A group of asymmetric difference schemes to approach the fourth-order parabolic equation is given. According to these schemes and the Crank-Nicolson scheme, an alternating segment Crank-Nicolson scheme with intrinsic parallelism is constructed. The truncation errors and the stability are discussed. Numerical simulations show that this new scheme has unconditional stability and high accuracy and convergency, and it is in preference to the implicit scheme method. 1. Introduction With the rapid development of high-performance computers, the need to construct parallel algorithms has long been desired. In recent years, the alternating schemes have been studied extensively in the literature. In 1983, Evans and Abdullah first developed the Alternating Group Explicit (AGE) scheme [1] for parabolic equation, which shows that it is possible to design parallel difference method by constructing a new difference scheme. Afterward, using the explicit scheme, the implicit scheme, and the Crank-Nicolson scheme, the Alternating Segment Explicit-Implicit (ASE-I) scheme [2] and the Alternating Segment Crank-Nicolson (ASC-N) scheme [3] were proposed. The results of numerical examples show that these schemes are unconditionally stable and have high accuracy. Currently, the alternating technology has been extended to dispersive equation [4–8], convection-diffusion equation [9], Burgers equation [10], nonlinear three-order KdV equation [11] and fourth-order parabolic [12, 13]. The Kuramoto-Sivashinsky equation (K-S) [14, 15] is well-known as one of the mathematical equations which models the reaction-diffusion systems, flame propagation, and viscous flow problems. During recent years, many authors have focused on solving this equation numerically and analytically [16–18]. However, the parallel difference method for this equation has not been found. In this paper, we present the alternating segment Crank-Nicolson scheme for the following fourth-order parabolic equation: which is the high order part of the linear K-S equation. We hope that the result of this paper makes an essential contribution in this direction. We consider the following problem: With the initial condition and the boundary conditions where is a given function and and are constants. The plan of this paper is as follows. In Section 2, some basic schemes are given and the ASC-N scheme is developed. In Section 3, the error analysis and the stability analysis are discussed. In Section 4, numerical simulations are performed. Finally, a brief conclusion is given. 2. New Alternating Segment Crank-Nicolson Scheme 2.1.
References
[1]
D. J. Evans and A. R. B. Abdullah, “Group explicit method for parabolic equations,” International Journal of Computer Mathematics, vol. 14, no. 1, pp. 73–105, 1983.
[2]
B. L. Zhang, “Alternating segment explicitimplicit method for the diffusion equation,” Journal on Numerical Methods and Computer Applications, vol. 12, no. 4, pp. 245–251, 1991.
[3]
J. Chen and B. L. Zhang, “A class of alternating block Crank-Nicolson method,” International Journal of Computer Mathematics, vol. 45, pp. 89–112, 1991.
[4]
S. Zhu, G. Yuan, and L. Shen, “Alternating group explicit method for the dispersive equation,” International Journal of Computer Mathematics, vol. 75, no. 1, pp. 97–105, 2000.
[5]
S. Zhu and J. Zhao, “The alternating segment explicit-implicit scheme for the dispersive equation,” Applied Mathematics Letters, vol. 14, no. 6, pp. 657–662, 2001.
[6]
W. Q. Wang and S. J. Fu, “An unconditionally stable alternating segment difference scheme of eight points for the dispersive equation,” International Journal for Numerical Methods in Engineering, vol. 67, no. 3, pp. 435–447, 2006.
[7]
Q. J. Zhang and W. Q. Wang, “A new alternating group explicit-implicit algorithm with high accuracy for dispersive equation,” Applied Mathematics and Mechanics, vol. 29, no. 9, pp. 1221–1230, 2008.
[8]
W. Q. Wang and Q. J. Zhang, “A highly accurate alternating 6-point group method for the dispersive equation,” International Journal of Computer Mathematics, vol. 87, no. 7, pp. 1512–1521, 2010.
[9]
W. Q. Wang, “Alternating segment Crank-Nicolson method for solving convection-diffusion equation with variable coefficient,” Applied Mathematics and Mechanics, vol. 24, no. 1, pp. 32–42, 2003.
[10]
W. Q. Wang, “A class of alternating group method of Burgers’ equation,” Applied Mathematics and Mechanics, vol. 25, no. 2, pp. 236–244, 2004.
[11]
F. L. Qu and W. Q. Wang, “Alternating segment explicit-implicit scheme for nonlinear third-order KdV equation,” Applied Mathematics and Mechanics, vol. 28, no. 7, pp. 973–980, 2007.
[12]
G. Y. Guo and B. Liu, “Unconditional stability of alternating difference schemes with intrinsic parallelism for the fourth-order parabolic equation,” Applied Mathematics and Computation, vol. 219, no. 14, pp. 7319–7328, 2013.
[13]
G. Y. Guo, Y. S. Zhai, and B. Liu, “The alternating segment explicit-implicit scheme for the fourth-order parabolic equation,” Journal of Information and Computational Science, vol. 10, no. 10, pp. 3125–3133, 2013.
[14]
Y. Kuramoto and T. Tsuzuki, “On the formation of dissipative structures in reaction diffusion systems,” Progress of Theoretical and Experimental Physics, vol. 54, pp. 687–699, 1975.
[15]
G. I. Sivashinsky, “Nonlinear analysis of hydrodynamic instability in laminar flames. I: derivation of basic equations,” Acta Astronautica, vol. 4, no. 11-12, pp. 1177–1206, 1977.
[16]
R. C. Mittal and G. Arora, “Quintic B-spline collocation method for numerical solution of the Kuramoto-Sivashinsky equation,” Communications in Nonlinear Science and Numerical Simulation, vol. 15, no. 10, pp. 2798–2808, 2010.
[17]
B. Soltanalizadeh and M. Zarebnia, “Numerical analysis of the linear and nonlinear Kuramoto-Sivashinsky equation by using differential transformation method,” International Journal of Applied Mathematics and Mechanics, vol. 7, pp. 63–72, 2011.
[18]
M. Lakestania and M. Dehghanb, “Numerical solutions of the generalized Kuramoto-Sivashinsky equation using B-spline functions,” Applied Mathematical Modelling, vol. 36, no. 2, pp. 605–617, 2012.
[19]
R. B. Kellogg, “An alternating direction method for operator equations,” SIAM, vol. 12, no. 4, pp. 848–854, 1964.
