Summary
The paper analyzes a splitting technique into fast and slow dynamical components of ODE systems as suggested by Maas And Poperecently. Their technique is based on a real block — Schur decomposition of the Jacobian of the right hand side of the ODE. As a result of the analysis, a computationally cheap monitor for the possible necessary recovering of the splitting is derived by singular perturbation theory. Numerical experiments on moderate size, but challenging reaction kinetics problems document the efficiency of the new device within a linearly-implicit stiff integrator.
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References
G.Bader, U.Nowak, P.Deuflhard: An advanced simulation package for large chemical reaction systems. In: R.Aiken (ed.): Stiff Computation. Oxford Univ. Press, pp. 255–264 (1985)
P.Deuflhard, G.Bader, U.Nowak: LARKIN - a Software Package for the Numerical Simulation of LARge Systems Arising in Chemical Reaction KINetics. in [7], pp. 38–55 (1981)
P.Deuflhard, F.Bornemann: Numerische Mathematik II. Integration gewöhnlicher Differentialgleichungen. de Gruyter: Berlin, New York (1994)
P.Deuflhard, E.Hairer, J.Zugck: One-step and Extrapolation Methods for Differential-Algebraic Systems. Numer. Math., vol 51, pp. 501–516 (1987)
P.Deuflhard, A.Hohmann: Numerical Analysis. A First Course in Scientific Computation, de Gruyter: Berlin, New York (1993)
P.Deuflhard, U.Nowak: Efficient Numerical Simulation and Identification of Large Chemical Reaction Systems. Ber. Bunsenges. Phys. Chem., vol 90, pp. 940–946 (1986)
K.H.Ebert, P.Deuflhard, W.Jager (eds.) Modelling of Chemical Reaction Systems. Springer Series in Chem Phys, vol 18, Berlin, Heidelberg, New York (1981)
L.A.Farrow, D.Edelson: The steady-state approximation, fact or fiction? Int. J.Chem.Kin, vol 6, pp. 787ff. (1974)
J.Field, R.M.Noyes: Oscillations in Chemical Systems. IV: Limit cycle behaviour in a model of a real chemical reaction. J.Chem.Phys, vol 60, pp. 1877–1884 (1974)
G.H.Golub, Ch.F.van Loan: Matrix Computations. Johns Hopkins University Press (1985)
G.H.Golub, J.H.Wilkinson: Ill-conditioned Eigensystems and Computation of the Jordan Canonical Form. SIAM Review 18, pp. 578–619 (1976)
E.Hairer, G.Wanner: Solving Ordinary Differential Equations II. Stiff and Differential-Algebraic Problems. Springer Series in Comp. Math., vol 14, Berlin, Heidelberg, New York (1991)
J.Heroth: Adaptive Dimensionsreduktion chemischer Reaktionssysteme. Freie Universitat Berlin, Diploma thesis (1995)
F.C.Hoppensteadt, P.Alfeld, R.Aiken: Numerical Treatment of Rapid Chemical Kinetics by Perturbation and Projection Methods, in: [7], pp. 31–37 (1981)
G.Isbarn, H. J.Ederer, E.H.Ebert The Thermal Decomposition of n-Hexane: Kinetics, Mechanism and Simulation, in: [7], pp. 235–248 (1981)
LAPACK. User’s Guide, Philadelphia (1992)
U.Maas: Automatische Reduktion von Reaktionsmechanismen zur Simulation reaktiver Strömungen. Institut fur Technische Verbrennung der Universitat Stuttgart, Habilitation thesis, (1993)
U.Maas, S.B.Pope: Simplifying Chemical Kinetics: Intrinsic Low-Dimensional Manifolds in Composition Space. Combustion and Flame, vol 88, pp. 239–264, (1992)
R.E.O’Malley: Introduction to Singular Perturbations. Academic Press, New York (1974)
W.C.Rheinboldt: Differential-Algebraic Systems as Differential Equations on Manifolds. Math. Comp. vol 43, pp. 473–482 (1984)
A.B.Vasil’eva: Asymptotic Behaviour of Solutions to Certain Problems involving Nonlinear Differential Equations… Usp. Mat. Nauk (Russian) vol 18, pp. 15–86 (1963). Russian Math. Surveys, vol 18, Nr.3, pp.13–84 (Translation)
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Deuflhard, P., Heroth, J. (1996). Dynamic Dimension Reduction in ODE Models. In: Keil, F., Mackens, W., Voß, H., Werther, J. (eds) Scientific Computing in Chemical Engineering. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-80149-5_4
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DOI: https://doi.org/10.1007/978-3-642-80149-5_4
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