共查询到20条相似文献，搜索用时 15 毫秒
1.
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
2.
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
3.
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
4.
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
5.
6.
《Chinese Astronomy and Astrophysics》2007,31(2):172-186
In this paper, we analyze the linear stabilities of several symplectic integrators, such as the first-order implicit Euler scheme, the second-order implicit mid-point Euler difference scheme, the first-order explicit Euler scheme, the second-order explicit leapfrog scheme and some of their combinations. For a linear Hamiltonian system, we find the stable regions of each scheme by theoretical analysis and check them by numerical tests. When the Hamiltonian is real symmetric quadratic, a diagonalizing by a similar transformation is suggested so that the theoretical analysis of the linear stability of the numerical method would be simplified. A Hamiltonian may be separated into a main part and a perturbation, or it may be spontaneously separated into kinetic and potential energy parts, but the former separation generally is much more charming because it has a much larger maximum step size for the symplectic being stable, no matter this Hamiltonian is linear or nonlinear. 相似文献
7.
《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
8.
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
9.
A recurrent method of solving the formal integrals of symplectic integrators is given. The special examples show that there are no long-term variations in all integrals of the Hamiltonian system in addition to the energy one when symplectic integrators are used in the numerical studies of the system. As an application of the formal integrals, the relation between them and the linear stability of symplectic integrators is discussed. 相似文献
10.
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
11.
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
12.
We obtain thex - p
xPoincare phase plane for a two dimensional, resonant, galactic type Hamiltonian using conventional numerical integration,
a second order symplectic integrator and a map based on the averaged Hamiltonian. It is found that all three methods give
good results, for small values of the perturbation parameter, while the symplectic integrator does a better job than the mapping,
for large perturbations. The dynamical spectra are used to distinguish between regular and chaotic motion. 相似文献
13.
Toshio Fukushima 《Celestial Mechanics and Dynamical Astronomy》1999,73(1-4):231-241
This paper reviews three recent works on the numerical methods to integrate ordinary differential equations (ODE), which are
specially designed for parallel, vector, and/or multi-processor-unit(PU) computers. The first is the Picard-Chebyshev method
(Fukushima, 1997a). It obtains a global solution of ODE in the form of Chebyshev polynomial of large (> 1000) degree by applying
the Picard iteration repeatedly. The iteration converges for smooth problems and/or perturbed dynamics. The method runs around
100-1000 times faster in the vector mode than in the scalar mode of a certain computer with vector processors (Fukushima,
1997b). The second is a parallelization of a symplectic integrator (Saha et al., 1997). It regards the implicit midpoint rules
covering thousands of timesteps as large-scale nonlinear equations and solves them by the fixed-point iteration. The method
is applicable to Hamiltonian systems and is expected to lead an acceleration factor of around 50 in parallel computers with
more than 1000 PUs. The last is a parallelization of the extrapolation method (Ito and Fukushima, 1997). It performs trial
integrations in parallel. Also the trial integrations are further accelerated by balancing computational load among PUs by
the technique of folding. The method is all-purpose and achieves an acceleration factor of around 3.5 by using several PUs.
Finally, we give a perspective on the parallelization of some implicit integrators which require multiple corrections in solving
implicit formulas like the implicit Hermitian integrators (Makino and Aarseth, 1992), (Hut et al., 1995) or the implicit symmetric
multistep methods (Fukushima, 1998), (Fukushima, 1999).
This revised version was published online in July 2006 with corrections to the Cover Date. 相似文献
14.
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
15.
16.
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
17.
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
18.
An operator associated with third-order potential derivatives and a force gradient operator corresponding to second-order potential derivatives are used together to design a number of new fourth-order explicit symplectic integrators for the natural splitting of a Hamiltonian into both the kinetic energy with a quadratic form of momenta and the potential energy as a function of position coordinates.Numerical simulations show that some new optimal symplectic algorithms are much better than their non-optimal c... 相似文献
19.
We consider Sundman and Poincaré transformations for the long-time numerical integration of Hamiltonian systems whose evolution occurs at different time scales. The transformed systems are numerically integrated using explicit symplectic methods. The schemes we consider are explicit symplectic methods with adaptive time steps and they generalise other methods from the literature, while exhibiting a high performance. The Sundman transformation can also be used on non-Hamiltonian systems while the Poincaré transformation can be used, in some cases, with more efficient symplectic integrators. The performance of both transformations with different symplectic methods is analysed on several numerical examples. 相似文献
20.
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