具有零相位滞后及其导数的多步法族用于振荡常微分方程的数值积分
A Family of Multistep Methods with Zero Phase-Lag and Derivatives for the Numerical Integration of Oscillatory ODEs
AI总结：
本文开发了一个三步8步法族，通过消除相位滞后及其前r阶导数（r=1,2,3），提高了求解振荡常微分方程（包括薛定谔方程和N体问题）的效率，并分析了局部截断误差和稳定性。
摘要：
本文开发了一个三步8步法族，针对振荡常微分方程的数值积分进行了优化。我们消除了方法的相位滞后及其前r阶导数，其中r=1,2,3。我们表明，通过这种新技术，每消除一个相位滞后导数，方法的效率就会提高。对于薛定谔方程和N体问题的积分都是如此。进行了局部截断误差分析，对于薛定谔方程的情况，还显示了误差与能量的联系，揭示了零相位滞后导数的重要性。稳定性分析也表明，具有更多消失导数的方法具有更大的周期性区间。
英文摘要：
In this paper we develop a family of three 8-step methods, optimized for the numerical integration of oscillatory ordinary differential equations. We have nullified the phase-lag of the methods and the first r derivatives, where r=1,2,3. We show that with this new technique, the method gains efficiency with each derivative of the phase-lag nullified. This is the case for the integration of both the Schrodinger equation and the N-body problem. A local truncation error analysis is performed, which, for the case of the Schrodinger equation, also shows the connection of the error and the energy, revealing the importance of the zero phase-lag derivatives. Also the stability analysis shows that the methods with more derivatives vanished, have a bigger interval of periodicity.
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