Hamilton系统下基于相位误差的精细辛算法

Hamilton系统是一类重要的动力系统，辛算法（如生成函数法、SRK法、SPRK法、多步法等）是针对Hamilton系统所设计的具有保持相空间辛结构不变或保Hamilton函数不变的算法.但是，时域上，同阶的辛算法与Runge-Kutta法具有相同的数值精度，即辛算法在计算过程中也存在相位误差，导致时域上解的数值精度不高.经过长时间计算后，计算结果在时域上也会变得“面目全非”.为了提高辛算法在时域上解的精度，将精细算法引入到辛差分格式中，提出了基于相位误差的精细辛算法（HPD-symplectic method），这种算法满足辛格式的要求，因此在离散过程中具有保Hamilton系统辛结构的优良特性.同时，由于精细化时间步长，极大地减小了辛算法的相位误差，大幅度提高了时域上解的数值精度，几乎可以达到计算机的精度，误差为O（10-13）.对于高低混频系统和刚性系统，常规的辛算法很难在较大的步长下同时实现对高低频精确仿真，精细辛算法通过精细计算时间步长，在大步长情况下，没有额外增加计算量，实现了高低混频的精确仿真.数值结果验证了此方法的有效性和可靠性.

A Highly Precise Symplectic Direct Integration Method Based on Phase Errors for Hamiltonian Systems

Symplectic methods, including the generating function method, the symplectic Runge-Kutta (RK) method, the symplectic partitioned Runge-Kutta method, the multi-step method and so on, are applicable to Hamiltonian systems. They can preserve the symplectic structure in the phase space and the laws of the Hamiltonian system. But in the time domain, due to phase lags in the computing course, the RK methods and the symplectic methods have the same algebraic precision under the same algebraic order of schemes. After longtime computing, the numerical precision goes worse and worse in the time domain. To improve the precision, a new method combining the highly precise direct integration method with the symplectic difference scheme, called the HPD-symplectic method, was proposed. This method, proved to be symplectic, can preserve the symplectic structure. Moreover, the HPD-symplectic method can largely decrease the phase error in the time domain, and accordingly, improve the numerical precision even up to an error level of 10-13. For systems with mixed frequencies or rigid systems, the traditional symplectic methods can hardly work well, while the HPD-symplectic method can simulate the signals at both high and low frequencies well with large time steps but no additional computation cost. The results of numerical examples demonstrate the reliability and effectiveness of the proposed method.