×

zbMATH — the first resource for mathematics

A semi-implicit numerical scheme for reacting flow. II: Stiff, operator-split formulation. (English) Zbl 0958.76061
From the summary: We construct a stiff operator-split projection scheme for simulation of unsteady two-dimensional reaction flows with detailed kinetics. The scheme is based on compressible conservation equations for an ideal gas mixture in the zero-Mach-number limit. The equations of motion are spatially discretized by using second-order centered differences, and are advanced in time using a new stiff predictor-corrector approach. The new scheme is a modified version of the additive stiff scheme introduced in part I [the authors, ibid. 143, No. 2, 381-402, Art. No. CP975856 (1998; Zbl 0936.76064)].
The performance and behavior of the operator-split scheme are first analyzed based on tests for a nonlinear reaction-diffusion equation in one space dimension, followed by computations with a detailed \(C_1C_2\) methane-air mechanism in one and two dimensions. The tests are used to verify that the scheme is effectively second order in time, and to suggest guidelines for selecting integration parameters, including the number of fractional diffusion steps and tolerance levels in the stiff integration. For two-dimensional simulations with the present reaction mechanism, flame conditions, and resolution parameters, speedup factors of about 5 are achieved over the previous additive scheme, and about 25 over the original explicit scheme.

MSC:
76M20 Finite difference methods applied to problems in fluid mechanics
76M22 Spectral methods applied to problems in fluid mechanics
76V05 Reaction effects in flows
92E20 Classical flows, reactions, etc. in chemistry
80A32 Chemically reacting flows
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Aiken, R, Stiff computation, (1985)
[2] Hairer, E; Wanner, G, Solving ordinary differential equations, vol. II, stiff and differential-algebraic problems, (1996) · Zbl 0859.65067
[3] Frenklach, M; Wang, H; Goldenberg, M; Smith, G.P; Golden, D.M; Bowman, C.T; Hanson, R.K; Gardiner, W.C; Lissianski, V, Grimech—an optimized detailed chemical reaction mechanism for methane combustion, (1995)
[4] Najm, H.N; Wyckoff, P.S; Knio, O.M, A semi-implicit numerical scheme for reacting flow. I stiff chemistry, J. comput. phys., 143, 381, (1998) · Zbl 0936.76064
[5] Smooke, M.D; Puri, I.K; Seshadri, K, A comparison between numerical calculations and experimental measurements of the structure of a counterflow diffusion flame burning diluted methane in diluted air, Proceedings, twenty-first symposium (international) on combustion, 1783-1792, (1986)
[6] Curtiss, C.F; Hirschfelder, J.O, Integration of stiff equations, Proc. nat. acad. sci. U.S.A., 38, 235, (1952) · Zbl 0046.13602
[7] Gear, C.W, Numerical initial value problems in ordinary differential equations, (1971) · Zbl 0217.21701
[8] Hindmarsh, A.C, GEAR: ordinary differential equation system solver, (1974)
[9] Hindmarsh, A.C, GEARB: solution of ordinary differential equations having banded Jacobian, (1977)
[10] Hindmarsh, A.C, LSODE and LSODI: two new initial value ordinary equation solvers, ACM-signum, 15, 10, (1980)
[11] A. C. Hindmarsh, ODEPACK: A systematized collection of ODE solvers, in Scientific Computing, edited by R. SteplemanNorth-Holland, Amsterdam, 1983, pp. 55-64.
