Geometric Integration Methods For Generalized Hamiltonian Systems And Furthermore For General Nonlinear Dynamic Systems

Geometric integration has advantages over classical integration either in higher precision or in preservation of the invariant of the systems. And it has backward error analysis property that can be used to analyze the long time behavior and the stability of the numerical method. In this paper the geometric integration methods for solving generalized Hamiltonian systems and furthermore for solving general nonlinear dynamic system are investigated.Firstly, Lie series method to solve general nonlinear dynamic system is presented, and the implementation method is given in detail. It is an expansion of the Talor series method. In the other hand, if dynamic differential equation is expressed in the form of linear differential operator action, its solver can be obtained by inversing the Laplace transform of the resolvent of its infinitesimal generator (differential operator). In such a way the Lie series method is again obtained. The Lie series method is a Lie group method for an autonomous dynamic system. In addition, another numerical solution for solving nonlinear dynamic equations is presented based on numerical Laplace transform inversion.Secondly, based on Lie series method, Lie group methods for solving generalized Hamiltonian systems and generalized Hamiltonian systems with dissipation are presented. Generalized Hamiltonian formalism is a proper description of dynamic system; it reveals some intrinsic symmetric properties of the dynamic systems. So it is much more important to study generalized Hamiltonian systems theoretically and practically. In this paper, arbitrary high-order explicit integration methods for solving generalized Hamiltonian systems are developed based on the theory of the analytical solution of conservation systems, and the implementation of algorithms is discussed in detail. Then the generalized Hamiltonian systems with dissipation are differentiated between autonomous and non-autonomous systems for the convenience of the study. Autonomous systems are solved by using Lie series method and operator-splitting and composition method. For solving non-autonomous systems, the numerical methods are developed based on the Magnus series method and Fer expansion method. The methods in this paper preserve the canonical property of the exact solution of the original systems, so it is stable. Sometimes, the energy property of the systems, forexample, the Hamiltonian property, is more important. In this case, using discrete gradient, the numerical methods are given to solve generalized Hamiltonian systems and generalized Hamiltonian control systems, which preserve the Hamiltonian property.Furthermore, generalized Hamiltonian systems with constraints are studied on the pseudo-Poisson manifold. Generalized Hamiltonian systems with constraints are converted into unconstrained generalized Hamiltonian systems of ordinary differential equations. And some Lie group integration methods are given so as to preserve the intrinsic construction and constraint invariants. In the meantime, the stability and the error analysis of the constrain-invariants along solutions are investigated numerically and analytically. Furthermore the integrations are simplified by introducing Lagrange multipliers and using projection techniques. Because the discussion does not differentiate holonomic constraints and non-holonomic constraints, the methods in this paper can be used to solve dynamic systems with non-holonomic constraints.However, general nonlinear dynamic system cannot always be expressed as a generalized Hamiltonian system (with dissipation). In other ward, there exist the generalized Hamiltonian realization problems. Based on the classical Magnus or Fer expansion and Lie group integration methods for generalized Hamiltonian systems with dissipation, Lie group integrations for solving general nonlinear dynamic systems are studied in a deep-going way in two different points of view: one is to discuss the question in the form of operator action, and the nonlinear dynamic equation can be expressed as model of linear map action, then new algorithm can be designed based on the theory of linear differential equation; the other is to expand the configuration space to the Minkowski space, where the original nonlinear dynamic system is converted into an augmented dynamic system of Lie type and it is convenient to design the algorithm and program. Among them, the integration methods based on Magnus expansion involve a large numbers of commutators. Taking advantage of the time-symmetry property of Magnus series and using the technique of Lie series expansion, a minimum of commutators are involved in the algorithms. Approximate schemes of 4-th, 6-th and 8-th order are constructed which involve only 1, 4 and 10 different commutators, and the time-symmetry properties of the schemes are proved. It should be point out that these methods are much more accurate and effective for solving autonomous dynamic systems. A second-order method can achieve computation precision up to 4-th order. And numerical examples show that large step size can be used in calculation. In themeantime, the integration schemes of supper convergence based on Fer expansion are also attained. Then the Fer expansion methods are connected with Magnus expansion methods effectively and some techniques are given to simplify the construction of Fer expansion methods. Furthermore time-symmetric integrators of Fer type are constructed.Runge-Kutta/Munthe-Kaas (RKMK) methods are expanded Runge-Kutta methods for solving differential equation whose configuration space is on a Lie group. The differential equation evolving on a Lie group was converted into an equivalent differential equation evolving on the corresponding Lie algebra. The numerical solution of the equivalent differential equation can be obtained by using Runge-Kutta methods, then it can be pulled back to the original Lie group using exponential maps, and the numerical solution of original differential equation can be obtained. Exponential matrix is a mapping from the Lie algebra of matrix to the Lie group of matrix. Based on RKMK methods, a series of simple and effective integration methods of RKMK type are presented by combining Runge-Kutta methods with precise integration method in order to solve the nonlinear dynamic system and its augmented dynamic system in the Minkowski space. These methods are all Lie group methods.The numerical methods based on Magnus and Fer expansion involve a large number of exponential matrices too. The computation precision of the exponential matrices influences the accuracy of the schemes directly. Precise integration method can compute the exponential matrix quickly and effectively. In this paper, precise integration method for solving linear dynamic systems are expanded to solve general nonlinear dynamic systems. In the Minkowski space the nonlinear dynamic system is converted into an augmented dynamic system and it is very convenient to use the precise integration method. In this case, precise integration method is a Lie group method for autonomous dynamic systems. Additionally, precise integration method is ingenuously applied to solving nonlinear dynamic systems by means of expanding the dimension of the configuration space or combining the Runge-Kutta methods, which avoids the numerically non-stability of the matrix inversion and the non-existence of the inverse matrix.