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A difference scheme of Landau-Lifshitz (LL for short) equations is studied. Their convergence and stability are proved. Furthermore, a new solution of LL equation is given for testing our scheme. At the end, three subcases of this LL equation are concerned about, and some properties about these equations are shown by a numeric simulation way.

The LL equation [

Equation (

For universality, we consider the following system with the Gilbert damping term which covers the above situations. The periodic condition is about the space variables

Although the existence of the global attractor of (

For convenien, we discuss our scheme in one-dimensional case which the

Define the discrete inner product and norm as follows:

We define the following finite difference approximations of derivatives along the space direction:

Under the definition and the symbol setting as above, we now define a finite difference approximation of (

Obviously, according to difference approximation (

For studying the convergence and the stability, we first introduce several lemmas.

Considering mesh-function

Let

Let

One has

Assume that

We have, by (

Let

Under the same condition of Lemma

We first note that, by (

If

By (

One has

According to (

According to the lemmas mentioned above, we now come to discuss the convergence of difference equation (

Let

According to the condition, the solution of (

Take the inner product of (

Just like the method mentioned in [

By Lemmas

So according to the inequality given above, we have

Substitute (

According to (

By

By (

So we have

So by (

By discrete Sobolev's inequality [

Similarly, we have the stability theorem about the difference equation.

Let

In this section, we propose the numerical examples and the error analysis of the solutions. Three subcases of (

Conveniently for computation, first we rewrite (

According to (

(i) Setting

In accordance with (

The solution

From Figure

The error of

Observed from Figures

The error of

(ii) Setting

(a) Solution of

(iii) Let

When

The solution

This work is supported by NSFC (Grant No. 10771009 and No. 10861014), BSFC (Grant No. 1085001) of China, Funding Project for Academic Human Resources Development in Institutions of Higher Leading Under the Jurisdiction of Beijing Municipality (PHR-IHLB 200906103), Beijing Education Committee Funds and Ph.D. Innovation Project of Beijing University of Technology (Grant No. ykj-2010-3418 and No.ykj-2010-4151).