[12] Brown, P.N; Byrne, G.D; Hindmarsh, A.C, VODE: A variable coefficient ODE solver, SIAM J. sci. statist. comput., 10, 1038, (1989) · Zbl 0677.65075
[13] Vanderhouven, P.J; Sommeijer, B.P, Analysis of parallel diagonally implicit iteration of runge – kutta methods, Appl. numer. math., 11, 169, (1993) · Zbl 0787.65054
[14] Cong, N.H, A parallel DIRK method for stiff initial-value problems, J. comput. appl. math., 54, 121, (1994) · Zbl 0819.65109
[15] Zhong, X, Additive semi-implicit runge – kutta methods for computing high-speed nonequilibrium reactive flows, J. comput. phys., 128, 19, (1996) · Zbl 0861.76057
[16] Baeza, J.J.B; Plá, F.P; Ramos, G.R, Stiffness – adaptive Taylor method for the integration of non-stiff and stiff kinetic models, J. comput. chem., 13, 810, (1992)
[17] Sun, P; Chock, D.P; Winkler, S.L, An implicit – explicit hybrid solver for a system of stiff kinetic equations, J. comput. phys., 115, 515, (1994) · Zbl 0812.65061
[18] Chock, D.P; Winkler, S.L; Sun, P, Comparison of stiff chemistry solvers for air quality modeling, Environ. sci. technol., 28, 1882, (1994)
[19] Dabdub, D; Seinfeld, J.H, Extrapolation techniques used in the solution of stiff ODEs associated with chemical kinetics of air quality models, Atmos. environ., 29, 403, (1995)
[20] Elliot, S; Turco, R.P; Jacobson, M.Z, Tests on combined projection/forward differencing integration for stiff photochemical family systems at long time step, Computers chem., 17, 91, (1993)
[21] Hesstvedt, E; Hov, O; Isaksen, I.S.A, Quasi-steady-state approximations in air pollution modeling: comparison of two numerical schemes for oxidant prediction, Int. J. chem. kinetics, 10, 971, (1978)
[22] Young, T.R; Boris, J.P, A numerical technique for solving stiff ordinary differential equations associated with the chemical kinetics of reactive flow problems, J. phys. chem., 81, 2424, (1977)
[23] Verwer, J.G; van Loon, M, An evaluation of explicit pseudo-steady-state approximation schemes for stiff ODE systems from chemical kinetics, J. comput. phys., 113, 347, (1994) · Zbl 0810.65068
[24] Gong, W; Cho, H.-R, A numerical scheme for the integration of the gas-phase chemical rate equations in three-dimensional atmospheric models, Atmos. environ., 27A, 2591, (1993)
[25] Saylor, R.D; Ford, G.D, On the comparison of numerical methods for the integration of kinetic equations in atmospheric chemistry and transport models, Atmos. environ., 29, 2585, (1995)
[26] H. H. Robertson, The solution of a set of reaction rate equations, in Numerical Analysis: an Introduction, edited by J. WalshAcademic Press, New York, 1966, pp. 178-182.
[27] Radhakrishnan, K, New integration techniques for chemical kinetic rate equations. I. efficiency comparison, Combust. sci. technol., 46, 59, (1986)
[28] D’Angelo, Y; Larrouturou, B, Comparison and analysis of some numerical schemes for stiff complex chemistry problems, RAIRO modél. math. anal. numér., 29, 259, (1995) · Zbl 0829.76062
[29] Verwer, J.G, Gauss – seidel iteration for stiff ODES from chemical kinetics, SIAM J. sci. comput., 15, 1234, (1994) · Zbl 0804.65068
[30] Verwer, J.G; Blom, J.G; Spee, E.J, A comparison of stiff ODE solvers for atmospheric chemistry problems, Atmos. environ., 30, 49, (1996)
[31] Aro, C.J, CHEMSODE: A stiff ODE solver for the equations of chemical kinetics, Comput. phys. commun., 97, 304, (1996) · Zbl 0926.65072
[32] Courant, R; Friedrichs, K.O; Lewy, H, Über die partiellen differenzengleichungen der mathematischen physik, Math. ann., 100, 32, (1928) · JFM 54.0486.01
[33] Courant, R; Friedrichs, K.O; Lewy, H, Translation, on the partial difference equations of mathematical physics, IBM J. res. develop., 11, 215, (1967) · Zbl 0145.40402
[34] Anderson, D.A; Tannehill, J.C; Pletcher, R.H, Computational fluid mechanics and heat transfer, (1984) · Zbl 0569.76001
[35] Fletcher, C.A.J, Computational techniques for fluid dynamics, I, (1988) · Zbl 0706.76001
[36] Varah, J.M, Stability restrictions on second order, three level finite difference schemes for parabolic equations, SIAM J. numer. anal., 17, 300, (1980) · Zbl 0426.65048
[37] Ascher, U.M; Ruuth, S.J; Wetton, B.T.R, Implicit – explicit methods for time-dependent partial differential equations, SIAM J. numer. anal., 32, 797, (1995) · Zbl 0841.65081
[38] Hundsdorfer, W; Verwer, J.G, A note on splitting errors for advection – reaction equations, Appl. numer. math., 18, 191, (1995) · Zbl 0833.65099
[39] Verwer, J.G; Blom, J.G; Hundsdorfer, W, An implicit – explicit approach for atmospheric transport-chemistry problems, Appl. numer. math., 20, 191, (1996) · Zbl 0853.76092
[40] Sun, P, A pseudo-non-time-splitting method in air quality modeling, J. comput. phys., 127, 152, (1996) · Zbl 0859.65133
[41] Verwer, J.G; Spee, E.J; Blom, J.G; Hundsdorfer, W.H, A second order rosenbrock method applied to photochemical dispersion problems, SIAM J. sci. comput., 20, 1456, (1999) · Zbl 0928.65116
[42] Frank, J; Hundsdorfer, W; Verwer, J.G, On the stability of implicit – explicit linear multistep methods, Appl. numer. math., 25, 193, (1997) · Zbl 0887.65094
[43] Knoth, O; Wolke, R, Implicit – explicit runge – kutta methods for computing atmospheric reactive flows, Appl. numer. math., 28, 327, (1998) · Zbl 0934.76058
[44] Verwer, J.G; Simpson, D, Explicit methods for stiff ODEs from atmospheric chemistry, Appl. numer. math., 18, 413, (1995) · Zbl 0839.65078
[45] Knoth, O; Wolke, R, An explicit – implicit numerical approach for atmospheric chemistry-transport modeling, Atmos. environ., 32, 1785, (1998)
[46] Najm, H.N; Knio, O.M; Paul, P.H; Wyckoff, P.S, A study of flame observables in premixed methane – air flames, Combust. sci. technol., 140, 369, (1998)
[47] Hundsdorfer, W.H, Numerical solution of advection – diffusion – reaction equations, (1996)
[48] E. J. Spee, Coupling advection and chemical kinetics in a global atmospheric test model, in Air Pollution III, edited by H. Poweret al. Computational Mechanics, Southampton-Boston, 1995, Vol. 1, pp. 319-326.
