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

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Received 30 December 2015; accepted 25 January 2016; published 28 January 2016

1. 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

(1.1)

where 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:

(1.2)

with the initial condition

(1.3)

subject to the -periodic boundary conditions

(1.4)

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 [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.

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:

For, denote

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 [5] [6] .

Lemma 1. For, we have

(2.1)

(2.2)

(2.3)

We discretize problems (1.2)-(1.4) by the following finite difference scheme: we approximate, by

(2.4)

(2.5)

(2.6)

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:

(3.1)

Taking the inner product of (3.1) with, it follows from Lemma 1 that

This implies,

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. [15] [16] . Let and be positive and satisfy

then

Lemma 3. [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 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

(4.1)

(4.2)

(4.3)

where and are truncation errors of difference schemes (2.4)-(2.6) and there exists a constant such that

(4.4)

(4.5)

Let and subtracting (2.4)-(2.6) from (4.1)-(4.3), we obtain

(4.6)

(4.7)

(4.8)

We prove by inductive method that

(4.9)

From (4.5) and (4.7)-(4.8), we have

(4.10)

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

(4.11)

Thus,

(4.12)

where

For, taking in (4.6) the inner product with

(4.13)

Noting that from the Lipschitz condition of f

(4.14)

where

For, it follows from (4.13) and (4.14) that

Using (4.4), we get

This yields

Therefore, when

It follows easily from this inequality that

Applying Lemma 2, we obtain

Using (4.10), we get

and hence,

where is constant dependent on and. That means, by the induction principle (4.9) is true.

Second, we will prove that

(4.15)

From (4.7), we find

(4.16)

Using (4.5), we obtain

This implies that

where. Thus, we get. Similarly we find,. Then

(4.17)

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,

and hence,

(4.18)

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

(5.1)

with the initial condition

(5.2)

where

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.

6. 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-

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 [3] [4] 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.

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