共查询到20条相似文献，搜索用时 468 毫秒
1.
《Chinese Astronomy and Astrophysics》2005,29(1):92-103
The symplectic integrator has been regarded as one of the optimal tools for research on qualitative secular evolution of Hamiltonian systems in solar system dynamics. An integrable and separate Hamiltonian system H = H0 + Σi=1N εiHi (εi ≪ 1) forms a pseudo third order symplectic integrator, whose accuracy is approximately equal to that of the first order corrector of the Wisdom-Holman second order symplectic integrator or that of the Forest-Ruth fourth order symplectic integrator. In addition, the symplectic algorithm with force gradients is also suited to the treatment of the Hamiltonian system H = H0(q,p) + εH1(q), with accuracy better than that of the original symplectic integrator but not superior to that of the corresponding pseudo higher order symplectic integrator. 相似文献
2.
For a Hamiltonian that can be separated into N+1(N\geq 2) integrable parts, four algorithms can be built for a symplectic integrator. This research compares these algorithms for the
first and second order integrators. We found that they have similar local truncation errors represented by error Hamiltonian
but rather different numerical stability. When the computation of the main part of the Hamiltonian, H
0, is not expensive, we recommend to use S
* type algorithm, which cuts the calculation of the H
0 system into several small time steps as Malhotra(1991) did. As to the order of the N+1 parts in one step calculation, we found that from the large to small would get a slower error accumulation.
This revised version was published online in July 2006 with corrections to the Cover Date. 相似文献
3.
几类辛方法的数值稳定性研究 总被引：1，自引：0，他引：1
主要对一阶隐式Euler辛方法M1、二阶隐式Euler中点辛方法M2、一阶显辛Euler方法M3和二阶leapfrog显辛积分器M4共4种辛方法及一些组合算法进行了通常意义下的线性稳定性分析．针对线性哈密顿系统,理论上找到每个数值方法的稳定区,然后用数值方法检验其正确性．对于哈密顿函数为实对称二次型的情况,为了理论推导便利,特推荐采用相似变换将二次型的矩阵对角化来研究辛方法的线性稳定性．当哈密顿分解为一个主要部分和一个小摄动次要部分且二者皆可积时,无论是线性系统还是非线性系统,这种主次分解与哈密顿具有动势能分解相比,明显扩大了辛方法的稳定步长范围． 相似文献
4.
We construct an explicit reversible symplectic integrator for the planar 3-body problem with zero angular momentum. We start with a Hamiltonian of the planar 3-body problem that is globally regularised and fully symmetry reduced. This Hamiltonian is a sum of 10 polynomials each of which can be integrated exactly, and hence a symplectic integrator is constructed. The performance of the integrator is examined with three numerical examples: The figure eight, the Pythagorean orbit, and a periodic collision orbit. 相似文献
5.
Xinhao Liao 《Celestial Mechanics and Dynamical Astronomy》1996,66(3):243-253
In this paper, following the idea of constructing the mixed symplectic integrator (MSI) for a separable Hamiltonian system, we give a low order mixed symplectic integrator for an inseparable, but nearly integrable, Hamiltonian system, Although the difference schemes of the integrators are implicit, they not only have a small truncation error but, due to near integrability, also a faster convergence rate of iterative solution than ordinary implicit integrators, Moreover, these second order integrators are time-reversible. 相似文献
6.
An explicit symplectic integrator is constructed for the problem of a rotating planetary satellite on a Keplerian orbit. The
spin vector is fixed perpendicularly to the orbital plane. The integrator is constructed according to the Wisdom-Holman approach:
the Hamiltonian is separated in two parts so that one of them is multiplied by a small parameter. The parameter depends on
the satellite’s shape or the eccentricity of its orbit. The leading part of the Hamiltonian for small eccentricity orbits
is similar to the simple pendulum and hence integrable; the perturbation does not depend on angular momentum which implies
a trivial ‘kick’ solution. In spite of the necessity to evaluate elliptic function at each step, the explicit symplectic integrator
proves to be quite efficient.
This revised version was published online in July 2006 with corrections to the Cover Date. 相似文献
7.
