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.

Construction of Symplectic Runge-Kutta Methods for Stochastic Hamiltonian Systems

2016 ◽  
Vol 21 (1) ◽  
pp. 237-270 ◽  
Author(s):  
Peng Wang ◽  
Jialin Hong ◽  
Dongsheng Xu
Keyword(s):  
Case Studies ◽  
Hamiltonian Systems ◽  
Multiplicative Noise ◽  
Additive Noise ◽  
Symplectic Methods ◽  
Computational Tools ◽  
Stochastic Hamiltonian Systems ◽  
Runge Kutta ◽  
Strong Order

AbstractWe study the construction of symplectic Runge-Kutta methods for stochastic Hamiltonian systems (SHS). Three types of systems, SHS with multiplicative noise, special separable Hamiltonians and multiple additive noise, respectively, are considered in this paper. Stochastic Runge-Kutta (SRK) methods for these systems are investigated, and the corresponding conditions for SRK methods to preserve the symplectic property are given. Based on the weak/strong order and symplectic conditions, some effective schemes are derived. In particular, using the algebraic computation, we obtained two classes of high weak order symplectic Runge-Kutta methods for SHS with a single multiplicative noise, and two classes of high strong order symplectic Runge-Kutta methods for SHS with multiple multiplicative and additive noise, respectively. The numerical case studies confirm that the symplectic methods are efficient computational tools for long-term simulations.

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 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 regularised symplectic integrator for collisional planetary systems

2019 ◽  
Vol 628 ◽  
pp. A32 ◽  
Author(s):  
Antoine C. Petit ◽  
Jacques Laskar ◽  
Gwenaël Boué ◽  
Mickaël Gastineau
Keyword(s):  
Hybrid Methods ◽  
Planetary Systems ◽  
High Order ◽  
Symplectic Integrators ◽  
Step Size ◽  
Time Step ◽  
Close Encounters ◽  
The Stability ◽  
Strong Scattering

We present a new mixed variable symplectic (MVS) integrator for planetary systems that fully resolves close encounters. The method is based on a time regularisation that allows keeping the stability properties of the symplectic integrators while also reducing the effective step size when two planets encounter. We used a high-order MVS scheme so that it was possible to integrate with large time-steps far away from close encounters. We show that this algorithm is able to resolve almost exact collisions (i.e. with a mutual separation of a fraction of the physical radius) while using the same time-step as in a weakly perturbed problem such as the solar system. We demonstrate the long-term behaviour in systems of six super-Earths that experience strong scattering for 50 kyr. We compare our algorithm to hybrid methods such as MERCURY and show that for an equivalent cost, we obtain better energy conservation.

scholarly journals REBOUNDx: a library for adding conservative and dissipative forces to otherwise symplectic N-body integrations

2019 ◽  
Vol 491 (2) ◽  
pp. 2885-2901 ◽  
Author(s):  
Daniel Tamayo ◽  
Hanno Rein ◽  
Pengshuai Shi ◽  
David M Hernandez
Keyword(s):  
Computational Cost ◽  
Dissipative Systems ◽  
Symplectic Methods ◽  
Splitting Schemes ◽  
Dissipative Forces ◽  
Binary Format ◽  
Carry Over ◽  
External Forces ◽  
Symplectic Schemes

ABSTRACT Symplectic methods, in particular the Wisdom–Holman map, have revolutionized our ability to model the long-term, conservative dynamics of planetary systems. However, many astrophysically important effects are dissipative. The consequences of incorporating such forces into otherwise symplectic schemes are not always clear. We show that moving to a general framework of non-commutative operators (dissipative or not) clarifies many of these questions, and that several important properties of symplectic schemes carry over to the general case. In particular, we show that explicit splitting schemes generically exploit symmetries in the applied external forces, which often strongly suppress integration errors. Furthermore, we demonstrate that so-called ‘symplectic correctors’ (which reduce energy errors by orders of magnitude at fixed computational cost) apply equally well to weakly dissipative systems and can thus be more generally thought of as ‘weak splitting correctors’. Finally, we show that previously advocated approaches of incorporating additional forces into symplectic methods work well for dissipative forces, but give qualitatively wrong answers for conservative but velocity-dependent forces like post-Newtonian corrections. We release REBOUNDx, an open-source C library for incorporating additional effects into REBOUNDN-body integrations, together with a convenient python wrapper. All effects are machine independent and we provide a binary format that interfaces with the SimulationArchive class in REBOUND to enable the sharing and reproducibility of results. Users can add effects from a list of pre-implemented astrophysical forces, or contribute new ones.

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.

Sign in / Sign up

Export Citation Format

Share Document