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C-N difference schemes for dissipative symmetric regularized long wave equations with damping term. (English) Zbl 1213.76125
Summary: We study the initial-boundary problem of dissipative symmetric regularized long wave equations with damping term. Crank-Nicolson nonlinear-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 the theoretical analysis.

MSC:
76M20Finite difference methods (fluid mechanics)
WorldCat.org
Full Text: DOI EuDML
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. · Zbl 0544.76170 · doi:10.1063/1.864487
[2] J. Albert, “On the decay of solutions of the generalized Benjamin-Bona-Mahony equations,” Journal of Mathematical Analysis and Applications, vol. 141, no. 2, pp. 527-537, 1989. · Zbl 0697.35116 · doi:10.1016/0022-247X(89)90195-9
[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. · Zbl 0689.35081 · doi:10.1016/0022-0396(89)90176-9
[4] T. Ogino and S. Takeda, “Computer simulation and analysis for the spherical and cylindrical ion-acoustic solitons,” Journal of the Physical Society of Japan, vol. 41, no. 1, pp. 257-264, 1976.
[5] V. G. Makhankov, “Dynamics of classical solitons (in nonintegrable systems),” Physics Reports C, vol. 35, no. 1, pp. 1-128, 1978. · doi:10.1016/0370-1573(78)90074-1
[6] P. A. Clarkson, “New similarity reductions and Painlevé analysis for the symmetric regularised long wave and modified Benjamin-Bona-Mahoney equations,” Journal of Physics A, vol. 22, no. 18, pp. 3821-3848, 1989. · Zbl 0711.35113 · doi:10.1088/0305-4470/22/18/020
[7] I. L. Bogolubsky, “Some examples of inelastic soliton interaction,” Computer Physics Communications, vol. 13, no. 3, pp. 149-155, 1977.
[8] B. L. Guo, “The spectral method for symmetric regularized wave equations,” Journal of Computational Mathematics, vol. 5, no. 4, pp. 297-306, 1987. · Zbl 0631.65084
[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. · Zbl 0729.65074 · doi:10.1007/BF02013752
[10] J. D. Zheng, “Pseudospectral collocation methods for the generalized SRLW equations,” Mathematica Numerica Sinica, vol. 11, no. 1, pp. 64-72, 1989. · Zbl 1002.65524
[11] Y. D. Shang and B. L. 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, 2005. · Zbl 1051.35008
[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, pp. 248-255, 2007. · Zbl 1142.65395
[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. · Zbl 1124.65080 · doi:10.1016/j.amc.2007.01.105
[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 A, vol. 26, no. 7, pp. 1039-1046, 2006. · Zbl 1115.65358
[15] 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. · Zbl 1164.65458
[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, no. 1, pp. 35-45, 2002. · Zbl 1056.35030
[17] Y. D. Shang and B. L. Guo, “Global attractors for a periodic initial value problem for dissipative generalized symmetric regularized long wave equations,” Acta Mathematica Scientia A, vol. 23, no. 6, pp. 745-757, 2003.
[18] B.-l. Guo and Y.-D. Shang, “Approximate inertial manifolds to the generalized symmetric regularized long wave equations with damping term,” Acta Mathematicae Applicatae Sinica, vol. 19, no. 2, pp. 191-204, 2003. · Zbl 1059.35105 · doi:10.1007/s10255-003-0095-1
[19] Y. D. Shang and B. L. 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. · Zbl 1144.76304 · doi:10.1007/BF02440077
[20] F. Shaomei, G. Boling, and Q. Hua, “The existence of global attractors for a system of multi-dimensional symmetric regularized wave equations,” Communications in Nonlinear Science and Numerical Simulation, vol. 14, no. 1, pp. 61-68, 2009. · Zbl 1221.35362 · doi:10.1016/j.cnsns.2007.07.001
[21] J. Hu, Y. Xu, and B. Hu, “A linear difference scheme for dissipative symmetric regularized long wave equations with damping term,” Boundary Value Problems, vol. 2010, Article ID 781750, 16 pages, 2010. · Zbl 1207.65110 · doi:10.1155/2010/781750 · eudml:224528
[22] S. K. Chung, “Finite difference approximate solutions for the Rosenau equation,” Applicable Analysis, vol. 69, no. 1-2, pp. 149-156, 1998. · Zbl 0904.65093 · doi:10.1080/00036819808840652
[23] B. Hu, Y. Xu, and J. Hu, “Crank-Nicolson finite difference scheme for the Rosenau-Burgers equation,” Applied Mathematics and Computation, vol. 204, no. 1, pp. 311-316, 2008. · Zbl 1166.65041 · doi:10.1016/j.amc.2008.06.051
[24] F. E. Browder, “Existence and uniqueness theorems for solutions of nonlinear boundary value problems,” Proceedings of Symposia in Applied Mathematics, vol. 17, pp. 24-49, 1965. · Zbl 0145.35302
[25] T. Wang and B. Guo, “A robust semi-explicit difference scheme for the Kuramoto-Tsuzuki equation,” Journal of Computational and Applied Mathematics, vol. 233, no. 4, pp. 878-888, 2009. · Zbl 1181.65117 · doi:10.1016/j.cam.2009.07.058