辛积分器中沿迹误差的一种补偿方法 总被引：2，自引：0，他引：2
辛积分器严格描述了一摄动Ｈａｍｉｌｔｏｎ系统的流，因而导致天体轨道的沿迹误差随时间呈线性增长趋势。本文利用这一特点，提出了一种对其沿迹误差进行估算的数值方法，从而达到了对数值结果进行沿迹误差补偿的目的，数值结果证实了此方法在较大积分步长和较长积分时间的数值计算中是有效的。 相似文献
8.
9.
This paper deals mainly with the application of the mixed leapfrog symplectic integrators with adaptive timestep to a conservative post-Newtonian Hamiltonian formulation with canonical spins for spinning compact binaries. The adaptive timestep depends on the two body separation r and the magnitude of the spins. Various numerical tests including a chaotic high-eccentricity orbit show that the fixed step symplectic integrators lost drastically the good long term behaviour in the test cases with large eccentricity, the adaptive timestep integrator is always superior to the constant step in the integral precision. 相似文献
10.
Haruo Yoshida 《Celestial Mechanics and Dynamical Astronomy》1993,56(1-2):27-43
In this paper various aspect of symplectic integrators are reviewed. Symplectic integrators are numerical integration methods for Hamiltonian systems which are designed to conserve the symplectic structure exactly as the original flow. There are explicit symplectic schemes for systems of the formH=T(p)+V(q), and implicit schemes for general Hamiltonian systems. As a general property, symplectic integrators conserve the energy quite well and therefore an artificial damping (excitation) caused by the accumulation of the local truncation error cannot occur. Symplectic integrators have been applied to the Kepler problem, the motion of minor bodies in the solar system and the long-term evolution of outer planets. 相似文献
11.
By Hamiltonian manipulation we demonstrate the existence of separable time‐transformed Hamiltonians in the extended phase‐space.
Due to separability explicit symplectic methods are available for the solution of the equations of motion. If the simple leapfrog
integrator is used, in case of two‐body motion, the method produces an exact Keplerian ellipse in which only the time‐coordinate
has an error. Numerical tests show that even the rectilinear N‐body problem is feasible using only the leapfrog integrator.
In practical terms the method cannot compete with regularized codes, but may provide new directions for studies of symplectic
N‐body integration.
This revised version was published online in July 2006 with corrections to the Cover Date. 相似文献
12.
Sławomir Breiter 《Celestial Mechanics and Dynamical Astronomy》1998,71(4):229-241
An explicit symplectic integrator is constructed for perturbed elliptic orbits of an arbitrary eccentricity. The perturbation
should be Hamiltonian, but it may depend on time explicitly. The main feature of the integrator is the use of KS variables
in the ten-dimensional extended phase space. As an example of its application the motion of an Earth satellite under the action
of the planet's oblateness and of lunar perturbations is studied. The results confirm the superiority of the method over a
classical Wisdom–Holman algorithm in both accuracy and computation time.
This revised version was published online in July 2006 with corrections to the Cover Date. 相似文献
13.
14.
We describe a parallel hybrid symplectic integrator for planetary system integration that runs on a graphics processing unit (GPU). The integrator identifies close approaches between particles and switches from symplectic to Hermite algorithms for particles that require higher resolution integrations. The integrator is approximately as accurate as other hybrid symplectic integrators but is GPU accelerated. 相似文献
15.
Kenneth R. Meyer Dieter S. Schmidt 《Celestial Mechanics and Dynamical Astronomy》1982,28(1-2):201-207
The fundamental matrix solutionT for the variational equations of a Hamiltonian system is symplectic. We use this fact to completeT when it is only partially known. We discuss three cases. The last one gives an easy proof for the method invented by Brown in his lunar theory.Paper presented at the 1981 Oberwolfach Conference on Mathematical Methods in Celestial Mechanics.Dedicated to Victor Szebehely. 相似文献
16.
Haruo Yoshida 《Celestial Mechanics and Dynamical Astronomy》2002,83(1-4):355-364
Symplectic integration methods conserve the Hamiltonian quite well because of the existence of the modified Hamiltonian as a formal conserved quantity. For a first integral of a given Hamiltonian system, the modified first integral is defined to be a formal first integral for the modified Hamiltonian. It is shown that the Runge-Lenz vector of the Kepler problem is not well conserved by symplectic methods, and that the corresponding modified first integral does not exist. This conclusion is given for a one-parameter family of symplectic methods including the symplectic Euler method and the Störmer/Verlet method. 相似文献
17.
