scholarly journals Adjustment of Force–Gradient Operator in Symplectic Methods

Mathematics ◽  
2021 ◽  
Vol 9 (21) ◽  
pp. 2718
Author(s):  
Lina Zhang ◽  
Xin Wu ◽  
Enwei Liang
Keyword(s):  
Spiral Structure ◽  
Symplectic Integrators ◽  
Symplectic Methods ◽  
Time Reversibility ◽  
Hamiltonian Problems ◽  
Gradient Operator ◽  
Numerical Tests ◽  
Integration Algorithms ◽  
Dynamical Features ◽  
Numerical Performance

Many force–gradient explicit symplectic integration algorithms have been designed for the Hamiltonian H=T(p)+V(q) with kinetic energy T(p)=p2/2 in the existing references. When a force–gradient operator is appropriately adjusted as a new operator, it is still suitable for a class of Hamiltonian problems H=K(p,q)+V(q) with integrable part K(p,q)=∑i=1n∑j=1naijpipj+∑i=1nbipi, where aij=aij(q) and bi=bi(q) are functions of coordinates q. The newly adjusted operator is not a force–gradient operator but is similar to the momentum-version operator associated to the potential V. The newly extended (or adjusted) algorithms are no longer solvers of the original Hamiltonian, but are solvers of slightly modified Hamiltonians. They are explicit symplectic integrators with symmetry or time reversibility. Numerical tests show that the standard symplectic integrators without the new operator are generally poorer than the corresponding extended methods with the new operator in computational accuracies and efficiencies. The optimized methods have better accuracies than the corresponding non-optimized counterparts. Among the tested symplectic methods, the two extended optimized seven-stage fourth-order methods of Omelyan, Mryglod and Folk exhibit the best numerical performance. As a result, one of the two optimized algorithms is used to study the orbital dynamical features of a modified Hénon–Heiles system and a spring pendulum. These extended integrators allow for integrations in Hamiltonian problems, such as the spiral structure in self-consistent models of rotating galaxies and the spiral arms in galaxies.

Splitting K-symplectic methods for non-canonical separable Hamiltonian problems

2016 ◽  
Vol 322 ◽  
pp. 387-399 ◽  
Author(s):  
Beibei Zhu ◽  
Ruili Zhang ◽  
Yifa Tang ◽  
Xiongbiao Tu ◽  
Yue Zhao
Keyword(s):  
Symplectic Methods ◽  
Hamiltonian Problems

SYMPLECTIC NUMERICAL METHODS FOR HAMILTONIAN PROBLEMS

1993 ◽  
Vol 04 (02) ◽  
pp. 385-392 ◽  
Author(s):  
J. M. SANZ-SERNA ◽  
M. P. CALVO
Keyword(s):  
Numerical Methods ◽  
Numerical Integration ◽  
Numerical Experiments ◽  
Symplectic Methods ◽  
Hamiltonian Problems ◽  
Integral Invariants

We consider symplectic methods for the numerical integration of Hamiltonian problems, i.e. methods that preserve the Poincaré integral invariants. Examples of symplectic methods are given and numerical experiments are reported.

scholarly journals Efficiency of Runge-Kutta Methods in Solving Simple Harmonic Oscillators

MATEMATIKA ◽  
2018 ◽  
Vol 34 (1) ◽  
pp. 1-12
Author(s):  
Annie Gorgey ◽  
Nor Azian Aini Mat
Keyword(s):  
Harmonic Oscillator ◽  
Numerical Solutions ◽  
Harmonic Oscillators ◽  
Symplectic Method ◽  
Symplectic Methods ◽  
Error Growth ◽  
Hamiltonian Problems ◽  
Simple Harmonic Oscillator ◽  
Short Period ◽  
Gauss Method

Symmetric methods such as the implicit midpoint rule (IMR), implicit trapezoidal rule (ITR) and 2-stage Gauss method are beneficial in solving Hamiltonian problems since they are also symplectic. Symplectic methods have advantages over non-symplectic methods in the long term integration of Hamiltonian problems. The study is to show the efficiency of IMR, ITR and the 2-stage Gauss method in solving simple harmonic oscillators (SHO). This study is done theoretically and numerically on the simple harmonic oscillator problem. The theoretical analysis and numerical results on SHO problem showed that the magnitude of the global error for a symmetric or symplectic method with stepsize h is linearly dependent on time t. This gives the linear error growth when a symmetric or symplectic method is applied to the simple harmonic oscillator problem. Passive and active extrapolations have been implemented to improve the accuracy of the numerical solutions. Passive extrapolation is observed to show quadratic error growth after a very short period of time. On the other hand, active extrapolation is observed to show linear error growth for a much longer period of time.

