论非常规离散化
On unconventional discretisations
AI总结：
该研究提出多种有别于经典算法的离散化技术，受特定原理启发，应用于逻辑斯谛方程和洛特卡 - 沃尔泰拉系统，探讨保持正性等问题及可积系统离散化，还讨论了离散时空的相关性。
AI中文摘要：
我们提出了各种离散化技术，这些技术有别于数值分析的经典算法，目的是即使在离散化步长不是非常小的情况下，也能保留我们正在离散化的微分系统解的物理行为。这些技术受米肯斯提出的原理启发，与广田和卡汉提出的技术相近。给出了这些方法在逻辑斯谛方程和洛特卡 - 沃尔泰拉系统中的详细应用。讨论了保持正性的重要性及实现方法，还探讨了所提离散化可能出现的问题。有一节专门讨论旨在保持可积性的可积系统的离散化。最后讨论了离散时空的可能相关性。
英文摘要：
We present various discretisation techniques that mark a departure from the classical algorithms of numerical analysis, in the aim of preserving the physical behaviour of the solutions to the differential systems we are discretising, even for discretisation steps that are not very small. These techniques are inspired by the principles laid down by Mickens and are close to those proposed by Hirota and Kahan. Detailed applications of these methods are presented for the logistic equation and for the Lotka-Volterra system. We discuss the importance of preserving positivity whenever the latter is an expected property of the solution and give a practical rule for achieving this. Possible problems arising from the discretisations we propose are also discussed. A section is devoted to discretisations of integrable systems aimed at preserving integrability. Finally we conclude this review with a discussion of the possible relevance of a discrete space-time.
