A method of Lie-symmetry GL(n,R) for solving non-linear dynamical systems

https://doi.org/10.1016/j.ijnonlinmec.2013.01.015Get rights and content

Abstract

The group-preserving scheme (GPS) developed in Liu [1] for solving a non-linear dynamical system x˙=f(x,t) was based on the Lie-symmetry SOo(n,1). Here we derive a more fundamental GL(n,R) dynamics for x, and develop a relevant Lie-group scheme based on the Lie-symmetry GL(n,R), which is a Lie-group set large enough to cope all non-linear ordinary differential equations (ODEs). We find that the first-order explicit scheme based on GL(n,R) is equivalent to the GPS. Moreover, when one uses an implicit scheme based on GL(n,R), it converges very fast at each time marching step and the accuracy is raised several orders than the explicit method. For the dynamical system endowed with a rotational vector field we also develop an implicit SO(n) Lie-group integration method. Several numerical examples are examined, showing that the GL(n,R) and SO(n) Lie-group schemes have superior efficiency, accuracy and stability.

Highlights

► The Lie-symmetry GL(n,R) can cope all non-linear dynamical systems. ► The non-linear dynamical system was solved by GL(n,R) integration method. ► SO(n) integration method was used on rotational vector field. ► The present implicit method is convergent very fast, effective and accurate.

Introduction

There are many numerical methods used to solve ordinary differential equations (ODEs). In general, those methods are effective. However, when the ODEs considered possess certain structure, like the first integrals, invariants, the Lie-symmetry in the tangent bundle, or a symplectic form on the cotangent bundle, the general purpose numerical methods cannot retain those properties very well, because they are not designed to do so. In the last few years there has been a substantial development in the geometric integrators of ODEs evolving on the Lie groups, and more generally on the homogeneous spaces as shown by Iserles et al. [2] and Hairer et al. [3].

For more than one century, the Lie groups played a decisive role in our understanding of the geometric structure of differential equations. It is believed that the concept of Lie groups, within their wider terminology and machinery of differential geometry, is very useful in devising superior numerical methods to integrate ODEs, and to retain the invariant property of dynamical system. By sharing the geometric structure and invariance with the original ODEs, the new methods are more accurate, more stable and more effective than conventional numerical methods.

In an attempt to retain the invariance of the underlying dynamical system in the Minkowski space, Liu [1] has developed numerical methods to integrate the augmented dynamical system of ordinary differential equations that evolve on a matrix Lie-group SOo(n,1). The Lie-group schemes apply to the problem of finding numerical approximations to the solution ofY˙=A(Y,t)Y,Y(0)=Y0,where the exact solution Y evolves in a matrix Lie group with A a matrix function on the associated Lie algebra. Nowadays, the method developed by Liu [1] is known as the group-preserving scheme (GPS), which can preserve the Lie-symmetry SOo(n,1) of the augmented dynamical system. Lee and Liu [4] have extended the first-order GPS to the fourth-order GPS. In fact, the GPS method is very effective to deal with the ODEs as shown by Liu [5] for stiff ODEs, and by Liu [6] for ODEs with constraints. Chen et al. [7] have modified the GPS to a time stepsize adaptive numerical scheme for ODEs. Some extensions of the GPS to other fields were undertaken by Chang et al. [8], Liu [9], [10], [11], [12], [13], [14], [15], and Liu et al. [16], [17].

Many non-linear mechanical problems of interest can be modeled by the system of differential equations whose solutions satisfy some invariants. In the past several decades, a particular attention has been paid to the development of the numerical methods which approximate the solution of such a type system while preserve invariants to a machinery precision; see e.g., Baumgarte [18], Führer and Leimkuhler [19], Ascher and Petzold [20], Ma¨rz [21], [22], Ascher et al. [23], Campbell and Moore [24], Ascher [25], Chan et al. [26], Arevalo et al. [27], and references therein.

We begin with the following n-dimensional non-linear dynamical system:x˙=f(x,t),x(0)=x0,tR,xRn.It has an important feature that the solution value at any given point on the trajectory determines the solution at all later points on the trajectory. In effect, the solutions x(t;x0) of the differential equations define a one-parameter mapping {ϕt}t0, which takes the initial point to later points along a trajectoryϕt(x0)=x(t;x0).We term the map ϕt:RnRn a flow map of the given system. In general, the flow map has the following property. If we solve the differential equations from a given initial point x0 up to a time t1, then solve from the resulting point forward t2 units of time, the effect is the same as solving the differential equations with the initial value x0 up to a time t1+t2. In terms of the flow mapping, that is, ϕt1ϕt2=ϕt1+t2. Such a mapping is referred to as a one-parameter semi-group. Ying and Candés [28] based on the semi-group have built up a phase flow method for autonomous ODEs. Indeed, the differential equations have such a semi-group property, but they do not necessarily have the Lie-group property.