[49] Verwer, J.G; Blom, J.G; van Loon, M; Spee, E.J, A comparison of stiff ODE solvers for atmospheric chemistry problems, Atmos. environ., 30, 49, (1995)
[50] Spee, E.J; de Zeeuw, P.M; Verwer, J.G; Blom, J.G; Hundsdorfer, W.H, Vectorization and parallelization of a numerical scheme for 3D global atmospheric transport-chemistry problems, (1996)
[51] Spee, E.J; Verwer, J.G; de Zeeuw, P.M; Blom, J.G; Hundsdorfer, W, A numerical study for global atmospheric transport-chemistry problems, Math. comput. simulation, 48, 177, (1998)
[52] Khan, L.A; Liu, P.L.-F, An operator splitting algorithm for coupled one-dimensional advection – diffusion – reaction equations, Comput. methods appl. mech. eng., 127, 181, (1995) · Zbl 0862.76060
[53] Sheng, Q, Solving linear partial differential equations by exponential splitting, IMA J. numer. anal., 9, 199, (1989) · Zbl 0676.65116
[54] Wright, J.P, Numerical instability due to varying time steps in explicit wave propagation and mechanics calculations, J. comput. phys., 140, 421, (1998) · Zbl 0920.65057
[55] LeVeque, R.J; Yee, H.C, A study of numerical methods for hyperbolic conservation laws with stiff source terms, J. comput. phys., 86, 187, (1990) · Zbl 0682.76053
[56] D. Lanser, and, J. G. Verwer, Analysis of operator splitting for advection – diffusion – reaction problems from air pollution modelling, CWI Report NM-R9805;, http://info4u.cwi.nl. · Zbl 0949.65090
[57] Strang, G, On the construction and comparison of difference schemes, SIAM J. numer. anal., 5, 506, (1968) · Zbl 0184.38503
[58] Burstein, S.Z; Mirin, A.A, Third order difference methods for hyperbolic equations, J. comput. phys., 5, 547, (1970) · Zbl 0223.65053
[59] Yoshida, H, Construction of higher order symplectic integrators, Phys. lett. A, 150, 262, (1990)
[60] Goyal, G; Paul, P.J; Mukunda, H.S; Deshpande, S.M, Time dependent operator-split and unsplit schemes for one dimensional premixed flames, Combust. sci. techol., 60, 167, (1988)
[61] Yang, B; Pope, S.B, An investigation of the accuracy of manifold methods and splitting schemes in the computational implementation of combustion chemistry, Combust. flame, 112, 16, (1998)
[62] Majda, A; Sethian, J, The derivation and numerical solution of the equations for zero Mach number combustion, Combust. sci. technol., 42, 185, (1985)
[63] Schlichting, H, Boundary-layer theory, (1979)
[64] Williams, F.A, Combustion theory, (1985)
[65] Kee, R.J; Rupley, F.M; Miller, J.A, Chemkin-II: A Fortran chemical kinetics package for the analysis of gas phase chemical kinetics, (1993)
[66] Chorin, A.J, A numerical method for solving incompressible viscous flow problems, J. comput. phys., 2, 12, (1967) · Zbl 0149.44802
[67] Bell, J.B; Marcus, D.L, A second-order projection method for variable density flows, J. comput. phys., 101, 334, (1992) · Zbl 0759.76045
[68] Almgren, A.S; Bell, J.B; Welcome, M.L, A conservative adaptive projection method for the variable density incompressible navier – stokes equations, J. comput. phys., 142, 1, (1998) · Zbl 0933.76055
[69] McMurtry, P.A; Jou, W.-H; Riley, J.J; Metcalfe, R.W, Direct numerical simulations of a reacting mixing layer with chemical heat release, Aiaa j., 24, 962, (1986)
[70] Rutland, C; Ferziger, J.H; Cantwell, B.J, Effects of strain, vorticity, and turbulence on premixed flames, (1989)
[71] Rutland, C.J; Ferziger, J.H, Simulations of flame – vortex interactions, Combust. flame, 84, 343, (1991)
[72] Mahalingam, S; Cantwell, B.J; Ferziger, J.H, Full numerical simulations of coflowing, axisymmetric jet diffusion flames, Phys. fluids A, 2, 720, (1990)
[73] Kim, J; Moin, P, Application of a fractional-step method to incompressible navier – stokes equations, J. comput. phys., 59, 308, (1985) · Zbl 0582.76038
[74] H. N. Najm, A conservative low Mach number projection method for reacting flow modeling, in Transport Phenomena in Combustion, edited by S. ChanTaylor & Francis, Washington, DC, 1996, Vol. 2, pp. 921-932.
