The Numerical Convergence of the Landau-Lifshitz Equations and Its Simulation
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.
1. Introduction and the Number Scheme
The LL equation  has aroused considerable interest among physicists and mathematicians. For the one-dimensional case, there have been many contributions to the study of the soliton solution, the interaction of solitary waves, and other properties of the solitary waves [2–4]. However, we point out here that it will be a more challenging task study the high-dimensional dynamics [5, 6] about LL equation. In the classical study of ferromagnetic chain, we often consider the following system (vector form): where “” denotes a 3-dimensional cross product. The isotropic Heisenberg chain  is the special case of (1.1). Moreover, another special case of (1.1) is the LL equation with an easy plane
Equation (1.2) has been studied by the inverse scattering method in . Its integrability theories  are established. As far as we know, no explicit solutions of (1.2) are given by various direct methods such as Jacobi elliptical function method . Therefore, the numerical method to study this equation and its more generalized form, which gives some of the visual characteristics of equations, is necessary. However, we do not intend to construct efficient (or optimal control) and high-precision algorithms [10–15] here. In this paper, we are concerned about the convergence of the discrete scheme. Furthermore, we hope that by means of numerical simulation, more intuitive understanding of the nature of the equation will be obtained.
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 where the spin vector is a 3-dimensional vector-valued unknown function with respect to space variables and time , and period about is . is continuous with respect to , and . is the initial value function which is also a 3-dimensional function. , is the -dimensional real value dominion. is damping coefficient. , Laplace operator .
Although the existence of the global attractor of (1.3) has been proved in , no exact solutions have been proposed as far as we know. In this paper, we give an exact solution of it, which can test the property of numerical scheme of (1.3). From a physical point of view, the value of is finite, just like the situation which is mentioned in [5, 16]. In this section, we just suppose that for convenience. So the first equation of problem (1.3) is equal to the following form: where , Furthermore, we consider a more general type of Landau-Lifshitz equation (as the variety of (1.4)): where is a continuous differential function. When , (1.6) takes the form of (1.4).
For convenien, we discuss our scheme in one-dimensional case which the -dimensional case can be discussed similarly. Firstly, we give some notion and symbol. Set , is a positive integer. We divide the region as a discrete mesh which in its space step and in its time step. ,. Here denotes the approximate value of on . Let denote the layer mesh function, that is, where .
Define the discrete inner product and norm as follows:
We define the following finite difference approximations of derivatives along the space direction: Similarly, we can define the derivatives along the time direction and the Laplace operator: .
Under the definition and the symbol setting as above, we now define a finite difference approximation of (1.6) by
Obviously, according to difference approximation (1.9), we can start from the zero layer to any layer about.
2. Existence of the Solution and the Stability of the Scheme
For studying the convergence and the stability, we first introduce several lemmas.
Lemma 2.1. Considering mesh-function , defined on mesh-points, one has the following relationship:(1),(2),(3),(4).
Lemma 2.2. Let stand for an antisymmetric matrix, is the three-dimension vector, then the following relationships hold:(1),(2),(3).
Let denote the solution of (1.6) (here will be regarded as a function with smoothness in some degree). denotes the value of on . denotes the value of -layer mesh about .
Definition 2.3. One has
Lemma 2.4. Assume that and are the solution of (1.6), then for any where .
Proof. We have, by () of Lemma 2.2, . According to the antiproperty of antisymmetric matrix, we have . Here, we set , hence
Let . By -inequality, we have
Lemma 2.5. Under the same condition of Lemma 2.4, one has
Lemma 2.6. If and are the solution of (1.6), then for any where
Proof. By () of Lemma 2.1, we have
Lemma 2.7. One has
Proof. According to () and () of Lemma 2.1, we have
Definition 2.8. .
According to the lemmas mentioned above, we now come to discuss the convergence of difference equation (1.9).
