辛算法的分类与发展

辛算法作为研究哈密顿系统长期定性演化的最佳积分工具,自问世以来就受到了很大的关注。通过对哈密顿函数的截断误差分析,可以从不同角度构造出较高精度的辛算法,也可以通过引入正规化技术实现自动调整积分步长和改善数值稳定性。从辛算法的表现形式可以将它分为显式和隐式两种。当哈密顿系统能够分解为几个可积部分且每部分的解能用时间显函数来表示时,可以构造显式算法。显式算法有非力梯度显式辛算法、力梯度辛算法、辛校正、类高阶辛算法四种。当哈密顿系统变量不能分离时,适合应用隐式辛算法和扩充相空间对称算法求解。分别对这些算法的构造方法及其适用的物理模型进行归纳对比,分析了各种辛算法的优劣性和发展趋势,对如何选择辛算法高效高精度地解决实际问题提供了一定的理论和数值计算依据。     

关于太阳系小天体轨道演化的算法研究

太阳系小天体的运动对应—哈密顿(Hamilton)系统,对其轨道演化的数值研究宜采用哈密顿算法(即辛算法)。本文将仔细讨论这一问题,并以主带小行星的运动为例,较系统地介绍几种辛算法对应的显式辛差分格式。     

几类辛方法的数值稳定性研究

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．     

辛算法在动力天文中的应用（Ⅲ）

文［１］和文［２］从哈密顿系统的整体结构保持一角度阐明了辛算法［３－６］的主要功能，本文将从定量的角度进一步表明辛算法的另一独特优点－可以控制天体运动沿迹误差的快速增长，并对可分离哈密顿系统的显式辛差分格式稍加改进，推广应用到一般动力系统，该系统含有小耗散项或小的不可分离项，计算结果表明，效果极佳，因此，辛算法与传统的数值解法相比，确有很多优点。     

约束条件和数值积分

自治的哈密顿系统存在约束条件，例如能量积分或广义相对论中的4速度大小为常数，它能否在数值积分过程中始终满足将直接影响数值稳定性．在牛顿力学中哈密顿系统的动能一般为椭圆型，直接运用约束条件对方程进行降阶存在开平方判断正负号的困难，导致应用高精度的经典数值积分器时能量存在耗散．然而相对论力学的度规为双曲型，利用约束条件有可能实行方程降阶．在时空具有一定对称性的情况下，能够找到整个时空的一个全局变换使变换后的度规的主对角线某一元素为零，于是从约束方程中不需开平方能够解出某一动量，顺利实现运动方程的降阶．相对论力学中另一个可以降阶的模型是Mixmaster宇宙模型．数值实验表明将经典算法用于降阶后的运动方程能够严格地满足约束，但不一定能保持辛结构。     

自旋致密双星后牛顿哈密顿系统轨道动力学数值研究

由中子星或黑洞构成的旋转致密双星后牛顿哈密顿系统属于相对论二体问题,该系统不但含有丰富的共振和混沌等动力学现象,而且成为探测引力波的理想天然波源.引力体的轨道动力学性质会在引力波中得到反映.因此,实际天体的混沌性既可能是对引力波探测的挑战,又可望是获得观测效应的机遇.本学位论文正是在这样的国际学术氛围下数值研究旋转致密双星后牛顿保守哈密顿动力学问题.基于最小二乘法原理我们构造了单和双标度因子等几种流形改正方法,分别对旋转致密双星后牛     

一个膺三阶辛积分器

在太阳系动力学中,辛积分器已成为研究哈密顿系统的长期定性演化的最佳工具.对于可积分离的哈密顿系统H=H0+∑i=1N∈iHi(∈≤1),构造了一个膺三阶辛积分器.它大约相当于Wisdom-Holman二阶辛积分器的一次校正或Forest-Ruth四阶辛算法的精度.此外,含力梯度的辛算法也适合处理哈密顿系统H=Ho(q,P)+∈H1(q),其精度好于原辛积分器,但不优越于相应膺高阶辛积分器.     

保持Runge-Lenz向量的数值方法

对孤立积分和能够保持Runge-Lenz向量的梯形公式进行详尽讨论．孤立积分就是限制粒子运动区域的不变量，具有n个自由度的自治可积哈密顿系统且只有n个互相对合的独立孤立积分，并且其他孤立积分的存在对粒子的运动是有意义的，Kepler二体系统存在能量积分、角动量积分和Runge-Lenz向量．对于平面运动情况，这三类积分中只有3个独立孤立积分；而对于三维空间情形，该三类积分仅有5个是独立的．就前者而言，Kepler二体平面运动积分构成该系统中的对称群SO（3），经过Levi-Civita变换，它可以转化为二维各向同性谐振子系统中的对称群，而该对称群能够被梯形公式准确保持，另一方面，对于后者梯形公式对这三类积分的严格保持还可以在5个Kepler轨道根数n、e、i、Ω和w上得到体现。     

