标题： 曲率时空中的显式辛积分器与自适应时间步长 显示英文标题

标题： Explicit symplectic integrators with adaptive time steps in curved spacetimes

摘要： 最近，我们小组开发了适用于非分裂为多个显式可积部分的弯曲时空的显式辛方法，而是通过适当的时间变换实现的。 这种经过时间变换的显式辛积分器原则上应采用自适应时间步长，但在实际实现中往往困难。 事实上，如果时间变换函数使得时间变换后的哈密顿量具有所需的分裂形式，并在足够大的距离处趋于1或常数，则它们表现良好。 然而，在这种情况下，它们不满足步长选择的要求。 基于Preto$\&$Saha提出的步长控制技术，非自适应时间步长的时间变换显式辛方法被略微调整为自适应方法。 与非自适应方法相比，自适应方法仅增加两个额外步骤，计算成本几乎不变。 它们的实现简单。 对黑洞附近粒子和光子的若干动力学模拟表明，自适应方法通常能提高非自适应方法的效率。 由于这一优良特性，新的自适应方法被应用于研究Schwarzschild-Melvin时空中视界外粒子和光子的混沌动力学。 新方法广泛适用于所有对应于具有预期分裂形式的哈密顿量或时间变换哈密顿量的弯曲时空。 此外，也可以应用于反向光线追踪方法，以研究光子的运动和黑洞的阴影。 显示英文摘要

摘要： Recently, our group developed explicit symplectic methods for curved spacetimes that are not split into several explicitly integrable parts, but are via appropriate time transformations. Such time-transformed explicit symplectic integrators should have employed adaptive time steps in principle, but they are often difficult in practical implementations. In fact, they work well if time transformation functions cause the time-transformed Hamiltonians to have the desired splits and approach 1 or constants for sufficiently large distances. However, they do not satisfy the requirement of step-size selections in this case. Based on the step-size control technique proposed by Preto $\&$ Saha, the nonadaptive time step time-transformed explicit symplectic methods are slightly adjusted as adaptive ones. The adaptive methods have only two additional steps and a negligible increase in computational cost as compared with the nonadaptive ones. Their implementation is simple. Several dynamical simulations of particles and photons near black holes have demonstrated that the adaptive methods typically improve the efficiency of the nonadaptive methods. Because of the desirable property, the new adaptive methods are applied to investigate the chaotic dynamics of particles and photons outside the horizon in a Schwarzschild-Melvin spacetime. The new methods are widely applicable to all curved spacetimes corresponding to Hamiltonians or time-transformed Hamiltonians with the expected splits. Also application to the backwards ray-tracing method for studying the motion of photons and shadows of black holes is possible.