[75] Najm, H.N; Wyckoff, P.S, Premixed flame response to unsteady strain-rate and curvature, Combust. flame, 110, 92, (1997)
[76] H. N. Najm, O. M. Knio, and, P. S. Wyckoff, Response of stoichiometric and rich methane – air flames to unsteady strain-rate and curvature, submitted for publication.
[77] Kee, R.J; Grcar, J.F; Smooke, M.D; Miller, J.A, A Fortran program for modeling steady laminar one-dimensional premixed flames, (1993)
[78] Ashurst, W.T; McMurtry, P.A, Flame generation of vorticity: vortex dipoles from monopoles, Combust. sci. technol., 66, 17, (1989)
[79] Poinsot, T; Veynante, D; Candel, S, Quenching processes and premixed turbulent combustion diagrams, J. fluid mech., 228, 561, (1991)
[80] T. Mantel, Fundamental, Mechanisms in Premixed Flame Propagation via Vortex-Flame Interactions—Numerical Simulations, Annual research briefs, Center for Turbulence Research, Stanford University/NASA Ames Research Center, 1994.
[81] M. Hilka, D. Veynante, M. Baum, and T. J. Poinsot, Simulation of flame – vortex interactions using detailed and reduced chemical kinetics, in Proceedings, Tenth Symposium on Turbulent Shear Flows, Penn. State University, University Park, PA, 1995, Vol. 2, Section 19, pp. 19-24.
[82] Jarosinski, J; Lee, J.H; Knystautas, R, Interaction of a vortex ring and a laminar flame, Proceedings, twenty-second symposium (international) on combustion, 505-514, (1988)
[83] Roberts, W.L; Driscoll, J.F; Drake, M.C; Goss, L.P, Images of the quenching of a flame by a vortex—to quantify regimes of turbulent combustion, Combust. flame, 94, 58, (1993)
[84] Samaniego, J.-M, Stretch-induced quenching in flame-vortex interactions, (1993)
[85] Mueller, C.J; Driscoll, J.F; Sutkus, D.J; Roberts, W.L; Drake, M.C; Smooke, M.D, Effect of unsteady stretch rate on OH chemistry during a flame – vortex interaction: to assess flamelet models, Combust. flame, 100, 323, (1995)
[86] Nguyen, Q.-V; Paul, P.H, The time evolution of a vortex – flame interaction observed via planar imaging of CH and OH, Proceedings, twenty-sixth symposium (international) on combustion, 357-364, (1996)
[87] Mueller, C.J; Driscoll, J.F; Reuss, D.L; Drake, M.C; Rosalik, M.E, Vorticity generation and attenuation as vortices convect through a premixed flame, Combust. flame, 112, 342, (1998)
[88] Najm, H.N; Paul, P.H; Mueller, C.J; Wyckoff, P.S, On the adequacy of certain experimental observables as measurements of flame burning rate, Combust. flame, 113, 312, (1998)
[89] Paul, P.H; Najm, H.N, Planar laser-induced fluorescence imaging of flame heat release rate, Proceedings, 27th symposium (international) on combustion, 43-50, (1998)
[90] Verwer, J.G, Explicit runge – kutta methodsfor parabolic partial differential equations, Appl. numer. math., 22, 359, (1996) · Zbl 0868.65064
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. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.