×

The \(G\)-scheme: a framework for multi-scale adaptive model reduction. (English) Zbl 1169.65325

Summary: The numerical solution of mathematical models for reaction systems in general, and reacting flows in particular, is a challenging task because of the simultaneous contribution of a wide range of time scales to the system dynamics. However, the dynamics can develop very-slow and very-fast time scales separated by a range of active time scales. An opportunity to reduce the complexity of the problem arises when the fast/active and slow/active time scales gaps becomes large.
We propose a numerical technique consisting of an algorithmic framework, named the \(G\)-scheme, to achieve multi-scale adaptive model reduction along-with the integration of the differential equations (DEs). The method is directly applicable to initial-value ordinary DSsand (by using the method of lines) partial DEs. We assume that the dynamics is decomposed into active, slow, fast, and when applicable, invariant subspaces. The \(G\)-scheme introduces locally a curvilinear frame of reference, defined by a set of orthonormal basis vectors with corresponding coordinates, attached to this decomposition. The evolution of the curvilinear coordinates associated with the active subspace is described by non-stiff DEs, whereas that associated with the slow and fast subspaces is accounted for by applying algebraic corrections derived from asymptotics of the original problem. Adjusting the active DEs dynamically during the time integration is the most significant feature of the \(G\)-scheme, since the numerical integration is accomplished by solving a number of DEs typically much smaller than the dimension of the original problem, with corresponding saving in computational work. To demonstrate the effectiveness of the \(G\)-scheme, we present results from illustrative as well as from relevant problems.

MSC:

