# A general algorithm for improving ILDMs

@article{Nafe2002AGA, title={A general algorithm for improving ILDMs}, author={Joerg Nafe and Ulrich Maas}, journal={Combustion Theory and Modelling}, year={2002}, volume={6}, pages={697 - 709} }

The method of intrinsic low-dimensional manifolds (ILDMs) has proven to be an efficient tool for the simplification of chemical kinetics. Nevertheless, there are still some open questions with respect to an efficient calculation and implementation of ILDMs. In this paper, we focus on the efficient calculation of ILDMs and present a refinement method for ILDMs. The method is based on an evolution equation of the manifolds towards a steady state solution which then represents the slow manifold… Expand

#### 46 Citations

The extension of the ILDM concept to reaction–diffusion manifolds

- Mathematics
- 2007

In the present work, the method of simplifying chemical kinetics based on Intrinsic Low-Dimensional Manifolds (ILDMs) is modified to deal with the coupling of reaction and diffusion processes.… Expand

Higher order corrections in the approximation of low-dimensional manifolds and the construction of simplified problems with the CSP method

- Mathematics
- 2005

In systems of stiff Ordinary Differential Equations (ODEs) both fast and slow time scales are encountered. The fast time scales are responsible for the development of low-dimensional manifolds on… Expand

Implementation of ILDMs based on a Representation in Generalized Coordinates

- Mathematics
- 2006

The usage of detailed chemical kinetics for the mathematical modeling of com- bustion processes is desirable, but leads to a high computational effort. In the present work one of the methods of… Expand

An efficient iterative algorithm for the approximation of the fast and slow dynamics of stiff systems

- Mathematics, Computer Science
- J. Comput. Phys.
- 2006

The relation between the iterative algorithms based on the computational singular perturbation (CSP) and the invariance equation (IE) methods is examined and it is shown that this equivalence of the two methods can be expressed by simple algebraic relations. Expand

An Approach for approximating slow and fast dynamics of multi-scale dynamical systems

- Mathematics
- 2011

In this paper, we introduce a fictitious dynamics for describing the only fast relaxation of a stiff ordinary differential equation (ODE) system towards a stable low-dimensional invariant manifold in… Expand

An Iterative Method for the Approximation of Fibers in Slow-Fast Systems

- Mathematics, Computer Science
- SIAM J. Appl. Dyn. Syst.
- 2014

A method for iteratively improving slow manifolds can be used to approximate the fiber directions and is applied to general finite-dimensional real analytic systems where it also approximates the curvature of the fibers. Expand

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

- Physics, Medicine
- The Journal of chemical physics
- 2006

It is proved that the ILDM and FET-approximated (FETA) manifolds are identical for the one-dimensional slow manifold of any planar system and in higher-dimensional spaces, the ILV and FETA manifolds agree to numerical accuracy almost everywhere. Expand

Approximation of slow and fast dynamics in multiscale dynamical systems by the linearized Relaxation Redistribution Method

- Mathematics, Computer Science
- J. Comput. Phys.
- 2012

The demanding problem of constructing SIM of any dimensions is recast into the remarkably simpler task of solving a properly devised ODE system by stiff numerical schemes available in the literature. Expand

The structure of slow invariant manifolds and their bifurcational routes in chemical kinetic models

- Mathematics, Computer Science
- Comput. Chem. Eng.
- 2007

It is shown that the existence of a global slow manifold along with its properties are controlled by a transcritical bifurcation of the points at infinity, that can be readily identified by analyzing the Poincare projected (Pp) system. Expand

Invariant manifolds and lattice Boltzmann method for combustion

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- 2009

The lattice Boltzmann (LB) method is a rel- atively novel approach to numerical flow simulations, and recent studies have proved that it is highly competitive to traditional methods when simulating… Expand

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