摘要:
当Hamilton函数分解为可积和不可积两部分时,前者能用分析方法给出解析解,而后者可借助一阶半隐Euler法或二阶隐中点法等数值求解,将这种解析和数值解法组合能构造二阶混合辛积分器.理论分析表明Euler嵌入法的稳定区要小于中点嵌入法的.再分别以圆形限制性三体问题和相对论自旋致密双星后牛顿Hamilton构型为例,详细比较了两嵌入法的性能特点.二者的数值精度、稳定性及计算效率与Hamilton的分解方式和轨道类型有关.就圆形限制性三体问题而言,当Hamilton采用势能和含坐标与动量混合项在内的动能分解时,无论对有序还是混沌轨道两嵌入法的数值性能基本相同;然而,当将含坐标与动量混合项做一个独立部分而其余的为另一个部分去分解Hamilton时Euler嵌入法数值稳定性劣于中点嵌入法.针对自旋致密双星来说,Euler嵌入法比中点嵌入法也具有明显的计算效率优势;当两嵌入法均保持数值稳定时二者精度没有显著差别;可是,在混沌存在等情况下Euler嵌入法大多变得数值不稳定.综合表明中点嵌入法具有较好的数值稳定性,尤其值得推荐用来求解后牛顿Hamilton动力学问题.
Abstract:
When a Hamiltonian can be split into integrable and nonintegrable parts, the former part is solved analytically, and the latter one is integrated numerically by means of implicit symplectic integrators such as the first-order semi-implicit Euler method or the second-order implicit midpoint rule. These analytical and numerical solutions are used to construct a second-order mixed symplectic integrator with the semi-implicit Euler method and one with the implicit midpoint rule. A theoretical analysis shows that the Euler mixed integrator is inferior to the midpoint one in the sense of numerical stability. Numerical simulations of the circularly-restricted three-body problem also support this fact. It is further shown through numerical integrations of the post-Newtonian Hamiltonian of spinning compact binaries that the qualities of the Euler mixed integrator and the midpoint mixed method do depend on the type of orbits. Especially for chaotic orbits, the Euler mixed integrator often becomes unstable. In addition, the Euler mixed integrator has an advantage over the midpoint mixed method in computational efficiency, and is almost equivalent to the latter in the numerical accuracy if the two mixed integrators are stable. In spite of this, the midpoint mixed integrator is worth recommending for the study of the dynamics of post-Newtonian Hamiltonians of spinning compact binaries.
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