Received 30 December 2015; accepted 25 January 2016; published 28 January 2016
In the solidification of a dilute binary alloy, a planer solid-liquid interface is often to be instable, spontaneously assuming a cellular structure. This situation enables one to derive an asymptotic nonlinear equation which directly describes the dynamic of the onset and stabilization of cellular structure
where is a positive constant, (see   ). Equation (1.1) is referred as the Sivashinsky equation.
In this article, we introduce the mathematical model for a finite difference discretization to the solution of the periodical boundary of two-dimensional Sivashinsky equation:
with the initial condition
subject to the -periodic boundary conditions
where, is the Laplacian operator, and is a given -periodic smooth function.
Several numerical methods have been proposed in the literature for discretizing Sivashinsky equation. A semi-implicit finite difference scheme and a linearized finite difference method for the Sivashinsky equation in one-dimensional have been proposed respectively in   . A semidiscrete approximation of the two dimensional Sivashinsky equation with lumped-mass method and optimal order error bounds for the piecewise linear approximation are derived in  . There are many papers that have already been published to study the finite difference method for fourth-order nonlinear equation, for example  - and so on.
In this work, we investigate a linearized three level difference scheme for two-dimensional Sivashinsky equations. The remainder of this paper is organized as follows. In Section 2, a linearized difference scheme for (1.2) is derived. The unique solvability of the approximate solutions is shown in Section 3. A second order convergent linearized difference scheme is proved in Section 4. At last section, some numerical examples are presented to improve the theoretical results.
2. Linearized Difference Scheme
To solve the periodic initial-value problems (1.2)-(1.4), one can restrict it on a bounded domain . For a positive integer N, let time-step, , , and ,. We define a partition of by the rectangles with, , , , such that where and are positive constants. The optimal choice for is. Denote
We define the space of periodic grid functions on as:
Further, define operators, and, respectively, as
For and define the inner product
and Sobolev norms (or seminorms)
Define as the space of functions which are of class with respect to and class with respect to t.
It follows from summation by parts that the following Lemma holds   .
Lemma 1. For, we have
We discretize problems (1.2)-(1.4) by the following finite difference scheme: we approximate, by
3. Solvability of the Difference Scheme
Next, we will discuss the unique solvability of the difference schemes (2.4)-(2.6).
Theorem 1. Difference schemes (2.4)-(2.6) have a unique solution.
Proof. It is obvious that and are uniquely determined by the initial conditions (2.5) and (2.6). Now, we suppose that () can be solved uniquely. Consider the homogeneous equation of (2.4) for:
Taking the inner product of (3.1) with, it follows from Lemma 1 that
That is, (3.1) has only a trivial solution. Thus, by the induction principle, (2.4) determines uniquely. This completes the proof.
4. Convergence of the Difference Scheme
For a smooth function u, we have
Therefore, the extrapolation just proposed will give second-order accuracy. To show the convergence of the difference scheme, we need the following Lemmas.
Lemma 2.   . Let and be positive and satisfy
Lemma 3.  . For any grid function v on there is a positive constant c independent h such that
The main result of this article is the following Theorem.
Theorem 2. Assume the solution of of (1.2)-(1.4) belong to. Then, the solution of difference schemes (2.4)-(2.6) converges to the solution of the problems (1.2)-(1.4) with the convergence order of in the discrete -norm.
Proof. Define the net function
Therefore, From Taylor expansion, we have for
where and are truncation errors of difference schemes (2.4)-(2.6) and there exists a constant such that
Let and subtracting (2.4)-(2.6) from (4.1)-(4.3), we obtain
We prove by inductive method that
From (4.5) and (4.7)-(4.8), we have
It follows from (4.10) that (4.9) is valid for and. Now suppose that (4.9) is true for n from 0 to l. Therefore, for h sufficiently small
For, taking in (4.6) the inner product with
Noting that from the Lipschitz condition of f
For, it follows from (4.13) and (4.14) that
Using (4.4), we get
It follows easily from this inequality that
Applying Lemma 2, we obtain
Using (4.10), we get
where is constant dependent on and. That means, by the induction principle (4.9) is true.
Second, we will prove that
From (4.7), we find
Using (4.5), we obtain
This implies that
where. Thus, we get. Similarly we find,. Then
Taking now in (4.6) the inner product with, we obtain for
Using the differentiability of f and the Cauchy Schwartz inequality, we obtain
This yields by (4.4)
It follows from (4.9) that
Here, by above,
Applying Lemma 3, (4.9) and (4.16)-(4.18), we obtain
This completes the proof.
