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A computationally-efficient, semi-implicit, iterative method for the time-integration of reacting flows with stiff chemistry. (English) Zbl 1349.80041
Summary: A semi-implicit preconditioned iterative method is proposed for the time-integration of the stiff chemistry in simulations of unsteady reacting flows, such as turbulent flames, using detailed chemical kinetic mechanisms. Emphasis is placed on the simultaneous treatment of convection, diffusion, and chemistry, without using operator splitting techniques. The preconditioner corresponds to an approximation of the diagonal of the chemical Jacobian. Upon convergence of the sub-iterations, the fully-implicit, second-order time-accurate, Crank-Nicolson formulation is recovered. Performance of the proposed method is tested theoretically and numerically on one-dimensional laminar and three-dimensional high Karlovitz turbulent premixed \(N\)-heptane/air flames. The species lifetimes contained in the diagonal preconditioner are found to capture all critical small chemical timescales, such that the largest stable time step size for the simulation of the turbulent flame with the proposed method is limited by the convective CFL, rather than chemistry. The theoretical and numerical stability limits are in good agreement and are independent of the number of sub-iterations. The results indicate that the overall procedure is second-order accurate in time, free of lagging errors, and the cost per iteration is similar to that of an explicit time integration. The theoretical analysis is extended to a wide range of flames (premixed and non-premixed), unburnt conditions, fuels, and chemical mechanisms. In all cases, the proposed method is found (theoretically) to be stable and to provide good convergence rate for the sub-iterations up to a time step size larger than 1 {\(\mu\)}s. This makes the proposed method ideal for the simulation of turbulent flames.

80M20 Finite difference methods applied to problems in thermodynamics and heat transfer
65M06 Finite difference methods for initial value and initial-boundary value problems involving PDEs
65M22 Numerical solution of discretized equations for initial value and initial-boundary value problems involving PDEs
80A32 Chemically reacting flows
Full Text: DOI
[1] Smooke, M.; Mitchell, R.; Keyes, D., Numerical-solution of 2-dimensional axisymmetric laminar diffusion flames, Combust. Sci. Technol., 67, 4-6, 85-122, (1989) · Zbl 0696.65087
[2] Bisetti, F., Integration of large chemical kinetic mechanisms via exponential methods with Krylov approximations to Jacobian matrix functions, Combust. Theory Model., 16, 3, 387-418, (2012) · Zbl 1262.80089
[3] Najm, H. N.; Wyckoff, P. S.; Knio, O. M., A semi-implicit numerical scheme for reacting flow: I. stiff chemistry, J. Comput. Phys., 143, 2, 381-402, (1998) · Zbl 0936.76064
[4] Maas, U.; Pope, S. B., Simplifying chemical kinetics: intrinsic low-dimensional manifolds in composition space, Combust. Flame, 88, 3, 239-264, (1992)
[5] Ju, Y., Lower-upper scheme for chemically reacting flow with finite rate chemistry, AIAA J., 33, 8, 1418-1425, (1995) · Zbl 0845.76058
[6] Yoo, C. S.; Lu, T.; Chen, J. H.; Law, C. K., Direct numerical simulations of ignition of a Lean n-heptane/air mixture with temperature inhomogeneities at constant volume: parametric study, Combust. Flame, 158, 9, 1727-1741, (2011)
[7] Perini, F.; Galligani, E.; Reitz, R. D., A study of direct and Krylov iterative sparse solver techniques to approach linear scaling of the integration of chemical kinetics with detailed combustion mechanisms, Combust. Flame, 161, 5, 1180-1195, (2014)
[8] Yoo, C. S.; Luo, Z.; Kim, H.; Chen, J. H., A DNS study of ignition characteristics of a Lean iso-octane/air mixture under HCCI and SACI conditions, Proc. Combust. Inst., 34, 2985-2993, (2013)
[9] Bhagatwala, A.; Chen, J. H.; Lu, T., Direct numerical simulations of HCCI/SACI with ethanol, Combust. Flame, 161, 1826-1841, (2014)
[10] Xuan, Y.; Blanquart, G., Numerical modeling of sooting tendencies in a laminar co-flow diffusion flame, Combust. Flame, 160, 9, 1657-1666, (2013)
[11] Aspden, A.; Day, M.; Bell, J., Lewis number effects in distributed flames, Proc. Combust. Inst., 33, 1473-1480, (2011)
[12] Bisetti, F.; Blanquart, G.; Mueller, M. E.; Pitsch, H., On the formation and early evolution of soot in turbulent nonpremixed flames, Combust. Flame, 159, 1, 317-335, (2012)
[13] Lignell, D. O.; Chen, J. H.; Smith, P. J.; Lu, T.; Law, C. K., The effect of flame structure on soot formation and transport in turbulent nonpremixed flames using direct numerical simulation, Combust. Flame, 151, 1-2, 2-28, (2007)