黑洞旋转能量的电磁提取及其在天体物理中的应用

该文着重介绍了两种大尺度磁场提取黑洞旋转能量的机制,即BZ机制和MC机制,以及BZMC共存模型在天体物理中的应用。BZ机制对应于连接黑洞与遥远天体物理负载的"开放"磁力线,而MC机制对应于连接黑洞与吸积盘的"闭合"磁力线。在BZ过程中大尺度磁场把黑洞的旋转能量以Poynting能流的形式输送到天体物理负载,成为驱动黑洞系统的喷流和伽马射线暴的中心发动机。在MC过程中能量和角动量通过大尺度磁场在黑洞与吸积盘之间转移。BZMC共存模型在高能天体物理中的应用包括以下几个方面:1)对活动星系核与黑洞双星的陡发射指数的拟合;2)对黑洞X射线双星的高频QPO与喷流的相关性的解释;3)对伽马射线暴的能量、时标的拟合以及对伽马射线暴与超新星成协的解释。最后对另外两种大尺度磁场提能机制,即BP机制和PC机制也作了简略介绍。     

近地小行星轨道演化的数值研究与辛算法有效性的探讨

本文采用改进的显式辛算法（symplecticalgorithm）和嵌套的RKF7（8）积分器对43颗已命名（或编号）的近地小行星的轨道演化进行数值研究．在力学模型上,除考虑各大行星的引力振动外,还增加了后牛顿效应,而在算法上则着重探索辛算法在近地小行星轨道演化研究中的应用前景,特别是当小行星与某一大行星靠近时辛算法的有效性．本文的结果可为了解近地小行星的轨道演化状况和对它们进行监测提供可靠的信息．     

Non-axisymmetric relativistic Bondi–Hoyle accretion on to a Schwarzschild black hole

We present the results of an exhaustive numerical study of fully relativistic non-axisymmetric Bondi–Hoyle accretion on to a moving Schwarzschild black hole. We have solved the equations of general relativistic hydrodynamics with a high-resolution shock-capturing numerical scheme based on a linearized Riemann solver. The numerical code was previously used to study axisymmetric flow configurations past a Schwarzschild black hole. We have analysed and discussed the flow morphology for a sample of asymptotically high Mach number models. The results of this work reveal that initially asymptotic uniform flows always accrete on to the hole in a stationary way, which closely resembles the previous axisymmetric patterns. This is in contrast with some Newtonian numerical studies where violent flip-flop instabilities were found. As discussed in the text, the reason can be found in the initial conditions used in the relativistic regime, as they cannot exactly duplicate the previous Newtonian setups where the instability appeared. The dependence of the final solution on the inner boundary condition as well as on the grid resolution has also been studied. Finally, we have computed the accretion rates of mass and linear and angular momentum.     

The runaway instability of thick discs around black holes I. The constant angular momentum case

We present results from a numerical study of the runaway instability of thick discs around black holes. This instability is an important issue for most models of cosmic gamma-ray bursts, where the central engine responsible for the initial energy release is such a system consisting of a thick disc surrounding a black hole. We have carried out a comprehensive number of time-dependent simulations aimed at exploring the appearance of the instability. Our study has been performed using a fully relativistic hydrodynamics code. The general relativistic hydrodynamic equations are formulated as a hyperbolic flux-conservative system and solved using a suitable Godunov-type scheme. We build a series of constant angular momentum discs around a Schwarzschild black hole. Furthermore, the self-gravity of the disc is neglected and the evolution of the central black hole is assumed to be that of a sequence of exact Schwarzschild black holes of varying mass. The black hole mass increase is thus determined by the mass accretion rate across the event horizon. In agreement with previous studies based on stationary models, we find that by allowing the mass of the black hole to grow the disc becomes unstable. Our hydrodynamical simulations show that for all disc-to-hole mass ratios considered (between 1 and 0.05), the runaway instability appears very fast on a dynamical time-scale of a few orbital periods, typically a few 10 ms and never exceeding 1 s for our particular choice of the mass of the black hole (2.5 M) and a large range of mass fluxes  ( m 10^-3 M s^-1)  . The implications of our results in the context of gamma-ray bursts are briefly discussed.     

Black hole spin dependence of general relativistic multi-transonic accretion close to the horizon

We introduce a novel formalism to investigate the role of the spin angular momentum of astrophysical black holes in influencing the behavior of low angular momentum general relativistic accretion. We propose a metric independent analysis of axisymmetric general relativistic flow, and consequently formulate the space and time dependent equations describing the general relativistic hydrodynamic accretion flow in the Kerr metric. The associated stationary critical solutions for such flow equations are provided and the stability of the stationary transonic configuration is examined using an elegant linear perturbation technique. We examine the properties of infalling material for both prograde and retrograde accretion as a function of the Kerr parameter at extremely close proximity to the event horizon. Our formalism can be used to identify a new spectral signature of black hole spin, and has the potential of performing the black hole shadow imaging corresponding to the low angular momentum accretion flow.     

