高阶三部分分裂辛积分方案

High order three part split symplectic integrators: Efficient techniques for the long time simulation of the disordered discrete nonlinear Schroedinger equation

While symplectic integration methods based on operator splitting are well established in many branches of science, high order methods for Hamiltonian systems that split in more than two parts have not been studied in great detail. Here, we present several high order symplectic integrators for Hamiltonian systems that can be split in exactly three integrable parts. We apply these techniques, as a practical case, for the integration of the disordered, discrete nonlinear Schroedinger equation (DDNLS) and compare their efficiencies. Three part split algorithms provide effective means to numericall…

Symplectic integration of deviation vectors and chaos determination. Application to the Hénon-Heiles model and to the restricted three-body problem

In this work we propose a new numerical approach to distinguish between regular and chaotic orbits in Hamiltonian systems, based on the simultaneous integration of both the orbit and the deviation vectors using a symplectic scheme, hereby called {\em global symplectic integrator}. In particular, the proposed method allows us to recover the correct orbits character with very large integration time steps, small energy losses and short CPU times. To illustrate the numerical performances of the global symplectic integrator we will apply it to two well-known and widely studied problems: the Hénon-H…

Explicit symplectic methods in black hole spacetimes

Many Hamiltonian problems in the Solar System are separable or separate into two analytically solvable parts, and thus give a great chance to the development and application of explicit symplectic integrators based on operator splitting and composing. However, such constructions cannot in general be available for curved spacetimes in general relativity and modified theories of gravity, because these curved spacetimes correspond to nonseparable Hamiltonians without the two part splits. Recently, several black hole spacetimes such as the Schwarzschild black hole were found to allow the construct…

基准测试：Tao辛积分方法

Benchmark: Tao's symplectic integration method

本研究针对Molei Tao提出的新型辛积分方法开展基准测试，对比其与Störmer-Verlet、Runge-Kutta等方法的性能，验证了该方法通用性强、无需隐式方程的优势。

On the symplectic integration of the discrete nonlinear Schrödinger equation with disorder

We present several methods, which utilize symplectic integration techniques based on two and three part operator splitting, for numerically solving the equations of motion of the disordered, discrete nonlinear Schrödinger (DDNLS) equation, and compare their efficiency. Our results suggest that the most suitable methods for the very long time integration of this one-dimensional Hamiltonian lattice model with many degrees of freedom (of the order of a few hundreds) are the ones based on three part splits of the system's Hamiltonian. Two part split techniques can be preferred for relatively small…

Symplectic Integration without Roundoff Error

Most numerical integration algorithms are not designed specifically for Hamiltonian systems and do not respect their characteristic properties, which include the preservation of phase space volume with time. This can lead to spurious damping or excitation. Methods that do preserve all the Hamiltonian properties, i.e., for which the time-forward map is symplectic, are called symplectic integration algorithms or SIAs. Although such integrators are symplectic in theory, they are not symplectic if implemented using finite-precision arithmetic. This paper explains how to eliminate this problem by u…