On the arbitrarily long-term stability of conservative methods

TL;DR

This paper proves that conservative numerical methods for autonomous ODEs maintain bounded global error over arbitrarily long times, given certain conditions, ensuring long-term stability in simulations.

Contribution

It establishes the long-term stability of conservative methods for autonomous ODEs under broad conditions, including finite precision scenarios.

Findings

Global error remains bounded over long times with small enough steps.

Stability holds even with finite machine precision until a large, quantifiable time.

Numerical experiments confirm the theoretical long-term stability results.

Abstract

We show the arbitrarily long-term stability of conservative methods for autonomous ODEs. Given a system of autonomous ODEs with conserved quantities, if the preimage of the conserved quantities possesses a bounded locally nite neighborhood, then the global error of any conservative method with the uniformly bounded displacement property is bounded for all time, when the uniform time step is taken suciently small. On nite precision machines, the global error still remains bounded and independent of time until some arbitrarily large time determined by machine precision and tolerance. The main result is proved using elementary topological properties for discretized conserved quantities which are equicontinuous. In particular, long-term stability is also shown using an averaging identity when the discretized conserved quantities do not explicitly depend on time steps. Numerical results are presented to illustrate the long-term stability result.