Using adaptive proper orthogonal decomposition to solve the reaction-diffusion equation. (English) Zbl 1157.80003
In order to reduce the computational cost of reacting flow simulations, the authors apply a Proper Orthogonal Decomposition (POD) to reduce the size of large system of ordinary differential equations governing the chemical reactions. Then, they combine POD with an adaptive strategy to construct locally valid basis vectors capturing the local chemical dynamics of the system.
Reviewer: Bernard Ducomet (Bruyères le Châtel)
MSC:
| 80A25 | Combustion |
Software:
VODEReferences:
| [1] | Atwell, J. A.; King, B. B., Proper orthogonal decomposition for reduced basis feedback controllers for parabolic equations, Math. Comput. Modelling, 33, 1-19 (2001) · Zbl 0964.93032 |
| [2] | Berkooz, G.; Holmes, P.; Lumley, J. L., The proper orthogonal decomposition in the analysis of turbulent flows, Annu. Rev. Fluid Mech., 25, 539-575 (1993) |
| [3] | Bodenstein, M.; Lind, S. C., Geschwindigkeit der Bildung des Bromwasserstoffs aus seinen Elementen, Z. Phys. Chem., Stoechiom. Verwandtschaftsl., 57, 168 (1906) |
| [4] | Brown, P. N.; Byrne, G. D.; Hindmarsh, A. C., VODE - a variable-coefficient ODE solver, SIAM J. Sci. Stat. Comput., 10, 5, 1038-1051 (1989) · Zbl 0677.65075 |
| [5] | Bui-Thanh, T.; Damodaran, A.; Willcox, K., Aerodynamic data reconstruction and inverse design using proper orthogonal decomposition, AIAA J., 42, 8, 1505-1516 (2004) |
| [6] | Chapman, D.; Underhill, L., The interaction of chlorine and hydrogen. The influence of mass, J. Chem. Soc. Trans., 103, 496-508 (1913) |
| [7] | Davis, M. J., Low-dimensional manifolds in reaction-diffusion equations. 1. Fundamental aspects, J. Phys. Chem. A, 110, 16, 5235-5256 (2006) |
| [8] | Davis, M. J., Low-dimensional manifolds in reaction-diffusion equations. 2. Numerical analysis and method development, J. Phys. Chem. A, 110, 16, 5257-5272 (2006) · Zbl 0900.82077 |
| [9] | Deane, A. E.; Kevrekidis, I. G.; Karniadakis, G. E.; Orszag, S. A., Low-dimensional models for complex-geometry flows - Application to grooved channels and circular-cylinders, Phys. Fluids, 3, 10, 2337-2354 (1991) · Zbl 0746.76021 |
| [10] | Elezgaray, J.; Arneodo, A., Crisis-induced intermittent bursting in reaction-diffusion chemical-systems, Phys. Rev. Lett., 68, 714-717 (1992) |
| [11] | Fraser, S. J., The steady-state and equilibrium approximations – a geometrical picture, J. Chem. Phys., 88, 4732-4738 (1988) |
| [12] | Graham, M. D.; Kevrekidis, I. G., Alternative approaches to the Karhunen-Loève decomposition for model reduction and data analysis, Comput. Chem. Engrg., 20, 5, 495-506 (1996) |
| [13] | Holmes, P.; Lumley, J. L.; Berkooz, G., Turbulence, Coherent Structures, Dynamical Systems and Symmetry (1996), Cambridge Univ. Press: Cambridge Univ. Press Cambridge, UK · Zbl 0890.76001 |
| [14] | Homescu, C.; Petzold, L. R.; Serban, R., Error estimation for reduced-order models of dynamical systems, SIAM J. Numer. Anal., 43, 4, 1693-1714 (2005) · Zbl 1096.65076 |
| [15] | Homescu, C.; Petzold, L. R.; Serban, R., Error estimation for reduced-order models of dynamical systems, SIAM Rev., 49, 2, 277-299 (2007) · Zbl 1117.65104 |
| [16] | Humble, R. A.; Scarano, F.; van Oudheusden, B. W., Unsteady flow organization of compressible planar base flows, Phys. Fluids, 19, 7 (2007) · Zbl 1182.76336 |
| [17] | Joyner, M. L., A numerical study of the POD method in NDE, Appl. Math. Comp., 174, 1, 732-754 (2006) · Zbl 1093.78010 |
| [18] | Keck, J. C., Rate-controlled constrained-equilibrium theory of chemical reactions in complex systems, Prog. Energy Combust. Sci., 16, 2, 125-154 (1990) |
| [19] | Keck, J. C.; Gillespie, D., Rate-controlled partial-equilibrium method for treating reacting gas mixtures, Combust. Flame, 17, 2, 237-241 (1971) |
| [20] | Kenney, C. S.; Laub, A. J., Small-sample statistical condition estimates for general matrix functions, SIAM J. Sci. Comput., 15, 36-61 (1994) · Zbl 0801.65042 |
| [21] | Kunisch, K.; Volkwein, S., Galerkin proper orthogonal decomposition methods for a general equation in fluid dynamics, SIAM J. Numer. Anal., 40, 492-515 (2002) · Zbl 1075.65118 |
| [22] | Lam, S., Using CSP to understand complex chemical kinetics, Comb. Sci. Tech., 89, 375-404 (1993) |
| [23] | Lam, S.; Goussis, D., Understanding complex chemical kinetics with computational singular perturbation, Proc. Comb. Inst., 22, 931 (1988) |
| [24] | Lam, S.; Goussis, D., A CSP method for simplifying kinetics, Int. J. Chem. Kinetics, 26, 461-486 (1994) |
