A Tutorial: Adaptive Runge-Kutta Integration for Stiff Systems : Comparing the Nos\'e and Nos\'e-Hoover Oscillator Dynamics

TL;DR

This paper demonstrates how adaptive Runge-Kutta integrators effectively solve stiff differential equations in mechanical systems, using Nosé and Nosé-Hoover oscillator dynamics as illustrative examples, emphasizing visualization and numerical stability.

Contribution

It introduces the application of adaptive integrators to stiff mechanical problems, highlighting their effectiveness in resolving numerical difficulties in Nosé oscillator dynamics.

Findings

Adaptive integrators successfully handle stiff Nosé oscillator equations.

Visualization enhances understanding of complex dynamical structures.

Nosé-Hoover dynamics are smoother and easier to simulate than Nosé dynamics.

Abstract

"Stiff" differential equations are commonplace in engineering and dynamical systems. To solve them we need flexible integrators that can deal with rapidly-changing righthand sides. This tutorial describes the application of "adaptive" [ variable timestep ] integrators to "stiff" mechanical problems encountered in modern applications of Gibbs' 1902 statistical mechanics. Linear harmonic oscillators subject to nonlinear thermal constraints can exhibit either stiff or smooth dynamics. Two closely-related examples, Nos\'e's 1984 dynamics and Nos\'e-Hoover 1985 dynamics, are both based on Hamiltonian mechanics, as was ultimately clarified by Dettmann and Morriss in 1996. Both these dynamics are consistent with Gibbs' canonical ensemble. Nos\'e's dynamics is "stiff" and can present severe numerical difficulties. Nos\'e-Hoover dynamics, though it follows exactly the same trajectory, is "smooth" and relatively trouble-free. Our tutorial emphasises the power of adaptive integrators to resolve stiff problems like the Nos\'e oscillator. The solutions obtained illustrate the power of computer graphics to enrich numerical solutions. Adaptive integration with computer graphics are basic to an understanding of dynamical systems and statistical mechanics. These tools lead naturally into the visualization of intricate fractal structures formed by chaos as well as elaborate knots tied by regular nonchaotic dynamics. This work was invited by the American Journal of Physics.