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Invariant grids for reaction kinetics

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  • Gorban, Alexander N.
  • Karlin, Iliya V.
  • Zinovyev, Andrei Yu.

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 a formulation of the condition of invariance as an equation, and its solution by Newton iterations. A grid-based version of MIM is developed (the method of invariant grids). We describe the Newton method and the relaxation method for the invariant grids construction. The problem of the grid correction is fully decomposed into the problems of the grid's nodes correction. The edges between the nodes appear only in the calculation of the tangent spaces. This fact determines high computational efficiency of the method of invariant grids. The method is illustrated by two examples: the simplest catalytic reaction (Michaelis–Menten mechanism), and the hydrogen oxidation. The algorithm of analytical continuation of the approximate invariant manifold from the discrete grid is proposed. Generalizations to open systems are suggested. The set of methods covered makes it possible to effectively reduce description in chemical kinetics.

Suggested Citation

  • Gorban, Alexander N. & Karlin, Iliya V. & Zinovyev, Andrei Yu., 2004. "Invariant grids for reaction kinetics," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 333(C), pages 106-154.
    • Handle: RePEc:eee:phsmap:v:333:y:2004:i:c:p:106-154
    DOI: 10.1016/j.physa.2003.10.043
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    References listed on IDEAS

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    1. Gorban, Alexander N. & Karlin, Iliya V., 1994. "General approach to constructing models of the Boltzmann equation," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 206(3), pages 401-420.
    2. Gorban, Alexander N. & Karlin, Iliya V. & Zmievskii, Vladimir B. & Nonnenmacher, T.F., 1996. "Relaxational trajectories: global approximations," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 231(4), pages 648-672.
    3. Gorban, A.N & Karlin, I.V & Zmievskii, V.B & Dymova, S.V, 2000. "Reduced description in the reaction kinetics," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 275(3), pages 361-379.
    4. Zmievski, Vladimir B. & Karlin, Iliya V. & Deville, Michel, 2000. "The universal limit in dynamics of dilute polymeric solutions," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 275(1), pages 152-177.
    5. Grmela, Miroslav, 2002. "Reciprocity relations in thermodynamics," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 309(3), pages 304-328.
    6. Dukek, G. & Karlin, Iliya V. & Nonnenmacher, T.F., 1997. "Dissipative brackets as a tool for kinetic modeling," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 239(4), pages 493-508.
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    Cited by:

    1. Gorban, Alexander N. & Karlin, Iliya V., 2004. "Uniqueness of thermodynamic projector and kinetic basis of molecular individualism," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 336(3), pages 391-432.
    2. Otero-Muras, Irene & Szederkényi, Gábor & Hangos, Katalin M. & Alonso, Antonio A., 2008. "Dynamic analysis and control of biochemical reaction networks," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 79(4), pages 999-1009.

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