Transforming Stiffness and Chaos

TL;DR

This work analyzes whether transforming differential equations to asymptotically stable forms can mitigate numerical difficulties from stiffness and chaos in explicit time-stepping. By introducing local Lyapunov exponents (LLEs) and a local stiffness/chaos diagnostic, it shows that chaotic ODEs, exemplified by the Lorenz 1984 system, can be reformulated to achieve substantially higher accuracy or longer time steps via a back-and-forth transformation, while stiff problems generally transfer their difficulty to transformed variables. Time-spectral methods like the GWRM prove robust to both stiffness and chaos, offering dramatic efficiency gains in stiff nonlinear problems such as Robertson kinetics and explosive combustion. The results suggest a practical framework for reducing numerical chaoticity in ODEs, with potential extensions to PDEs, while clarifying that stiffness remains a fundamental barrier to similar transformations.

Abstract

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.

Figures (11)

• Figure 1: Comparison, for solution of Eq. (13) at $\epsilon=0.001$ accuracy, between maximum time step $(\Delta t)_{max}$ for numerical resolution and maximum time step $(\Delta t)_{stiff}$ for resolving stiffness of perturbed solution $u(t;\epsilon)=1+t+0.05e^{-300t}$. It is seen that the ODE is locally stiff for $t<0.004$, where $Q>1$.
• Figure 2: GWRM solutions $u(t)$ of Eq. (17) for $d=0.1$, $0.05$ and $0.01$.
• Figure 3: Illustration of stiffness of Eq. (17) for $d=0.01$. Eq. (19) is solved in the two intervals $[105,110]$ and $[115,120]$. The time step for $\epsilon=0.001$ accuracy becomes limited by stiffness sufficiently beyond $t=1/d=100$ where the solution curve $u(t)$ approaches a straight line; thus $(\Delta t)_{stiff}<(\Delta t)_{max}$, or $Q>1$ here.
• Figure 4: Solution of the Robertson equations (20), employing accuracy $\epsilon=0.001$. Note that $y(t)$ has been magnified a factor $10^4$. The solution was obtained by a GWRM solver.
• Figure 5: Solution of the Lorenz equations (48) with parameters $a=0.25$, $b=4.0$, $F=8.0$ and $G=1.0$. The initial conditions are $(x,y,z)=(0.96,-1.1,0.5)$. From top to bottom; $x(t)+8$, $y(t)+4$, $z(t)$. A high accuracy, time-spectral GWRM code was used.