Newton--Cotes formulae for long-time integration
01 Sep 2003-Journal of Computational and Applied Mathematics (North-Holland)-Vol. 158, Iss: 1, pp 75-82
TL;DR: The connection between closed Newton-Cotes differential methods and symplectic integrators is considered in this article, where the authors apply the symplectic schemes in order to solve Hamilton's equations of motion which are linear in position and momentum.
About: This article is published in Journal of Computational and Applied Mathematics.The article was published on 2003-09-01 and is currently open access. It has received 161 citations till now. The article focuses on the topics: Symplectic integrator & Moment map.
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TL;DR: The closed Newton–Cotes formulae are studied and written as symplectic multilayer structures and trigonometrically-fitted symplectic methods are developed and applied to the well-known radial Schrodinger equation to investigate the efficiency of the proposed method.
155 citations
01 Jan 2009
TL;DR: In this paper, the authors investigated the connection between closed Newton-Cotes formulae, trigonometrically-fitted methods, and symplectic integrators and the efficient integration of the Schrodinger equation.
Abstract: In this paper, we investigate the connection between • closed Newton-Cotes formulae, • trigonometrically-fitted methods, • symplectic integrators and • efficient integration of the Schrodinger equation. The study of multistep symplectic integrators is very poor although in the last decades several one step symplectic integrators have been produced based on symplectic geometry (see the relevant literature and the references here). In this paper we study the closed Newton-Cotes formulae and we write them as symplectic multilayer structures. Based on the closed Newton-Cotes formulae, we also develop trigonometrically-fitted symplectic methods. An error analysis for the one-dimensional Schrodinger equation of the new developed methods and a comparison with previous developed methods is also given. We apply the new symplectic schemes to the well-known radial Schrodinger equation in order to investigate the efficiency of the proposed method to these type of problems.
153 citations
TL;DR: This work studies how the vanishing of the phase-lag and its derivatives optimizes the efficiency of the hybrid two-step method for the numerical solution of the radial Schrodinger equation and related problems with periodic or oscillating solutions.
Abstract: We use a methodology of optimization of the efficiency of a hybrid two-step
method for the numerical solution of the radial Schrodinger equation and related problems with
periodic or oscillating solutions. More specifically, we study how the vanishing of the phase-lag
and its derivatives optimizes the efficiency of the hybrid two-step method.
139 citations
TL;DR: In this paper, a modified Runge-Kutta-Nystrom method of third algebraic order is developed, which has phase-lag and amplification error of order infinity.
Abstract: In this paper, a new modified Runge-Kutta-Nystrom method of third algebraic order is developed. The new modified RKN method has phase-lag and amplification error of order infinity, also the first derivative of the phase lag is of order infinity. Numerical results indicate that the new method presented in this paper, is much more efficient than other methods of the same algebraic order, for the numerical integration of orbital problems.
131 citations
TL;DR: In this article, the authors presented a new optimized symmetric eight-step semi-embedded predictor-corrector method (SEPCM) with minimal phase-lag, which is based on the symmetric multistep method of Quinlan-Tremaine (1), with eight steps and eighth algebraic order and is constructed to solve IVPs with oscillating solutions.
Abstract: In this paper we present a new optimized symmetric eight-step semi-embedded predictor-corrector method (SEPCM) with minimal phase-lag. The method is based on the symmetric multistep method of Quinlan-Tremaine (1), with eight steps and eighth algebraic order and is constructed to solve IVPs with oscillating solutions. We compare the new method to some recently constructed optimized methods and other methods from the literature. We measure the efficiency of the methods and conclude that the new method with minimal phase-lag is the most efficient of all the compared methods and for all the problems solved.
126 citations
References
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Book•
01 Jan 1974
TL;DR: In this paper, Newtonian mechanics: experimental facts investigation of the equations of motion, variational principles Lagrangian mechanics on manifolds oscillations rigid bodies, differential forms symplectic manifolds canonical formalism introduction to pertubation theory.
Abstract: Part 1 Newtonian mechanics: experimental facts investigation of the equations of motion. Part 2 Lagrangian mechanics: variational principles Lagrangian mechanics on manifolds oscillations rigid bodies. Part 3 Hamiltonian mechanics: differential forms symplectic manifolds canonical formalism introduction to pertubation theory.
11,008 citations
Book•
01 Jan 1994
TL;DR: Examples of Hamiltonian Systems, symplectic integration, and Numerical Methods: Checking preservation of area: Jacobians, and Necessity of the symplecticness conditions.
