The functional equation truncation method for approximating slow invariant manifolds: A rapid method for computing intrinsic low-dimensional manifolds

Abstract

A slow manifold is a low-dimensional invariant manifold to which trajectories nearby are rapidly attracted on the way to the equilibrium point. The exact computation of the slow manifold simplifies the model without sacrificing accuracy on the slow time scales of the system. The Maas-Pope intrinsic low-dimensional manifold (ILDM) [Combust. Flame 88, 239 (1992)] is frequently used as an approximation to the slow manifold. This approximation is based on a linearized analysis of the differential equations and thus neglects curvature. We present here an efficient way to calculate an approximation equivalent to the ILDM. Our method, called functional equation truncation (FET), first develops a hierarchy of functional equations involving higher derivatives which can then be truncated at second-derivative terms to explicitly neglect the curvature. We prove that the ILDM and FET-approximated (FETA) manifolds are identical for the one-dimensional slow manifold of any planar system. In higher-dimensional spaces, the ILDM and FETA manifolds agree to numerical accuracy almost everywhere. Solution of the FET equations is, however, expected to generally be faster than the ILDM method.