AAM  >> Vol. 4 No. 3 (August 2015)

    一维Sine-Gordon方程四阶紧致有限体积方法
    A Fourth-Order Compact Finite Volume Scheme for 1D Sine-Gordon Equations

  • 全文下载: PDF(550KB) HTML   XML   PP.262-270   DOI: 10.12677/AAM.2015.43032  
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作者:  

刘盎然,高巍,李宏:内蒙古大学数学科学学院,内蒙古 呼和浩特

关键词:
Sine-Gordon有限体积紧格式Runge-Kutta方法Sine-Gordon Compact Method Finite Volume Method Runge-Kutta Method

摘要:

本文提出一个高阶方法来数值求解非线性Sine-Gordon方程。空间离散上应用四阶的紧致有限体积格式,时间离散应用三阶SSP Runge-Kutta (RK)方法。数值实验表明该算法是求解一维Sine-Gordon方程的较为高效的方法。

In this work, we propose a compact finite volume method for solving the one-dimensional nonlinear sine-Gordon equation. The third-order SSP Runge-Kutta (RK) scheme is used for temporal disretization. Numerical experiments show that the present scheme is an efficient algorithm for solving the one-dimensional Sine-Gordon equation.

文章引用:
刘盎然, 高巍, 李宏. 一维Sine-Gordon方程四阶紧致有限体积方法[J]. 应用数学进展, 2015, 4(3): 262-270. http://dx.doi.org/10.12677/AAM.2015.43032

参考文献

[1] Perring, J.K. and Skyrme, T.H.R. (1962) A model unified field equation. Nuclear Physics, 31, 550-555.
http://dx.doi.org/10.1016/0029-5582(62)90774-5
[2] Whitham, G.B. (1999) Linear and nonlinear waves. Wi-ley-Interscience, New York.
http://dx.doi.org/10.1002/9781118032954
[3] Guo, B.Y., Pascual, P.J., Rodriguez, M.J. and Vazquez, L. (1986) Numerical solution of the sine-Gordon equation. Applied Mathematics and Computation, 18, 1-14.
http://dx.doi.org/10.1016/0096-3003(86)90025-1
[4] Strauss, W.A. and Vázquez, L. (1978) Numerical solution of a nonlinear Klein-Gordon equation. Journal of Computational Physics, 28, 271-278.
http://dx.doi.org/10.1016/0021-9991(78)90038-4
[5] Dehghan, M. (2006) Finite difference procedures for solving a problem arising in modeling and design of certain optoelectronic devices. Mathematics and Computers in Simulation, 71, 16-30.
http://dx.doi.org/10.1016/j.matcom.2005.10.001
[6] Dehghan, M. (2005) On the solution of an initial-boundary value problem that combines Neumann and integral condition for the wave equation. Numerical Methods for Partial Differential Equations, 21, 24-40.
http://dx.doi.org/10.1002/num.20019
[7] Dehghan, M. and Shokri, A. (2008) A numerical method for one-dimensional nonlinear sine-Gordon equation using collocation and radial basis functions. Numerical Methods for Partial Differential Equations, 24, 687-698.
http://dx.doi.org/10.1002/num.20289
[8] Ramos, J.I. (2001) The Sine-Gordon equation in the finite line. Applied Mathematics and Computation, 124, 45-93.
http://dx.doi.org/10.1016/S0096-3003(00)00080-1
[9] Lu, X. (2001) Symplectic computation of solitary waves for general sine-Gordon equations. Mathematics and Computers in Simulation, 55, 519-532.
http://dx.doi.org/10.1016/S0378-4754(00)00300-1
[10] Batiha, B., Noorani, M.S.M. and Hashim, I. (2007) Nu-merical solution of sine-Gordon equation by variational iteration method. Physics Letters A, 370, 437-440.
http://dx.doi.org/10.1016/j.physleta.2007.05.087
[11] Bratsos, A.G. and Twizell, E.H. (1996) The solution of the Sine-Gordon equation using the method of lines. International Journal of Computer Mathematics, 61, 271-292.
http://dx.doi.org/10.1080/00207169608804516
[12] Bratsos, A.G. (2008) A fourth order numerical scheme for the one-dimensional sine-Gordon equation. International Journal of Computer Mathematics, 85, 1083-1095.
http://dx.doi.org/10.1080/00207160701473939
[13] Piller, M. and Stalio, E. (2004) Finite-volume compact schemes on staggered grids. Journal of Computational Physics, 197, 299-340.
http://dx.doi.org/10.1016/j.jcp.2003.10.037
[14] Li, S. and Vu-Quoc, L. (1995) Finite difference calculus invariant structure of a class of algorithms for the Klein- Gordon equation. SIAM Journal on Numerical Analysis, 32, 1839-1875.
http://dx.doi.org/10.1137/0732083
[15] 李庆扬, 王能超 (2006) 数值分析. 第四版, 华中科技大学出版社, 武汉.
[16] Gottlieb, S., Shu, C.-W. and Tadmor, E. (2001) Strong stability-preserving high-order time discretization methods. SIAM Review, 43, 89-112.
http://dx.doi.org/10.1137/S003614450036757X
[17] Wei, G.W. (2000) Discrete singular convolution for the Sine-Gordon equation. Physica D, 137, 247-259.
http://dx.doi.org/10.1016/S0167-2789(99)00186-4