Abstract: Hamiltonian Systems. Examples of Hamiltonian Systems. Symplecticness. The solution operator. Preservation of area. Checking preservation of area: Jacobians. Checking preservation of area: differential forms. Symplectic transformations. Conservation of volume. Numerical Methods. Numerical integrators. Stiff problems. Runge-Kutta methods. Partitioned Runge-Kutta methods. Runge-Kutta-Nystrom methods. Composition of methods - adjoints. Order conditions. Order conditions for Runge-Kutta methods. The local error in Runge-Kutta methods. Order conditions for PRK methods. The local error in Partitioned Runge-Kutta methods. Order conditions for Runge-Kutta-Nystrom methods. The local error in Runge-Kutta-Nystrom mehthods. Implementation. Variable step sizes. Embedded pairs. Numerical experience with variable step sizes. Implementing implicit methods. Fourth-order Gauss method. Symplectic integration. Symplectic methods. Symplectic Runge-Kutta methods. Symplectic partitioned Runge-Kutta methods. Symplectic Runge-Kutta-Nystrom methods. Necessity of the symplecticness conditions. Symplectic order conditions. Prelimiaries. Order conditions for symplectic RK methods. Order conditions for symplectic PRK methods. Order conditions for symplectic RKN methods. Homogenous form of the order conditions. Available symplectic methods. Symplecticness of the Gauss methods. Diagonally implicity Runge-Kutta methods. Other symplectic Runge-Kutta methods. Explicit partitioned Runge-Kutta methods. Available symplectic Runge-Kutta-Nystrom methods. Numerical experiments. A comparison of symplectic integrators. Variable step sizes for symplectic methods. Conclusions and recommendations. Properties of symplectic integrators. Backward error interpretation. An alternative approach. Conservation of energy. KAM theory. Generating functions. The concept of generating function. Hamilton-Jacobi equations. Integrators based on generating functions. Generating functions for RK methods. Canonical order theory. Lie formalism. The Poisson bracket. Lie operators and Lie series. The Baker-Campbell-Hausdorff formula. Application to fractional-step methods. Extension to the non-Hamilton case. High-order methods. High-order Lie methods. High-order Runge-Kutta-Nystrom methods. A comparison of order 8 symplectic integrators. Extensions. Partitioned Runge-Kutta methods for nonseparable Hamiltonian systems. Canonical B-series. Conjugate symplectic methods. Trapezoidal rule. Constrained systems. General Poisson structures. Multistep methods. Partial differential equations. Reversable systems. Volume preserving flows.
1,327 citations
TL;DR: In this paper, a generalized derivation of explicit symplectic difference methods with any finite order of accuracy in the quantum system is presented, and the choice of coefficients and order for the best efficiency in multistep symplectic methods and Newton-Cotes differential methods are studied.
Abstract: Two kinds of numerical methods with a high order of accuracy are developed in this paper. In the general classical Hamiltonian system, it was claimed that no explicit n‐step symplectic difference method with the nth order of accuracy can be achieved if n is larger than 4. We show that there is no such constraint in the quantum system. We also exploit to investigate the high order Newton–Cotes differential methods in the quantum system. For the first time, we work out the generalized derivation of explicit symplectic difference methods with any finite order of accuracy in the quantum system. We point out that different coefficients in the same multistep symplectic method will lead to quite different results. The choices of coefficients and order of accuracy for the best efficiency in multistep symplectic methods and Newton–Cotes differential methods are studied. The connections between explicit symplectic difference structure, Newton–Cotes differential schemes, and other methods are presented. Numerical tests on the model system have also been carried out. The comparison shows that the explicit symplectic difference methods and the Newton–Cotes differential methods are both accurate and efficient.
51 citations
10 May 2002
48 citations
TL;DR: Recently, Zhu et al. as mentioned in this paper converted open Newton-Cotes differential methods into a multilayer symplectic structure so that multistep VPIs of a Hamiltonian system are obtained.
Abstract: Open Newton–Cotes differential methods that possess the characteristics of multilayer symplectic structures are shown in this paper. In numerical simulation, volume-preservation plays an important role in solving the Hamiltonian system. In this regard, developing a numerical integrator that preserves the volume in the phase space of a Hamiltonian system is a great challenge to the researchers in this field. Symplectic integrators were proven to be good candidates for volume-preserving integrators (VPIs) in the past ten years. Several one-step (single-stage or multistages) symplectic integrators have been developed based on the symplectic geometric theory. However, multistep VPIs have seldom been investigated by other researchers for the lack of an advanced theory. Recently, Zhu et al. converted open Newton–Cotes differential methods into a multilayer symplectic structure so that multistep VPIs of a Hamiltonian system are obtained. Mainly, their work has concentrated on the issue of achieving both the accu...
23 citations