[14] Lignell, D. O.; Chen, J. H.; Smith, P. J., Three-dimensional direct numerical simulation of soot formation and transport in a temporally evolving nonpremixed ethylene jet flame, Combust. Flame, 155, 1-2, 316-333, (2008)
[15] Chen, J. H., Petascale direct numerical simulation of turbulent combustion - fundamental insights towards predictive models, Proc. Combust. Inst., 33, 1, 99-123, (2011)
[16] Aspden, A.; Day, M.; Bell, J., Turbulence-flame interactions in Lean premixed hydrogen: transition to the distributed burning regime, J. Fluid Mech., 680, 287-320, (2011) · Zbl 1241.76435
[17] Savard, B.; Bobbitt, B.; Blanquart, G., Structure of a high karlovitz n-C_{7}H_{16} premixed turbulent flame, Proc. Combust. Inst., 35, 1377-1384, (2015)
[18] Muller, J.-M., Elementary functions: algorithms and implementation, (2005), Birkhäuser
[19] D’Angelo, Y.; Larrouturou, B., Comparison and analysis of some numerical schemes for stiff complex chemistry problems, Modél. Math. Anal. Numér., 29, 3, 259-301, (1995) · Zbl 0829.76062
[20] Ropp, D. L.; Shadid, J. N.; Ober, C. C., Studies of the accuracy of time integration methods for reaction-diffusion equations, J. Comput. Phys., 194, 2, 544-574, (2004) · Zbl 1039.65069
[21] Ober, C. C.; Shadid, J. N., Studies on the accuracy of time-integration methods for the radiation-diffusion equations, J. Comput. Phys., 195, 2, 743-772, (2004) · Zbl 1053.65082
[22] Lu, T.; Law, C. K., A criterion based on computational singular perturbation for the identification of quasi steady state species: a reduced mechanism for methane oxidation with NO chemistry, Combust. Flame, 154, 4, 761-774, (2008)
[23] Lu, T.; Law, C. K.; Yoo, C. S.; Chen, J. H., Dynamic stiffness removal for direct numerical simulations, Combust. Flame, 156, 8, 1542-1551, (2009)
[24] Lu, T.; Law, C. K., A directed relation graph method for mechanism reduction, Proc. Combust. Inst., 30, 1, 1333-1341, (2005)
[25] Pepiot-Desjardins, P.; Pitsch, H., An efficient error-propagation-based reduction method for large chemical kinetic mechanisms, Combust. Flame, 154, 1, 67-81, (2008) · Zbl 1158.80325
[26] Peters, N., Reducing mechanisms, (Smooke, M. D., Reduced Kinetic Mechanisms and Asymptotic Approximations for Methane-Air Flames, Lect. Notes Phys., vol. 384, (1991), Springer Berlin, Heidelberg), 48-67
[27] Gou, X.; Sun, W.; Chen, Z.; Ju, Y., A dynamic multi-timescale method for combustion modeling with detailed and reduced chemical kinetic mechanisms, Combust. Flame, 157, 6, 1111-1121, (2010)
[28] Lam, S.; Coussis, D., Understanding complex chemical kinetics with computational singular perturbation, (Symposium (International) on Combustion, vol. 22, (1989), Elsevier), 931-941
[29] Bagrinovskii, K.; Godunov, S., Difference schemes for multidimensional problems, Dokl. Akad. Nauk USSR, 115, 431-433, (1957), (in Russian) · Zbl 0087.12201
[30] Strang, G., On the construction and comparison of difference schemes, SIAM J. Numer. Anal., 5, 3, 506-517, (1968) · Zbl 0184.38503
[31] Day, M.; Bell, J., Numerical simulation of laminar reacting flows with complex chemistry, Combust. Theory Model., 4, 4, 535-556, (1999) · Zbl 0970.76065
[32] Almgren, A.; Bell, J.; Colella, P.; Howell, L.; Welcome, M., A conservative adaptive projection method for the variable density incompressible Navier-Stokes equations, J. Comput. Phys., 142, 1-46, (1998) · Zbl 0933.76055
[33] Knio, O. M.; Najm, H. N.; Wyckoff, P. S., A semi-implicit numerical scheme for reacting flow: II. stiff, operator-split formulation, J. Comput. Phys., 154, 2, 428-467, (1999) · Zbl 0958.76061
[34] Yu, R.; Yu, J.; Bai, X.-S., An improved high-order scheme for DNS of low Mach number turbulent reacting flows based on stiff chemistry solver, J. Comput. Phys., 231, 16, 5504-5521, (2012) · Zbl 1428.76084
[35] Yoshida, H., Construction of higher order symplectic integrators, Phys. Lett. A, 150, 5, 262-268, (1990)
[36] Kennedy, C. A.; Carpenter, M. H., Additive Runge-Kutta schemes for convection-diffusion-reaction equations, Appl. Numer. Math., 44, 1-2, 139-181, (2003) · Zbl 1013.65103
[37] Kassam, A.-K.; Trefethen, L. N., Fourth-order time-stepping for stiff pdes, SIAM J. Sci. Comput., 26, 4, 1214-1233, (2005) · Zbl 1077.65105
[38] Sandu, A.; Verwer, J.; Loon, M. V.; Carmichael, G.; Potra, F.; Dabdub, D.; Seinfeld, J., Benchmarking stiff ODE solvers for atmospheric chemistry problems I: implicit vs explicit, EUMAC: European Modelling of Atmospheric Constituents, Atmos. Environ., 31, 19, 3151-3166, (1997)
[39] Sandu, A.; Verwer, J.; Blom, J.; Spee, E.; Carmichael, G.; Potra, F., Benchmarking stiff ODE solvers for atmospheric chemistry problems II: rosenbrock solvers, Atmos. Environ., 31, 20, 3459-3472, (1997)
[40] 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
[41] L.R. Petzold, A description of DASSL: a differential/algebraic system solver, Sandia National Laboratories Report, SAND282-8637.
