# Quasi-equilibrium grid algorithm: Geometric construction for model reduction

@article{Chiavazzo2008QuasiequilibriumGA, title={Quasi-equilibrium grid algorithm: Geometric construction for model reduction}, author={Eliodoro Chiavazzo and Iliya V. Karlin}, journal={J. Comput. Phys.}, year={2008}, volume={227}, pages={5535-5560} }

The method of invariant grid (MIG) is an iterative procedure for model reduction in chemical kinetics which is based on the notion of Slow Invariant Manifold (SIM) [A.N. Gorban, I.V. Karlin, Method of invariant manifold for chemical kinetics, Chem. Eng. Sci. 58 (2003) 4751-4768; E. Chiavazzo, A.N. Gorban, I.V. Karlin, Comparison of invariant manifolds for model reduction in chemical kinetics, Commun. Comput. Phys. 2(5) (2007) 964-992; A.N. Gorban, I.V. Karlin, A.Y. Zinovyev, Invariant grids for… Expand

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#### References

SHOWING 1-10 OF 47 REFERENCES

Invariant grids for reaction kinetics

- Mathematics, Chemistry
- 2003

Abstract In this paper, we construct low-dimensional manifolds of reduced description for equations of chemical kinetics from the standpoint of the method of invariant manifold (MIM). MIM is based on… Expand

Method of invariant grid for model reduction of hydrogen combustion

- Physics, Mathematics
- 2007

The Method of Invariant Grid (MIG) is a model reduction technique based on the concept of slow invariant manifold (SIM). The MIG approximates the SIM by a set of nodes in the concentration space… Expand

Asymptotic analysis of two reduction methods for systems of chemical reactions

- Mathematics
- 2002

Abstract This paper concerns two methods for reducing large systems of chemical kinetics equations, namely, the method of intrinsic low-dimensional manifolds (ILDMs) due to Maas and Pope [Combust.… Expand

Comparison of invariant manifolds for model reduction in chemical kinetics

- Mathematics
- 2007

A modern approach to model reduction in chemical kinetics is often based on the notion of slow invariant manifold. The goal of this paper is to give a com- parison of various methods of construction… Expand

Geometry of the steady-state approximation: Perturbation and accelerated convergence methods

- Chemistry
- 1990

The time evolution of two model enzyme reactions is represented in phase space Γ. The phase flow is attracted to a unique trajectory, the slow manifold M, before it reaches the point equilibrium of… Expand

Method of invariant manifold for chemical kinetics

- Mathematics, Physics
- 2003

In this paper, we review the construction of low-dimensional manifolds of reduced description for equations of chemical kinetics from the standpoint of the method of invariant manifold (MIM). The MIM… Expand

A more accurate projection in the rate-controlled constrained-equilibrium method for dimension reduction of combustion chemistry

- Mathematics
- 2004

The rate-controlled constrained-equilibrium (RCCE) method for dimension reduction of combustion chemistry is revisited from a geometric viewpoint. A constrained equilibrium manifold (CEM) is defined… 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

Invariant Manifolds for Physical and Chemical Kinetics

- Physics
- 2005

Introduction.- The Source of Examples.- Invariance Equation in the Differential Form.- Film Extension of the Dynamics: Slowness as Stability.- Entropy, Quasi-Equilibrium and Projector Field.- Newton… Expand

On the geometry of transient relaxation

- Chemistry
- 1991

Coupled chemical reactions are often described by (stiff) systems of ordinary differential equations (ODEs) with widely separated relaxation times. In the phase space Γ of species concentration… Expand