On convergence of higher order schemes for the projective integration method for stiff ordinary differential equations
Provides theoretical guarantees for a numerical method used in multi-scale simulations, but the result is incremental as it extends existing convergence proofs to higher-order schemes.
The paper proves convergence for higher-order projective integration methods applied to stiff ODEs with a slow manifold, showing error contributions from microsolver length, macrosolver accuracy, and distance from the slow manifold, with stability conditions preventing divergence.
We present a convergence proof for higher order implementations of the projective integration method (PI) for a class of deterministic multi-scale systems in which fast variables quickly settle on a slow manifold. The error is shown to contain contributions associated with the length of the microsolver, the numerical accuracy of the macrosolver and the distance from the slow manifold caused by the combined effect of micro- and macrosolvers, respectively. We also provide stability conditions for the PI methods under which the fast variables will not diverge from the slow manifold. We corroborate our results by numerical simulations.