5. Numerical Experiments
In this section, we give some numerical experiments to verify our theoretical results that are given in the previous sections. For that purpose, we consider the following periodic inhomogeneous Sivashinsky equation
with the initial condition
For which the exact solution is
In the runs, we use the same spacing h in each direction, , and compute the maximum norm errors of the numerical solution
The convergence order in spatial direction is defined as
when is sufficiently small. The convergence order in temporal direction is defined as
when h is sufficiently small. We also define the rate of convergence
when both h and are sufficiently small.
By computing the problems (5.1)-(5.2) with the difference schemes (2.4)-(2.6), we carry out the spatial and temporal convergence in the sense of the maximum norm. Table 1 and Table 2 give the errors between numerical solutions and exact solutions for spatial and temporal convergence, respectively. Once again, we conclude from Tables 1-3, that the difference schemes (2.4)-(2.6) are convergent with the convergence order of two both in space and in time. This is in accordance with Theorem 2.
In this paper, we use the discrete energy method to study the convergence of a linearized difference scheme for solving the two-dimensional Sivashinsky equation. The convergence is proved to be second order in the maxi-
Table 1. The spatial convergence orders in maximum norm for difference schemes (2.1)-(2.3) to the inhomogeneous Sivashinsky Equations (5.1) and (5.2), with.
Table 2. The temporal convergence orders in maximum norm for difference schemes (2.1)-(2.3) to the inhomogeneous Sivashinsky Equations (5.1) and (5.2), with.
Table 3. The maximum norm errors and convergence orders for difference schemes (2.1)-(2.3) to the inhomogeneous Sivashinsky Equations (5.1) and (5.2).
mum norm, which extends the result in   where they only prove the second order convergence of the difference scheme for one-dimensional Sivashinsky equation in the discrete -norm. For obtaining the approximate solution for the two dimensional Sivashinsky equation by finite element Galerkin method, one must need polynomials of the degree. It means that they have to construct minimum 10 node triangle for approximating the solution. Computationally, it is very expensive and difficult to impose inter-element continuity condition. If the boundary is curved, imposition of boundary conditions causes some more difficulties. Therefore, based on the linearized difference schemes (2.4)-(2.6), this article proposes a recipe to eradicate such numerical difficulties.
 Sivashinsky, G.I. (1983) On Cellular Instability in the Solidification of a Dilute Binary Alloy. Physica 8D, North-Holland Publishing Company, Amsterdam, 243-248.
 Omrani, K. (2003) A Second-Order Splitting Method for a Finite Difference Scheme for the Sivashinsky Equation. Applied Mathematics Letters, 16, 441-445.
 Khiari, N., Achouri, T., Mohamed, M.L.B. and Omrani, K. (2007) Finite Difference Approximate Solutions for the Cahn-Hilliard Equation. Numerical Methods for Partial Differential Equations, 23, 437-455.
 Atouani, N. and Omrani, K. (2015) On the Convergence of Conservative Difference Schemes for the 2D Generalized Rosenau-Korteweg de Vries Equation. Applied Mathematics and Computation, 250, 832-847.
 Atouani, N. and Omrani, K. (2015) A New Conservative High-Order Accurate Difference Scheme for the Rosenau Equation. Applicable Analysis, 94, 2435-2455.
 Omrani, K., Abidi, F., Achouri, T. and Khiari, N. (2008) A New Conservative Finite Difference Scheme for the Rosenau Equation. Applied Mathematics and Computation, 201, 35-43.
 Khiari, N. and Omrani, K. (2011) Finite difference Discretization of the Extended Fisher-Kolmogorov Equation in Two Dimension. Computers and Mathematics with Applications, 62, 4151-4160.
 Pan, X.T., Wang, T.C., Zhang, L.M. and Guo, B.L., (2012) On the Convergence of a Conservative Numerical Scheme for the Usual Rosenau-RLW Equation. Applied Mathematical Modelling, 36, 3371-3378.
 Pan, X. and Zhang, L. (2012) Numerical Simulation for General Rosenau-RLW Equation: An Average Linearized Conservative Scheme. Mathematical Problems in Engineering, 2012, Article ID: 517818.
 Zhao, X, Liu, F and Liu, B (2014) Finite Difference Discretization of a Fourth-Order Parabolic Equation Describing Crystal Surface Growth. Applicable Analysis, 94, 1964-1975.