常微分方程数值解与哥伦布之蛋：三个困难问题的三个简单想法
Numerical Solution of ODEs and the Columbus' Egg: Three Simple Ideas for Three Difficult Problems
AI总结：
本文针对常微分方程数值解中的三个主要困难，提出了三个简单而新颖的解决方案，并进行了综述。
摘要：
在计算机上，我们求解离散问题而非连续问题。必须确保实时（即步长h非无穷小）获得的离散问题解是连续问题解的良好近似。然而，由于离散世界远比连续世界丰富（后者是前者的极限情况），为分析连续问题行为而设计的经典定义和技术往往不足以处理离散情况，需要新的特定工具。通常，坚持遵循连续设定中已有的路径会导致时间和精力的浪费，而新的特定工具却能更轻松、更优雅地解决问题。本文综述了常微分方程数值解中遇到的三个主要困难及其提出的新颖解决方案。
英文摘要：
On computers, discrete problems are solved instead of continuous ones. One must be sure that the solutions of the former problems, obtained in real time (i.e., when the stepsize h is not infinitesimal) are good approximations of the solutions of the latter ones. However, since the discrete world is much richer than the continuous one (the latter being a limit case of the former), the classical definitions and techniques, devised to analyze the behaviors of continuous problems, are often insufficient to handle the discrete case, and new specific tools are needed. Often, the insistence in following a path already traced in the continuous setting, has caused waste of time and efforts, whereas new specific tools have solved the problems both more easily and elegantly. In this paper we survey three of the main difficulties encountered in the numerical solutions of ODEs, along with the novel solutions proposed.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