Closed Newton-Cotes Trigonometrically-Fitted Formulae for Numerical Integration of the Schrödinger Equation

2007 ◽  
Vol 3 (1) ◽  
pp. 45-57 ◽  
Author(s):  
T.E. Simos
Keyword(s):  
Numerical Integration ◽  
Symplectic Geometry ◽  
Efficient Solution ◽  
Symplectic Integrators ◽  
Multilayer Structures ◽  
Symplectic Methods ◽  
One Dimensional ◽  
One Step ◽  
Differential Methods ◽  
Symplectic Schemes

In this paper we investigate the connection between closed Newton-Cotes formulae, trigonometrically-fitted differential methods, symplectic integrators and efficient solution of the Schr¨odinger equation. Several one step symplectic integrators have been produced based on symplectic geometry, as one can see from the literature. However, the study of multistep symplectic integrators is very poor. Zhu et. al. [1] has studied the symplectic integrators and the well known open Newton-Cotes differential methods and as a result has presented the open Newton-Cotes differential methods as multilayer symplectic integrators. The construction of multistep symplectic integrators based on the open Newton-Cotes integration methods was investigated by Chiou and Wu [2]. In this paper we investigate the closed Newton-Cotes formulae and we write them as symplectic multilayer structures. We also develop trigonometrically-fitted symplectic methods which are based on the closed Newton-Cotes formulae. We apply the symplectic schemes to the well known one-dimensional Schr¨odinger equation in order to investigate the efficiency of the proposed method to these type of problems.

scholarly journals High-order symplectic integrators for planetary dynamics and their implementation in rebound

2019 ◽  
Vol 489 (4) ◽  
pp. 4632-4640 ◽  
Author(s):  
Hanno Rein ◽  
Daniel Tamayo ◽  
Garett Brown
Keyword(s):  
Large Time ◽  
Good Accuracy ◽  
Long Term Evolution ◽  
High Order ◽  
Symplectic Integrators ◽  
Symplectic Methods ◽  
Available N ◽  
Term Evolution ◽  
Planetary Dynamics

ABSTRACT Direct N-body simulations and symplectic integrators are effective tools to study the long-term evolution of planetary systems. The Wisdom–Holman (WH) integrator in particular has been used extensively in planetary dynamics as it allows for large time-steps at good accuracy. One can extend the WH method to achieve even higher accuracy using several different approaches. In this paper, we survey integrators developed by Wisdom et al., Laskar & Robutel, and Blanes et al. Since some of these methods are harder to implement and not as readily available to astronomers compared to the standard WH method, they are not used as often. This is somewhat unfortunate given that in typical simulations it is possible to improve the accuracy by up to six orders of magnitude (!) compared to the standard WH method without the need for any additional force evaluations. To change this, we implement a variety of high-order symplectic methods in the freely available N-body integrator rebound. In this paper, we catalogue these methods, discuss their differences, describe their error scalings, and benchmark their speed using our implementations.

CLOSED NEWTON–COTES TRIGONOMETRICALLY-FITTED FORMULAE FOR LONG-TIME INTEGRATION

2003 ◽  
Vol 14 (08) ◽  
pp. 1061-1074 ◽  
Author(s):  
T. E. SIMOS
Keyword(s):  
Symplectic Geometry ◽  
Equations Of Motion ◽  
Time Integration ◽  
Symplectic Integrators ◽  
Symplectic Methods ◽  
Long Time ◽  
One Step ◽  
Long Time Integration ◽  
Differential Methods ◽  
Symplectic Schemes

The connection between closed Newton–Cotes, trigonometrically-fitted differential methods and symplectic integrators is investigated in this paper. It is known from the literature that several one-step symplectic integrators have been obtained based on symplectic geometry. However, the investigation of multistep symplectic integrators is very poor. Zhu et al.2 presented the well known open Newton–Cotes differential methods as multilayer symplectic integrators. Chiou and Wu2 also investigated the construction of multistep symplectic integrators based on the open Newton–Cotes integration methods. In this paper we investigate the closed Newton–Cotes formulae and we write them as symplectic multilayer structures. After this we construct trigonometrically-fitted symplectic methods which are based on the closed Newton–Cotes formulae. We apply the symplectic schemes in order to solve Hamilton's equations of motion which are linear in position and momentum. We observe that the Hamiltonian energy of the system remains almost constant as integration procceeds.

