Abstract and Applied Analysis
Volume 2012 (2012), Article ID 106343, 22 pages
http://dx.doi.org/10.1155/2012/106343
Research Article

Optimal Error Estimate of Chebyshev-Legendre Spectral Method for the Generalised Benjamin-Bona-Mahony-Burgers Equations

Tinggang Zhao,1,2 Xiaoxian Zhang,1 Jinxia Huo,1 Wanghui Su,1 Yongli Liu,1 and Yujiang Wu2

1School of Mathematics, Lanzhou City University, Lanzhou 730070, China
2School of Mathematics and Statistics, Lanzhou University, Lanzhou 730000, China

Received 29 December 2011; Revised 6 April 2012; Accepted 15 April 2012

Academic Editor: M. Victoria Otero-Espinar

Copyright © 2012 Tinggang Zhao 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.

Abstract

Combining with the Crank-Nicolson/leapfrog scheme in time discretization, Chebyshev-Legendre spectral method is applied to space discretization for numerically solving the Benjamin-Bona-Mahony-Burgers (gBBM-B) equations. The proposed approach is based on Legendre Galerkin formulation while the Chebyshev-Gauss-Lobatto (CGL) nodes are used in the computation. By using the proposed method, the computational complexity is reduced and both accuracy and efficiency are achieved. The stability and convergence are rigorously set up. Optimal error estimate of the Chebyshev-Legendre method is proved for the problem with Dirichlet boundary condition. The convergence rate shows “spectral accuracy.” Numerical experiments are presented to demonstrate the effectiveness of the method and to confirm the theoretical results.