang and L M Zhang, “Pseudo-compact conservative finite difference approximate solution for the symmetric regularized long wave equation,” Acta Mathematica Scientia Series A, vol 26, no 7, pp 1039–1046, 2006 15 T C Wang,... T C Wang, L M Zhang, and F Q Chen, “Pseudo-compact conservative finite difference approximate solutions for symmetric regularized- long- wave equations, ” Chinese Journal of Engineering Mathematics, vol 25, no 1, pp 169–172, 2008 16 Y Shang, B Guo, and S Fang, Long time behavior of the dissipative generalized symmetric regularized long wave equations, ” Journal of Partial Differential Equations, vol 15,... spectral method for symmetric regularized wave equations, ” Journal of Computational Mathematics, vol 5, no 4, pp 297–306, 1987 9 J D Zheng, R F Zhang, and B Y Guo, “The Fourier pseudo-spectral method for the SRLW equation,” Applied Mathematics and Mechanics, vol 10, no 9, pp 801–810, 1989 10 J D Zheng, “Pseudospectral collocation methods for the generalized SRLW equations, ” Mathematica Numerica Sinica,... 527–537, 1989 3 C J Amick, J L Bona, and M E Schonbek, “Decay of solutions of some nonlinear wave equations, ” Journal of Differential Equations, vol 81, no 1, pp 1–49, 1989 4 T Ogino and S Takeda, “Computer simulation and analysis for the spherical and cylindrical ionacoustic solitons,” Journal of the Physical Society of Japan, vol 41, no 1, pp 257–264, 1976 5 V G Makhankov, “Dynamics of classical solitons... XZD0910-09-1 The work of Youcai Xu was supported by the Youth Research Foundation of Sichuan University no 2009SCU11113 References 1 C E Seyler and D L Fenstermacher, A symmetric regularized- long- wave equation,” Physics of Fluids, vol 27, no 1, pp 4–7, 1984 2 J Albert, “On the decay of solutions of the generalized Benjamin-Bona-Mahony equations, ” Journal of Mathematical Analysis and Applications, vol 141, no... 0.05, a wave figure comparison of u and ρ at various time steps is as in Figures 1 and 2 From Table 1, it is easy to find that the difference scheme in this paper is second-order convergent Figures 1 and 2 show that the height of wave crest is more and more low with time elapsing due to the effect of damping and dissipativeness It simulates that the continue energy E t of problem 1.4 – 1.7 in Lemma 1.1... in non-integrable systems ,” Physics Reports Section C, vol 35, no 1, pp 1–128, 1978 6 P A Clarkson, “New similarity reductions and Painlev´ analysis for the symmetric regularised long e wave and modified Benjamin-Bona-Mahoney equations, ” Journal of Physics A, vol 22, no 18, pp 3821– 3848, 1989 7 I L Bogolubsky, “Some examples of inelastic soliton interaction,” Computer Physics Communications, vol 13, . Y. D. Shang and B. Guo, “Global attractors for a periodic initial value problem for dissipative generalized symmetric regularized long wave equations, ” Acta Mathematica Scientia. Series A, vol. 23,. 2003. 18 B. Guo and Y. Shang, “Approximate inertial manifolds to the generalized symmetric regularized long wave equations with damping term,” Acta Mathematicae Applicatae Sinica, vol. 19, no pseudospectral methods for the generalized symmetric regularized long wave equations, ” Acta Mathematicae Applicatae Sinica,vol.26,no.4,pp. 590–604, 2003. 12 Y. Bai and L. M. Zhang, A conservative
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