[42] Brenan, K. E.; Campbell, S. L., A description of DASSL: a differential/algebraic equation solver, (Scientific Computing, (1983), North-Holland Amsterdam, The Netherlands), 65-68
[43] Robertson, H., The solution of a set of reaction rate equations, (Numerical Analysis: An Introduction, (1966)), 178-182
[44] 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, 19, 2585-2593, (1995)
[45] Aro, C. J.; Rodrigue, G. H., Preconditioned time differencing for stiff ODEs in diurnal atmospheric kinetics, Comput. Phys. Commun., 92, 1, 27-53, (1995) · Zbl 0908.65059
[46] Hawkes, E.; Sankaran, R.; Sutherland, C.; Chen, J., Direct numerical simulation of turbulent combustion: fundamental insights towards predictive models, J. Phys. Conf. Ser., 16, 65-79, (2005)
[47] Mueller, M. A.; Kim, T. J.; Yetter, R. A.; Dryer, F. L., Flow reactor studies and kinetic modeling of the H_{2}/O_{2} reaction, Int. J. Chem. Kinet., 31, 2, 113-125, (1999)
[48] Li, J.; Zhao, Z.; Kazakov, A.; Dryer, F. L., An updated comprehensive kinetic model of hydrogen combustion, Int. J. Chem. Kinet., 36, 10, 566-575, (2004)
[49] Sankaran, R.; Hawkes, E.; Chen, J.; Lu, T.; Law, C. K., Structure of a spatially developing turbulent Lean methane-air bunsen flame, Proc. Combust. Inst., 31, 1291-1298, (2007)
[50] Yoo, C. S.; Richardson, E.; Sankaran, R.; Chen, J. H., A DNS study on the stabilization mechanism of a turbulent lifted ethylene jet flame in highly-heated coflow, Proc. Combust. Inst., 33, 1, 1619-1627, (2011)
[51] Brown, P. N.; Shumaker, D. E.; Woodward, C. S., Fully implicit solution of large-scale non-equilibrium radiation diffusion with high order time integration, J. Comput. Phys., 204, 2, 760-783, (2005) · Zbl 1060.82001
[52] Chen, Z.; Burke, M. P.; Ju, Y., Effects of Lewis number and ignition energy on the determination of laminar flame speed using propagating spherical flames, Proc. Combust. Inst., 32, 1, 1253-1260, (2009)
[53] Wang, H.; Frenklach, M., A detailed kinetic modeling study of aromatics formation in laminar premixed acetylene and ethylene flames, Combust. Flame, 110, 1-2, 173-221, (1997)
[54] Lu, T.; Law, C. K., Toward accommodating realistic fuel chemistry in large-scale computations, Prog. Energy Combust. Sci., 35, 2, 192-215, (2009)
[55] Blanquart, G.; Pepiot-Desjardins, P.; Pitsch, H., Chemical mechanism for high temperature combustion of engine relevant fuels with emphasis on soot precursors, Combust. Flame, 156, 3, 588-607, (2009)
[56] McNenly, M. J.; Whitesides, R. A.; Flowers, D. L., Faster solvers for large kinetic mechanisms using adaptive preconditioners, Proc. Combust. Inst., 35, 1, 581-587, (2015)
[57] Tranquilli, P.; Sandu, A., Rosenbrock-Krylov methods for large systems of differential equations, SIAM J. Sci. Comput., 36, 3, A1313-A1338, (2014) · Zbl 1320.65108
[58] Dennis, J. E.; Schnabel, R. B., Numerical methods for unconstrained optimization and nonlinear equations, vol. 16, (1996), SIAM · Zbl 0847.65038
[59] Park, C.; Yoon, S., Fully coupled implicit method for thermochemical nonequilibrium air at suborbital flight speeds, AIAA J., 28, 1, 31-39, (1991)
[60] Eberhardt, S.; Imlay, S., Diagonal implicit scheme for computing flows with finite rate chemistry, J. Thermophys. Heat Transf., 6, 2, 208-216, (1992)
[61] Desjardins, O.; Blanquart, G.; Balarac, G.; Pitsch, H., High order conservative finite difference scheme for variable density low Mach number turbulent flows, J. Comput. Phys., 227, 15, 7125-7159, (2008) · Zbl 1201.76139