A new algorithm is developed for long-term integrations of the N-body problem. The method uses symplectic integrations of the Hamiltonian equations of motion for each body. This allows one
to employ individual adaptive time-steps in computations. The efficiency of this technique is demonstrated by several tests
performed for typical problems of Solar System dynamics. 相似文献
18.
Jean-Marc Petit 《Celestial Mechanics and Dynamical Astronomy》1998,70(1):1-21
We investigate the numerical implementation of a symplectic integrator combined with a rotation (as in the case of an elongated
rotating primary). We show that a straightforward implementation of the rotation as a matrix multiplication destroys the conservative
property of the global integrator, due to roundoff errors. According to Blank et al. (1997), there exists a KAM-like theorem
for twist maps, where the angle of rotation is a function of the radius. This theorem proves the existence of invariant tori
which confine the orbit and prevent shifts in radius. We replace the rotation by a twist map or a combination of shears that
display the same kind of behaviour and show that we are able not only to recover the conservative properties of the rotation,
but also make it more efficient in term of computing time. Next we test the shear combination together with symplectic integrator
of order 2, 4, and 6 on a Keplerian orbit. The resulting integrator is conservative down to the roundoff errors. No linear
drift of the energy remains, only a divergence as the square root of the number of iterations is to be seen, as in a random
walk. We finally test the three symplectic integrators on a real case problem of the orbit of a satellite around an elongated
irregular fast rotating primary. We compare these integrators to the well-known general purpose, self-adaptative Bulirsch–Stoer
integrator. The sixth order symplectic integrator is more accurate and faster than the Bulirsch–Stoer integrator. The second-
and fourth- order integrators are faster, but of interest only when extreme speed is mandatory.
This revised version was published online in August 2006 with corrections to the Cover Date. 相似文献
19.
Patrick Michel Giovanni H. Valsecchi 《Celestial Mechanics and Dynamical Astronomy》1996,65(4):355-371
We discuss the efficiency of the so-called mixed-variable symplectic integrators for N-body problems. By performing numerical experiments, we first show that the evolution of the mean error in action-like variables is strongly dependent on the initial configuration of the system. Then we study the effect of changing the stepsize when dealing with problems including close encounters between a particle and a planet. Considering a previous study of the slow encounter between comet P/Oterma and Jupiter, we show that the overall orbital patterns can be reproduced, but this depends on the chosen value of the maximum integration stepsize. Moreover the Jacobi constant in a restricted three-body problem is not conserved anymore when the stepsize is changed frequently: over a 105 year time span, to keep a relative error in this integral of motion of the same order as that given by a Bulirsch-Stoer integrator requires a very small integration stepsize and much more computing time. However, an integration of a sample including 104 particles close to Neptune shows that the distributions of the variation of the elements over one orbital period of the particles obtained by the Bulirsch-Stoer integrator and the symplectic integrator up to a certain integration stepsize are rather similar. Therefore, mixed-variable symplectic integrators are efficient either for N-body problems which do not include close encounters or for statistical investigations on a big sample of particles. 相似文献
20.
当史瓦西黑洞周围存在渐近均匀的外部磁场时, 描述带电粒子在史瓦西黑洞附近运动的哈密顿系统会变为不可积系统. 类似于这样的相对论哈密顿系统不存在有显式分析解的2部分分离形式, 给显式辛算法的构建和应用带来困难. 近一年以来的系列工作提出将相对论哈密顿系统分解为具有显式分析解的2个以上分离部分形式, 成功解决了许多相对论时空构建显式辛算法的难题. 最近的工作回答了哈密顿系统显式可积分离数目对长期数值积分精度有何影响、哪种显式辛算法有最佳长期数值性能这两个问题, 指出哈密顿有最小可积分离数目即3部分分裂解形式并且应用于优化的4阶分段龙格库塔显式辛算法可取得最好精度. 由此选择上述数值积分方法并利用庞加莱截面、最大李雅普诺夫指数和快速李雅普诺夫指标研究在磁化史瓦西黑洞附近运动的带电粒子轨道动力学. 结果显示: 针对某特定的粒子能量和角动量, 较小的外部磁场很难形成混沌轨道; 较大的正磁场参数容易使轨道产生混沌, 并且随着磁场的增大, 轨道的混沌程度也随之加强; 粒子能量适当变大也可以加剧混沌程度, 但负磁场参数和粒子角动量变大都会减弱混沌. 相似文献