Resonant tidal disruption in galactic nuclei

It has recently been shown by Rauch 38 Tremaine that the rate of angular momentum relaxation in nearly Keplerian star clusters is greatly increased by a process termed 'resonant relaxation'; it was also argued, via a series of scaling arguments, that tidal disruption of stars in galactic nuclei containing massive black holes could be noticeably enhanced by this process. We describe here the results of numerical simulations of resonant tidal disruption which quantitatively test the predictions made by Rauch 38 Tremaine. The simulation method is based on an N -body routine incorporating cloning of stars near the loss cone and a semirelativistic symplectic integration scheme. Normalized disruption rates for resonant and non-resonant nuclei are derived at orbital energies both above and below the critical energy, and the corresponding angular momentum distribution functions are found. The black hole mass above which resonant tidal disruption is quenched by relativistic precession is determined. We also briefly describe the discovery of chaos in the Wisdom–Holman symplectic integrator applied to highly eccentric orbits and propose a modified integration scheme that remains robust under these conditions. We find that resonant disruption rates exceed their non-resonant counterparts by an amount consistent with the predictions; in particular, we estimate the net tidal disruption rate for a fully resonant cluster to be about twice that of its non-resonant counterpart. No significant enhancement in rates is observed outside the critical radius. Relativistic quenching of the effect is found to occur for hole masses M  >  M _Q  = (8 ± 3) × 10⁷  M . The numerical results combined with the observed properties of galactic nuclei indicate that for most galaxies the resonant enhancement to tidal disruption rates will be very small.     

Direct numerical simulations of the Blandford–Znajek effect

The time-dependent general relativistic equations of degenerate electrodynamics are solved numerically in order to study the mechanism of the electromagnetic extraction of the rotational energy of black holes. We performed a series of 2D runs for black holes with specific angular momentum, a , from 0.1 to 0.9 and for a monopole magnetic field assuming axisymmetry. In the inner region of the wind, the solution quickly settles to a steady state with an outgoing Poynting flux. In all cases the angular velocity of the magnetic field lines is almost half the angular velocity of the black hole. Thus, at least for the configuration considered, the Blandford–Znajek mechanism operates near its maximum power output.     

A symplectic integrator for the symmetry reduced and regularised planar 3-body problem with vanishing angular momentum

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.     

The external field of a rotating and charged body in the vector graviton metric theory

We discuss certain properties of the external field of a rotating and charged body in the frame of the vector graviton metric field theory. We find: 1) a black hole cannot have angular momentum or charge, that is, a rotating body whether charged or not, cannot be a black hole. The Kerr black hole and the Kerr-Newman black hole do not exist. 2) For a rotating and charged axisymmetric body, there exists a latitude-dependent critical distance r_k(θ), such that the radial force acting on a test particle is attractive or repulsive according as the particle is outside or inside the critical distance. The repulsive force means that a massive object cannot collapse indefinitely. Maximum redshift in this case comes from sources on the equator. 3) A test particle also experiences a force along the meridian.     

Thermal properties of charged dilaton black hole, energy and momentum in teleparallel equivalent of general relativity

Two different charged dilaton black holes in 4-dimension, within teleparallel equivalent of general relativity (TEGR), are derived. These solutions are related through local Lorentz transformation. The total energy of these black holes, using three different methods, the Hamiltonian method, the translational momentum 2-form and the Euclidean continuation method given by Gibbons and Hawking, is calculated. It is shown that the three methods give the same results. The value of energy is shown to depend on the mass M and charge q. The verification of the first law of thermodynamics is proved. Finally, it is shown that if the charge q is vanishing then, the total energy reduced to that of Schwarzschild’s black hole.     

Some general relativistic effects on the dynamics of fluid disks rotating around a Schwarzschild black hole

In this paper we present various classes of solutions for perfect fluid disks rotating around Schwarzschild black holes. We study the profiles of pressure, density and specific angular momentum and the formation of cusp-like structures at the inner edge of the disks. Using the trial function method, we calculate the frequency of the global axi-symmetric oscillations. We compare the results with those of the corresponding Newtonian calculations to find the general relativistic effects.     

Large Proper-Motion Infrared

When some magnetic field lines connect a Kerr black hole with a disk rotating around it, energy and angular momentum are transferred between them. If the black hole rotates faster than the disk, ca&solm0;GMH>0.36 for a thin Keplerian disk, then energy and angular momentum are extracted from the black hole and transferred to the disk (MH is the mass and aMH is the angular momentum of the black hole). This way, the energy originating in the black hole may be radiated away by the disk. The total amount of energy that can be extracted from the black hole spun down from ca&solm0;GMH=0.998 to ca&solm0;GMH=0.36 by a thin Keplerian disk is approximately 0.15MHc2. This is larger than approximately 0.09MHc2, which can be extracted by the Blandford-Znajek mechanism.