The functional equation truncation method for approximating slow invariant manifolds: a rapid method for computing...

Area of Science:

• Applied Mathematics
• Chemical Engineering
• Dynamical Systems

Background:

• Slow manifolds simplify complex systems by identifying low-dimensional invariant manifolds.
• The Maas-Pope intrinsic low-dimensional manifold (ILDM) is a common approximation but neglects system curvature.
• Accurate slow manifold computation is vital for model reduction without sacrificing accuracy on slow timescales.

Purpose of the Study:

• To introduce an efficient computational method for approximating slow manifolds.
• To present functional equation truncation (FET) as an alternative to the ILDM.
• To compare the accuracy and computational efficiency of FET with the ILDM.

Main Methods:

• Developed a hierarchy of functional equations involving higher derivatives.
• Truncated the functional equations at second-derivative terms to neglect curvature.
• Proved the equivalence of FET-approximated (FETA) and ILDM manifolds in specific cases.

Main Results:

• The FET method provides an approximation equivalent to the ILDM.
• FETA and ILDM manifolds are identical for one-dimensional slow manifolds in planar systems.
• In higher dimensions, FETA and ILDM manifolds agree with numerical accuracy across most of the domain.
• FET is expected to be computationally faster than the ILDM.

Conclusions:

• Functional Equation Truncation (FET) offers an efficient and accurate method for slow manifold approximation.
• FET provides a viable, potentially faster, alternative to the ILDM for model reduction.
• The FET method's ability to neglect curvature simplifies calculations while maintaining high accuracy.