Commenced in January 2007
Frequency: Monthly
Edition: International
Paper Count: 31742
Septic B-Spline Collocation Method for Numerical Solution of the Kuramoto-Sivashinsky Equation

Authors: M. Zarebnia, R. Parvaz


In this paper the Kuramoto-Sivashinsky equation is solved numerically by collocation method. The solution is approximated as a linear combination of septic B-spline functions. Applying the Von-Neumann stability analysis technique, we show that the method is unconditionally stable. The method is applied on some test examples, and the numerical results have been compared with the exact solutions. The global relative error and L∞ in the solutions show the efficiency of the method computationally.

Keywords: Kuramoto-Sivashinsky equation, Septic B-spline, Collocation method, Finite difference.

Digital Object Identifier (DOI):

Procedia APA BibTeX Chicago EndNote Harvard JSON MLA RIS XML ISO 690 PDF Downloads 1836


[1] Y. Xu, C.Wang Shu, Local discontinuous Galerkin methods for the Kuramoto-Sivashinsky equations and the Ito-type coupled KdV equations Comput, Methods Appl, Mech. Engrg, 195 (2006) 3430-3447.
[2] I.S. Yang, On traveling-wave solutions of the Kuramoto-Sivashinsky equation, Physica D, 110 (1997) 25-42.
[3] G. Akrivis, Y.S. Smyrlis, Implicit-explicit BDF methods for the Kuramoto-Sivashinsky equation, Applied Numerical Mathematic, 51 (2004) 151-169.
[4] A.V. Manickam, K.M. Moudgalya, A.K. Pani, Second-order splitting combined with orthogonal cubic spline collocation method for the Kuramoto- Sivashinsky equation, Comput Math Appl, 35 (1998) 5-25.
[5] P. Collet, J.P. Eckmann, H. Epstein, J. Stubbe, A global attracting set for the Kuramoto-Sivashinsky equation, Comm. Math. Phys, 152 (1993) 203-214.
[6] A.H. Khater, R.S. Temsah, Numerical solutions of the generalized Kuramoto-Sivashinsky equation by Chebyshev spectral collocation methods, Computers and Mathematics with Applications, 56 (2008) 1465- 1472.
[7] M. Uddin, S. Haq , S. Islam, A mesh-free numerical method for solution of the family of Kuramoto-Sivashinsky equations, Applied Mathematics and Computation, 212 (2009) 458-469.
[8] S.G. Rubin, R.A. Graves, Cubic spline approximation for problems in fluid mechanics. NASA TR R-436, Washington, DC, 1975.
[9] G.D. Smith, Numerical Solution of Patial Differential Method, Second Edition, Oxford University Press, 1978.
[10] R.D Richtmyer, K.W. Morton, Difference Methods for Initial-Value Problems, Inter science Publishers (John Wiley), New York, (1967).
[11] L. Huilin, M. Changfeng, Lattice Boltzmann method for the generalized Kuramoto-Sivashinsky equation, Physica A, 388 (2009) 1405-1412