Transforming Stiffness and Chaos

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.