A New Alternating Segment Crank-Nicolson Scheme for the Fourth-Order Parabolic Equation

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.


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][5][6][7][8], convectiondiffusion 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][17][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  0 () 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.

The Analysis of the Truncation Errors.
In order to analyze the truncation errors, we change the scheme  into the equivalent segment scheme of three levels; that is, adding scheme   at the point (  ,  +1 ) and scheme  at the point (  ,  +2 ) for two adjacent points of the ( + 1)st time level and ( + 2)nd time level, we obtain three-level scheme (17).Similarly, we obtain three-level schemes ( 18)-(24), respectively; From the Taylor series expansion at (  ,  +1 ), we obtain the following truncation error expressions for formulae ( 17)-(24): where We briefly discuss the truncation error analysis of the ASC-N scheme.Obviously, the truncation error of the Crank-Nicolson scheme is ( 2 + ℎ 2 ).On the same time level, the asymmetric schemes are used symmetrically in the space direction.The signs of the terms with the parameter ℎ in (25)-(32) are opposite, the effect of the terms with ℎ can be nearly canceled, and the truncation error at these boundary points is approximately ( + ℎ 2 ).

The Analysis of the Stability.
To prove the stability, we have to introduce the following Kellogg lemma in [19].
Proof.By eliminating U +1 from ( 14)-( 15), we obtain U +2 = GU  , where G is the growth matrix For any even number , there holds Since G and G 2 are all symmetric, G 1 + G This shows that the ASC-N scheme is unconditionally stable.

Numerical Simulations
To illustrate the convergency, stability, and accuracy, we perform the numerical experiment for (2) using the model The exact solution of this problem is We first examine the errors and convergence rate in space for the ASC-N scheme in this paper.Let 2 = 50; we compute the  2 -norm of the error  ℎ = ‖ − ‖  2 and the convergence rate  ℎ /ℎ 2 for different grid ratio.The results are given in Tables 1 and 2, which show that the convergence rate is approximately (ℎ 2 ).
Next, we compare the ASC-N scheme numerical solution with the exact solution.The absolute errors (Ae) and the percentage errors (Re) of numerical solution at  = 0.1 are displayed in Table 3 for different .From Table 3, we can see that the method given in this paper is unconditionally stable and has high accuracy.
In addition, we also compare the ASC-N solution with the implicit difference scheme (IMP) solution using the same mesh refinements.The results are displayed in Figures 3 and  4 at different time.From Figure 3 and Table 3, we can get that the ASC-N solution is nearly as good as the exact solution and is stable and reliable.From Figures 3 and 4, we also find that the ASC-N scheme has better accuracy than the IMP scheme.This is because the truncation error of the IMP scheme is ( + ℎ 2 ) and the truncation errors of the ASC-N scheme are also ( + ℎ 2 ) at those points computed by the asymmetric scheme, but the truncation errors at those points computed by the C-N scheme are ( 2 + ℎ 2 ).Consequently, we can obtain that these results agree with the theoretical analysis.
Finally, owing to the use of the asymmetric schemes, the ASC-N scheme changed the discrete problem of  order into some small (2 or  order) independent problems; the parallelism of the scheme is clarity.

Conclusion
In this paper, we first constructed a group of asymmetric schemes and the C-N scheme; based on the idea of the alternating method, we gave the ASC-N scheme for the fourth-order parabolic equation.The theoretical analysis and the numerical simulations show that the ASC-N scheme constructed in this paper has high accuracy and convergence, unconditional stability, and intrinsic parallelism.The idea of this scheme is helpful for the deep study of the K-S equation.

Table 1 :
The error and the convergence rates of the ASC-N scheme.

Table 2 :
The error and the convergence rates of the ASC-N scheme.