Stiff and chaotic differential equations are challenging for time-stepping numerical methods. For explicit methods, the required time step resolution significantly exceeds the resolution associated with the smoothness of the exact solution for specified accuracy. In order to improve efficiency, the question arises whether transformation to asymptotically stable solutions can be performed, for which neighbouring solutions converge towards each other at a controlled rate. Employing the concept of local Lyapunov exponents, it is demonstrated that chaotic differential equations can be successfully transformed to obtain high accuracy, whereas stiff equations cannot. For instance, the accuracy of explicit fourth order Runge-Kutta solution of the Lorenz chaotic equations can be increased by two orders of magnitude. Alternatively, the time step can be significantly extended with retained accuracy.

翻译：刚性微分方程与混沌微分方程对时间步进数值方法构成挑战。对于显式方法而言，在给定精度下所需的时间步长分辨率远超精确解光滑性所对应的分辨率。为提升效率，自然产生疑问：是否可实施向渐近稳定解的变换？此类变换可使相邻解以可控速率相互收敛。基于局部李雅普诺夫指数概念，本文证明混沌微分方程可成功实施变换以获取高精度，而刚性方程则无法实现。例如，洛伦兹混沌方程的显式四阶龙格-库塔求解精度可提升两个数量级，或者在保持精度的前提下显著延长时间步长。