| [25] | Lee, C. H.; Tran, H. T., Reduced-order-based feedback control of the Kuramoto-Sivashinsky equation, J. Comp. Appl. Math., 173, 1, 1-19 (2005) · Zbl 1107.93028 |
| [26] | Li, A. Q.; Dowell, E. H., Modal reduction of mathematical models of biological molecules, J. Comp. Phys., 211, 1, 262-288 (2006) · Zbl 1073.92008 |
| [27] | E.N. Lorenz, in: Technical Report 1, MIT Department of Meteorology Statistical Forecasting Project, 1956; E.N. Lorenz, in: Technical Report 1, MIT Department of Meteorology Statistical Forecasting Project, 1956 |
| [28] | Luo, Z.; Zhu, J.; Wang, R.; Navon, I. M., Proper orthogonal decomposition approach and error estimation of mixed finite element methods for the tropical pacific ocean reduced gravity model, Comp. Methods Appl. Mech. Engrg., 196, 4184-4195 (2007) · Zbl 1173.76348 |
| [29] | Maas, U.; Pope, S. B., Implementation of simplified chemical kinetics based on intrinsic low-dimensional manifolds, Proc. Comb. Inst., 24, 103-112 (1992) |
| [30] | Nie, Q. H.; Joshi, Y., Multiscale thermal modeling methodology for thermoelectrically cooled electronic cabinets, Numer. Heat Transfer, 53, 3, 225-248 (2008) |
| [31] | Pinnau, R.; Schulze, A., Model reduction techniques for frequency averaging in radiative heat transfer, J. Comput. Phys., 226, 1, 712-731 (2007) · Zbl 1124.35323 |
| [32] | Rathinam, M.; Petzold, L. R., A new look at proper orthogonal decomposition, SIAM J. Numer. Anal., 41, 5, 1893-1925 (2003) · Zbl 1053.65106 |
| [33] | Ravindran, S. S., A reduced-order approach for optimal control of fluids using proper orthogonal decomposition, Int. J. Numer. Methods Fluids, 34, 425-488 (2000) · Zbl 1005.76020 |
| [34] | Ren, Z.; Pope, S. B., Entropy production and element conservation in the quasi-steady-state approximation, Combust. Flame, 137, 251-254 (2004) |
| [35] | Ren, Z.; Pope, S. B.; Vladimirsky, A.; Guckenheimer, J. M., The invariant constrained equilibrium edge preimage curve method for the dimension reduction of chemical kinetics, J. Chem. Phys., 124 (2006), Art. No. 114111 |
| [36] | Roussel, M. R.; Fraser, S. J., Global analysis of enzyme inhibition kinetics, J. Phys. Chem., 97, 31, 8316-8327 (1993) |
| [37] | Schwer, D. A.; Lu, P.; Green, W. H., An adaptive chemistry approach to modeling complex kinetics in reacting flows, Combust. Flame, 133, 4, 451-465 (2003) |
| [38] | Shvartsman, S. Y.; Kevrekidis, I. G., Low-dimensional approximation and control of periodic solutions in spatially extended systems, Phys. Rev. E, 58, 1, 361-368 (1998) |
| [39] | Shvartsman, S. Y.; Kevrekidis, I. G., Nonlinear model reduction for control of distributed systems: a computer-assisted study, AIChE J., 44, 7, 1579-1595 (1998) |
| [40] | Shvartsman, S. Y.; Theodoropoulos, C.; Rico-Martinez, R.; Kevrekidis, I. G.; Titi, E. S.; Mountziaris, T. J., Order reduction for nonlinear dynamic models of distributed reacting systems, J. Process Control, 10, 177-184 (2000) |
| [41] | Singer, M. A.; Pope, S. B.; Najm, H. N., Operator-splitting with ISAT to model reacting flow with detailed chemistry, Combust. Theory Modelling, 10, 199-217 (2006) · Zbl 1121.80333 |
| [42] | Singer, M. A.; Pope, S. B.; Najm, H. N., Modeling unsteady reacting flow with operator splitting and ISAT, Combust. Flame, 147, 150-162 (2006) · Zbl 1121.80333 |
| [43] | Sirovich, L., Turbulence and the dynamics of coherent structures. I - Coherent structures, Quart. Appl. Math., 45, 3, 561-571 (1987) · Zbl 0676.76047 |
| [44] | Smith, G. P.; Golden, D. M.; Frenklach, M.; Moriarty, N. W.; Eiteneer, B.; Goldenberg, M.; Bowman, C. T.; K Hanson, R.; Song, S.; Gardiner, W. C.; Lissianski, V. V.; Qin, Z. |
| [45] | Smooke, M. D., Reduced Kinetic Mechanisms and Asymptotic Approximations for Methane-Air Flames, Lecture Notes in Physics, vol. 384 (1991), Springer: Springer Berlin |
| [46] | Strang, G., On the construction and comparison of different schemes, SIAM J. Numer. Anal., 5, 3, 506-517 (1968) · Zbl 0184.38503 |
| [47] | Tinney, C. E.; Jordan, P.; Hall, A. M.; Delville, J.; Glauser, M. N., A time-resolved estimate of the turbulence and sound source mechanisms in a subsonic jet flow, J. Turbulence, 8, 7, 1-20 (2007) |
| [48] | Turányi, T.; Tomlin, A. S.; Pilling, M. J., On the error of the quasi-steady-state approximation, J. Phys. Chem., 97, 163-172 (1993) |
| [49] | Willcox, K., Unsteady flow sensing and estimation via the gappy proper orthogonal decomposition, Computers and Fluids, 35, 2, 208-226 (2006) · Zbl 1160.76394 |
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