Theorem 2.9. Let be the solution of (1.6), is the solution of (1.9), , . For any positive integer , if , then there exists a positive constant which independent of and . If , , one has furthermore, where ,
Proof. According to the condition, the solution of (1.6) , substituting the according part of (1.9) with , we have
where is the error estimate of (1.9). According to (2.14) and (1.9), denote , we have
Take the inner product of (2.15) and , combine them with () and () of Lemma 2.1, we have
Just like the method mentioned in , we can assume that and . Conveniently, we can denote as and as . By Lemmas 2.4 and 2.6, (1.9) can change into the following form: where In the following part, we propose the estimation about of (2.17).
By Lemmas 2.5 and 2.7 and (2.15), we have where , , and .
So according to the inequality given above, we have
Substitute (2.16) into (2.15), let , according to the condition given in Theorem 2.9, we have where .
According to (2.17) and the condition given in Theorem 2.9,
By and Gronwall inequality, we have
So we have where by (2.25), hence
So by (2.23) and (2.26), when , , and , Theorem 2.9 is right. Similarly to , these presuppositions can be omitted according to the finite extensive method of nonlinear function. In fact, by (2.23) and , when , , according to the definition about and mentioned in  ( can be defined accordingly), we have and and . So this theorem and also hold when , , . Here, we finish the proof of the first inequality of Theorem 2.9.
Similarly, we have the stability theorem about the difference equation.
Theorem 2.10. Let be the solution of (1.9), is the solution of (1.9) under the disturbance . . For any positive constant , if , then there exist constant which is independent on and . When , , one has where .
3. Numerical Experiment and Its Error Analysis
Conveniently for computation, first we rewrite (1.9) as the following form:
According to (3.1), in the first step, we can get the value of on . Second, according to second equation of (3.1), we can also get the solution step by step. In each step of computation, the third equation of (3.1) will be used repeatedly.
(i) Setting and , we consider the following spin wave of : where . In fact, we found that (3.2) is the solution of (1.3), where . As far as we know, exact solutions of this case were still not constructed. For the simplicity, we have omitted the details of our constructing.
In accordance with (3.2), the initial and boundary conditions of this equation can be proposed. Let us first consider a domain with the Dirichlet boundary condition on the spin vector. We have implemented (3.1) where in numerical resolution. Figures 1(a) and 1(b) show the numerical solutions in time , and respectively.
From Figure 1, we observe that the numerical solutions exhibit an irregular changing at the beginning of the space steps. This will happen in the range of probably space steps in Figure 1(a) as well as mainly in steps in Figure 1(b). This can be seen more clearly in Figure 2 which exhibit the error about the solution in Figure 1 accordingly.
Observed from Figures 1 and 2, probably from the 10 or 20 space steps, the numerical solution is credible. But from then on, the creditable region about the solution is gradually shrinking when the time increase. At space step 50, by amplifying solution error image Figure 2, we found that the magnitude of error is approximately which is consist with our Theorem 2.9. These details can be seen in Figure 3.
(ii) Setting , and in (1.3), the solitary solution which proposed in  can be written as (3.3) (here we can set ). For this case, according to (3.3), the initial boundary conditions can be given in our numerical scheme (3.1). Furthermore, we mention here at (3.3) is not a periodic solution, but we can extend the problem into a periodic one. In fact, we take the truncated domain as . and the smoothness of the solution ensure the extend can be done. Thus the finite difference scheme (3.1) still can be used for computation. Figure 4 shows the numerical solution in the and the corresponding error image. Seen in Figure 4, we find that solution error is quite large. Therefore, we suspect that (3.3) which is proposed in  is not correct. Moreover, a simple symbol computation by Maple confirms our speculation:
When , Theorem 2.9 obviously can not give estimate about . But when we follow (3.2) to set initial boundary value, in addition to numerical viscosity began in the outside of space step, solution is very irregular. Therefore, we believe that the numerical solution is convergent. Nevertheless, for a better estimation, we should try to use other methods to estimate convergence rate of the discrete form solution. From Figure 5, we found that the evolution of and is similar to subcase (i) mentioned above; compared to subcase (i), the evolution of exhibits a larger undulate behavior.
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).
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