On the arbitrarily long-term stability of conservative methods
For numerical analysts and practitioners solving conservative ODEs, this provides a theoretical guarantee of long-term stability, though the result is incremental as it extends known stability properties to a broader class of methods.
The paper proves that conservative methods for autonomous ODEs with conserved quantities achieve arbitrarily long-term stability, with global error bounded for all time under small time steps, and on finite precision machines error remains bounded until arbitrarily large times determined by precision and tolerance.
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.