65L05 Numerical methods for initial value problems involving ordinary differential equations
34A34 Nonlinear ordinary differential equations and systems
80A30 Chemical kinetics in thermodynamics and heat transfer
80A32 Chemically reacting flows
76V05 Reaction effects in flows
65M20 Method of lines for initial value and initial-boundary value problems involving PDEs
34E15 Singular perturbations for ordinary differential equations
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Echekki, T.; Chen, J. H., Unsteady strain rate and curvature effects in turbulent premixed methane-air flames, Combust. Flame, 106, 184-202 (1996)
[2] Singh, S.; Rastigejev, Y.; Paolucci, S.; Powers, J. M., Viscous detonation in \(\text{H}_2 - \text{O}_2-Ar\) using intrinsic low-dimensional manifolds and wavelet adaptive multilevel representation, Combust. Theory Modell., 5, 163-184 (2001) · Zbl 1044.76512
[3] Segel, L. A.; Slemrod, M., The quasi steady state assumption: a case study in perturbation, SIAM Rev., 31, 446-447 (1989) · Zbl 0679.34066
[4] Yannacopoulos, A. N.; Tomlin, A. S.; Brindley, J.; Merkin, J. H.; Pilling, M. J., The error of the quasi steady state approximation in spatially distributed systems, Chem. Phys. Lett., 248, 63-70 (1996)
[5] Goussis, D. A., On the construction and use of reduced chemical kinetics mechanisms produced on the basis of given algebraic relations, J. Comput. Phys., 128, 261-273 (1996) · Zbl 0862.65088
[6] Massias, A.; Diamantis, D.; Mastorakos, E.; Goussis, D. A., An algorithm for the construction of global reduced mechanisms with CSP data, Combust. Flame, 117, 685-708 (1999) · Zbl 0939.80506
[7] Maas, U.; Pope, S. B., Simplifying chemical kinetics: intrinsic low dimensional manifolds in composition space, Combust. Flame, 88, 239-264 (1992)
[8] Pope, S. B., Computationally efficient implementation of combustion chemistry using in situ adaptive tabulation, Combust. Theory Modell., 1, 41-63 (1997) · Zbl 1046.80500
[9] Tonse, S. R.; Moriarty, N. W.; Brown, N. J.; Frenklach, M., PRISM: piecewise reusable implementation of solution mapping. An economical strategy for chemical kinetics, Israel J. Chem., 39, 97-106 (1999)
[10] Lu, T.; Law, C. K., Linear time reduction of large kinetic mechanisms with directed relation graph: \(n\)-heptane and iso-octane, Combust. Flame, 30, 24-36 (2006)
[11] P. Pepiot, H. Pitsch, Systematic reduction of large chemical mechanisms, in: Proceedings of the Fourth Joint Meeting of the US Sections of the Combustion Institute, 2005.; P. Pepiot, H. Pitsch, Systematic reduction of large chemical mechanisms, in: Proceedings of the Fourth Joint Meeting of the US Sections of the Combustion Institute, 2005. · Zbl 1158.80325
[12] Valorani, M.; Creta, F.; Goussis, D. A.; Najm, H. N.; Lee, J. C., An automatic procedure for the simplification of chemical kinetics mechanisms based on CSP, Combust. Flame, 146, 1-2, 29-51 (2006)
[13] Løvås, T.; Mauss, F.; Hasse, C.; Peters, N., Development of adaptive kinetics for application in combustion systems, Proc. Combust. Inst., 29, 1403-1410 (2002)
[14] P. Pepiot-Desjardins, H. Pitsch, An automatic strategy to develop reduced kinetic mechanisms for surrogate fuels, in: International Workshop on Model Reduction in Reacting Flow, September 3-5, 2007.; P. Pepiot-Desjardins, H. Pitsch, An automatic strategy to develop reduced kinetic mechanisms for surrogate fuels, in: International Workshop on Model Reduction in Reacting Flow, September 3-5, 2007.
[15] Oseledec, V. I., A multiplicative ergodic theorem, Lyapunov characteristic numbers for dynamical systems, Trans. Moscow Math. Soc., 19 (1997)
[16] Adrover, A.; Creta, F.; Giona, M.; Valorani, M.; Vitacolonna, V., Natural tangent dynamics with recurrent biorthogonalizations: a geometric computational approach to dynamical systems exhibiting slow manifolds and periodic/chaotic limit sets, Physica D - Nonlinear Phenom., 213, 2, 121-146 (2006) · Zbl 1111.34042
[17] Lam, S. H.; Goussis, D. A., Understanding complex chemical kinetics with computational singular perturbation, Proc. Comb. Inst., 22, 931-941 (1988)
[18] Lam, S. H., Using CSP to understand complex chemical kinetics, Combust. Sci. Technol., 89, 375-404 (1993)
[19] Lam, S. H.; Goussis, D. A., The CSP method for simplifying kinetics, Int. J. Chem. Kinet., 26, 461-486 (1994)
[20] Fenichel, N., Geometric singular perturbation theory for ordinary differential equations, J. Differen. Equat., 31, 53-98 (1979) · Zbl 0476.34034
[21] Bykov, V.; Goldfarb, I.; Gol’dshtein, V.; Sazhin, S.; Sazhina, E., System decomposition technique for spray modelling in CFD codes, Comput. Fluids, 3, 3, 601-610 (2007) · Zbl 1177.76279
[22] Adrover, A.; Creta, F.; Giona, M.; Valorani, M., Stretching-based diagnostics and reduction of chemical kinetic models with diffusion, J. Comput. Phys., 225, 1442-1471 (2007) · Zbl 1123.65119
[23] Valorani, M.; Goussis, D. A., Explicit time-scale splitting algorithm for stiff problems: auto-ignition of gaseous-mixtures behind a steady shock, J. Comput. Phys., 168, 1, 44-79 (2001) · Zbl 1037.76045
[24] Ortega, J. M.; Najm, H. N.; Ray, J.; Valorani, M.; Goussis, D. A.; Frenklach, M. Y., Adaptive chemistry computations of reacting flow, J. Phys.: Con. Ser., 78, 012054 (2007)
[25] Valorani, M.; Goussis, D. A.; Creta, F.; Najm, H. N., Higher order corrections in the approximation of low dimensional manifolds and the construction of simplified problems with the CSP method, J. Comput. Phys., 209, 754-786 (2005) · Zbl 1073.65057
[26] Zagaris, A.; Kaper, H. G.; Kaper, T. J., Analysis of the CSP reduction method for chemical kinetics, Nonlinear Sci., 14, 59-91 (2004) · Zbl 1053.92051
[27] Zagaris, A.; Kaper, H. G.; Kaper, T. J., Fast and slow dynamics for the computational singular perturbation method, Multiscale Model. Simul., 2, 613-638 (2004) · Zbl 1065.34049
[28] Elezgaray, J.; Arneodo, A., Crisis-induced intermittent bursting in reaction-diffusion chemical systems, Phys. Rev. Lett., 68, 714-717 (1992)
[29] Creta, F.; Adrover, A.; Cerbelli, S.; Valorani, M.; Giona, M., Slow manifold structure in explosive kinetics. 1. Bifurcations of points-at-infinity in prototypical models, J. Phys. Chem. A, 110, 50, 13447-13462 (2006)
[30] Brad, R. B.; Tomlin, A. S.; Fairweather, M.; Griffiths, J. F., The application of chemical reduction methods to a combustion system exhibiting complex dynamics, Proc. Combust. Inst., 31, 455-463 (2007)
[31] F. Creta, A. Adrover, M. Giona, M. Valorani, Geometric approaches to model reduction: from invariant to stretching-based subspaces, in: International Workshop on Model Reduction in Reacting Flow, Rome, 2007.; F. Creta, A. Adrover, M. Giona, M. Valorani, Geometric approaches to model reduction: from invariant to stretching-based subspaces, in: International Workshop on Model Reduction in Reacting Flow, Rome, 2007. · Zbl 1123.65119
[32] M. Valorani, S. Paolucci, Adaptive model reduction in chemical kinetics, in: R. Ragucci (Ed.), Italian Section of the Combustion Institute, Turin, Italy, 2008.; M. Valorani, S. Paolucci, Adaptive model reduction in chemical kinetics, in: R. Ragucci (Ed.), Italian Section of the Combustion Institute, Turin, Italy, 2008. · Zbl 1169.65325
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.