The main purpose of the Lie-group solver is providing a better algorithm that can better retain the orbit generated from the numerical solution on the manifold which is associated with the Lie-group. The retention of the Lie-group structure under discretization is a vital task in the recovery of qualitatively correct behavior in the minimization of numerical error. The theory of Lie-group and Lie-algebra has been developed for a long time. However, the Lie-group methods were developed only recently, which were initiated mainly by Munthe-Kaas [29], [30] and Iserles et al. [2], [31]. The Lie-group method is used to construct the numerical solution of differential equations evolving on a manifold. The numerical solutions have been designed by evolving on the same manifold as the analytical solutions are. Presently, there are some famous methods such as the Crouch–Grossman methods, the RKMK methods, the Magnus methods, the Fer methods, etc. The past studies clearly indicated that the Lie-group methods not only produce an improved qualitative behavior but also allow for a more accurate long term integration than that offered by the general purpose methods. Zhang and Deng [32] have extended the GPS by combining it with the above mentioned RKMK methods.

The idea of extending the Euler method by allowing for a multiplicity of evaluations of the vector field functions within each time step was originally proposed by Runge in 1895. Further contributions were made by Kutta in 1901, who completely characterized the set of Runge–Kutta method of order 4, which is known as the RK4 method today. In the construction of higher-order Lie-group methods, basically, there are two approaches: one is using the Lie algebra property of A, and another is using the Lie group property of G. The reader can refer those developments of geometric integrators, for example, Iserles and Nørsett [31], Munthe-Kaas [29], [30], Iserles et al. [2], and Hairer et al. [3]. The discussions of the Lie-group integration method for ODEs starting from the augmented dynamical system developed by Liu [1] were made by Zhang and Deng [32], [33] and Lee and Liu [4].

However, up to now there does not have the idea how to derive a Lie-symmetry in the state space xRn for the general non-linear dynamical system (not in the augmented state space as that done by Liu [1]). This paper will extend the group-preserving scheme developed by Liu [1] by using a new format of the ODEs expressed as a form in Eq. (1). We give a new form of Eq. (2) and derive the corresponding group-preserving scheme (GPS) based on a larger Lie-group symmetry. A different approach from that of Liu [1] is announced, which is a fundamental study of the Lie-group method for the general non-linear ODEs, no matter whether the ODEs possess special structures or not.

The remaining parts of this paper are arranged as follows. In Section 2 we divide the non-linear dynamical systems into two classes: a rotational vector field and a non-rotational vector field, and then we give two new formulations such that they can be re-formulated into the type as that in Eq. (1). This is a new starting point of our developments of the Lie-group GL(n,R) schemes in Section 3, and the Lie-group SO(n) scheme in Section 4. Several numerical examples are given in Section 5 to validate the performance of the newly developed Lie-group methods. Finally, the conclusions are drawn in Section 6.

Section snippets

The GL(n,R) structure of non-linear dynamical system

Lie-group is a differentiable manifold, which is endowed with a group structure that is compatible with the underlying topology of the manifold. The Lie-group solver can provide a better algorithm that retains the manifold which is associated with the Lie-group.

The general linear group is a Lie-group set, whose manifold is an open subset GL(n,R):={GRn×n|detG0} of the linear space consists of all n×n non-singular matrices. Thus, GL(n,R) is also an n×n-dimensional manifold. The group

A Riccati ODE and Lie-group G(t)

Letafx,nxx,hence, by Eqs. (15), (16) we havex˙=n·xa.

Liu [1] has derived the following ODEs system for n:n˙=fxfx·nn.Letw(t)n·x=xand Eq. (20) becomesx˙=w(t)a.

From Eqs. (19), (20), (21), (22) it follows that:w˙(t)=n·x˙+n˙·x=a·nn·x+fx·xfx·nn·x=y(t)w(t)+y(t)xy(t)x=y(t)w(t),wherey(t)=a·n.From Eqs. (23), (24), (25) we can recover Eq. (8). Here we have provided a further theoretical foundation of Eq. (8) from the state space representation. A deeper analysis is given below.

The SO(n) Lie-group scheme

In order to develop a numerical scheme from Eqs. (17), (18), we suppose that the coefficient matrix W is constant witha=f¯x¯,b=x¯x¯being two constant vectors, which can be obtained by taking the values of f and x at a suitable mid-point of t¯[t0=0,t], where tt0+h and h is a small time stepsize. Thus from Eqs. (17), (18) we havex˙=b·xaa·xb.Letwb·x,za·xand Eq. (59) becomesx˙=wazb.At the same time, from the above two equations we can derive the following ODEs for w and z:ddtwz=a·bb2a

Numerical examples

In order to assess the performance of the newly developed schemes for the integration of non-linear dynamical system (2) let us investigate the following examples.

Conclusions

In this paper, it is the first time that the non-linear dynamical system x˙=f(x,t) is re-formulated into a new system x˙=Ax with A=(f/x)(x/x). Such that we can develop a very powerful Lie-group integrator to find the solution of non-linear dynamical system based on GL(n,R). Only this Lie-symmetry group GL(n,R) is large enough to cover all the non-linear dynamical systems. In doing so we find that GL(n,R) is a more fundamental Lie-group that it can retrieve the group-preserving scheme based

Acknowledgments

The projects NSC-99-2221-E-002-074-MY3, NSC-100-2221-E-002-165-MY3 and the 2011 Outstanding Research Award from Taiwan's National Science Council, and the 2011 Taiwan Research Front Award from Thomson Reuters, granted to the author, are highly appreciated.

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