On convergence of higher order schemes for the projective integration method for stiff ordinary differential equations

TL;DR

This paper proves convergence and stability conditions for higher order projective integration methods applied to stiff multi-scale ODEs, demonstrating their effectiveness through theoretical analysis and numerical simulations.

Contribution

It provides a rigorous convergence proof and stability criteria for higher order projective integration schemes on stiff multi-scale systems.

Findings

Error depends on microsolver length, macro solver accuracy, and proximity to the slow manifold.

Stability conditions ensure fast variables remain close to the slow manifold.

Numerical simulations confirm theoretical convergence and stability results.

Abstract

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.