On The Numerical Solution of Two Dimensional Model of an Alloy Solidification Problem

In this paper, a linearized three level difference scheme is derived for two-dimensional model of an alloy solidification problem called Sivashinsky equation. Further, it is proved that the scheme is uniquely solvable and convergent with convergence rate of order two in a discrete L-norm. At last, numerical experiments are carried out to support the theoretical claims.


Introduction
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 α is a positive constant, (see [1] [2]).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: , L L -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 [3] [4].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 [5].There are many papers that have already been published to study the finite difference method for fourth-order nonlinear equation, for example [5]- [14] 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.

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 ( ) , , i i j j x x y y where 1 2 3 , , γ γ γ and ε are positive constants.The optimal choice for ε is We define the space of periodic grid functions on h Ω as: , , , , , , , .
V and n t V ∂ , respectively, as , ,

M M M M y i j y i j h i j
, , .

Solvability of the Difference Scheme
Next, we will discuss the unique solvability of the difference schemes (2. Taking the inner product of (3.1) with 1 , That is, (3.1) has only a trivial solution.Thus, by the induction principle, (2.4) determines 1 n U + uniquely.This completes the proof.

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.
 be positive and satisfy ( ) [17].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 ( ) 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 ( ) Proof.Define the net function ≤ ≤ Therefore, From Taylor expansion, we have for .
where , n i j F and , i j G are truncation errors of difference schemes (2.4)-(2.6)and there exists a constant 1 c such that ( ) .
Noting that from the Lipschitz condition of f ( ) ( ) where α > , it follows from (4.13) and (4.14) that ( ) It follows easily from this inequality that Applying Lemma 2, we obtain Using (4.10), we get and hence, ( ) where 5 c is constant dependent on 1 , T c and 4 c .That means, by the induction principle (4.9) is true.Second, we will prove that ( ) , 0 .
Taking now in (4.6) the inner product with n t E ∂ , we obtain for Using the differentiability of f and the Cauchy Schwartz inequality, we obtain ( ) This completes the proof.

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 In the runs, we use the same spacing h in each direction, 1 2 h h h = = , and compute the maximum norm errors of the numerical solution ( ) The convergence order in spatial direction is defined as when τ is sufficiently small.The convergence order in temporal direction is defined as , 2 log , , when h is sufficiently small.We also define the rate of convergence ( ) ( ) when both h and τ 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.

Conclusion
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-  mum norm, which extends the result in [3] [4] where they only prove the second order convergence of the difference scheme for one-dimensional Sivashinsky equation in the discrete 2 L -norm.For obtaining the approximate solution for the two dimensional Sivashinsky equation by finite element Galerkin method, one must need polynomials of the degree 3 ≥ .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 1  C 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.

Table 1 .
The spatial convergence orders in maximum norm for difference schemes (2.1)-(2.3) to the inhomogeneous Siva-