High-Order Symplectic Schemes for Stochastic Hamiltonian Systems

2014 ◽  
Vol 16 (1) ◽  
pp. 169-200 ◽  
Author(s):  
Jian Deng ◽  
Cristina Anton ◽  
Yau Shu Wong
Keyword(s):  
Hamiltonian Systems ◽  
High Order ◽  
Symplectic Integrators ◽  
Symplectic Methods ◽  
Computational Tools ◽  
Fully Implicit ◽  
Stochastic Hamiltonian Systems ◽  
Hamiltonian Functions ◽  
Symplectic Schemes

AbstractThe construction of symplectic numerical schemes for stochastic Hamiltonian systems is studied. An approach based on generating functions method is proposed to generate the stochastic symplectic integration of any desired order. In general the proposed symplectic schemes are fully implicit, and they become computationally expensive for mean square orders greater than two. However, for stochastic Hamiltonian systems preserving Hamiltonian functions, the high-order symplectic methods have simpler forms than the explicit Taylor expansion schemes. A theoretical analysis of the convergence and numerical simulations are reported for several symplectic integrators. The numerical case studies confirm that the symplectic methods are efficient computational tools for long-term simulations.

An optimized Forest–Ruth-like algorithm in extended phase space

2018 ◽  
Vol 29 (01) ◽  
pp. 1850006 ◽  
Author(s):  
Ya-Lin Wu ◽  
Xin Wu
Keyword(s):  
Phase Space ◽  
Extended Phase Space ◽  
Symplectic Integrator ◽  
Hamiltonian Problems ◽  
Extended Phase ◽  
Numerical Tests ◽  
Numerical Integrations ◽  
Phase Space Methods ◽  
Fifth Order

An optimized fourth-order Forest–Ruth-like symplectic algorithm, which is based on a minimum of the norm of fifth-order truncation terms, was originally designed for separable Hamiltonian problems. With the aid of extended phase space methods, this algorithm is further reformulated to explicitly solve inseparable Hamiltonian systems. Although an extra permutation substep is included and destroys symplecticity in general, the method is symmetric and therefore resembles a symplectic integrator in conservation of the original Hamiltonian. In fact, our numerical tests show that the optimized algorithm combined with the midpoint permutations always enhances the quality of numerical integrations in comparison with the corresponding nonoptimized counterpart for inseparable Hamiltonian problems. As a result, the optimized algorithm is worth recommending in application.

AN ADAPTED SYMPLECTIC INTEGRATOR FOR HAMILTONIAN PROBLEMS

2001 ◽  
Vol 12 (02) ◽  
pp. 225-234 ◽  
Author(s):  
JESÚS VIGO-AGUIAR ◽  
T. E. SIMOS ◽  
A. TOCINO
Keyword(s):  
Numerical Results ◽  
New Method ◽  
Symplectic Integrators ◽  
Order Method ◽  
Symplectic Integrator ◽  
Hamiltonian Problems ◽  
Trigonometric Fitting ◽  
Algebraic Order ◽  
Runge Kutta ◽  
Trigonometrically Fitted Method

In this paper, a new procedure for deriving efficient symplectic integrators for Hamiltonian problems is introduced. This procedure is based on the combination of the trigonometric fitting technique and symplecticness conditions. Based on this procedure, a simple modified Runge–Kutta–Nyström second algebraic order trigonometrically fitted method is developed. We present explicity the symplecticity conditions for the new modified Runge–Kutta–Nyström method. Numerical results indicate that the new method is much more efficient than the "classical" symplectic Runge–Kutta–Nyström second algebraic order method introduced in Ref. 1.

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