[62] Shunn, L.; Ham, F.; Moin, P., Verification of variable-density flow solvers using manufactured solutions, J. Comput. Phys., 231, 9, 3801-3827, (2012) · Zbl 1402.76107
[63] Williams, F. A., Combustion theory, (1985), Addison-Wesley
[64] Peters, N., Turbulent combustion, (2000), Cambridge University Press UK · Zbl 0955.76002
[65] Dworkin, S.; Smooke, M.; Giovangigli, V., The impact of detailed multicomponent transport and thermal diffusion effects on soot formation in ethylene/air flames, Proc. Combust. Inst., 32, 1, 1165-1172, (2009)
[66] Xuan, Y.; Blanquart, G., Effects of aromatic chemistry-turbulence interactions on soot formation in a turbulent non-premixed flame, Proc. Combust. Inst., 35, 2, 1911-1919, (2015)
[67] Xuan, Y.; Blanquart, G.; Mueller, M. E., Modeling curvature effects in diffusion flames using a laminar flamelet model, Combust. Flame, 161, 5, 1294-1309, (2014)
[68] Carroll, P. L.; Blanquart, G., A proposed modification to Lundgren’s physical space velocity forcing method for isotropic turbulence, Phys. Fluids, 25, 10, 105114, (2013)
[69] Verma, S.; Xuan, Y.; Blanquart, G., An improved bounded semi-Lagrangian scheme for the turbulent transport of passive scalars, J. Comput. Phys., 272, 0, 1-22, (2014) · Zbl 1349.76143
[70] Mueller, M. E.; Pitsch, H., LES model for sooting turbulent nonpremixed flames, Combust. Flame, 159, 6, 2166-2180, (2012)
[71] Carroll, P. L.; Blanquart, G., The effect of velocity field forcing techniques on the karman-howarth equation, J. Turbul., 15, 7, 429-448, (2014)
[72] Herrmann, M.; Blanquart, G.; Raman, V., Flux corrected finite volume scheme for preserving scalar boundedness in reacting large-eddy simulations, AIAA J., 44, 12, 2879-2886, (2006)
[73] Pierce, C. D., Progress-variable approach for large-eddy simulation of turbulent combustion, (2001), Stanford University, PhD thesis
[74] Maas, U.; Pope, S., Simplifying chemical kinetics: intrinsic low-dimensional manifolds in composition space, Combust. Flame, 88, 3-4, 239-264, (1992)
[75] Lam, S., Singular perturbation for stiff equations using numerical methods, (Recent Advances in the Aerospace Sciences, (1985), Springer), 3-19
[76] Falgout, R. D.; Yang, U. M., HYPRE: a library of high performance preconditioners, (Computational Science, ICCS 2002, (2002), Springer), 632-641 · Zbl 1056.65046
[77] Richardson, L. F., The approximate arithmetical solution by finite differences of physical problems involving differential equations, with an application to the stresses in a masonry dam, Philos. Trans. R. Soc. Lond. A, 210, 307-357, (1911) · JFM 42.0873.02
[78] Candler, G.; Subbareddy, P.; Nompelis, I., Decoupled implicit method for aerothermodynamics and reacting flows, AIAA J., 51, 5, 1245-1254, (2013)
[79] Candler, G.; Olynick, D., Hypersonic flow simulations using a diagonal implicit method, (Glowinski, R., Computing Methods in Applied Sciences and Engineering, (1991), Nova Science Publishers), 29-47
[80] Pitsch, H., Flamemaster: a C++ computer program for 0D combustion and 1D laminar flame calculations, (1998), available at:
[81] Hong, Z.; Davidson, D.; Hanson, R., An improved H_{2}/O_{2} mechanism based on recent shock tube/laser absorption measurements, Combust. Flame, 158, 4, 633-644, (2011)
[82] Smith, G. P.; Golden, D. M.; Frenklach, M.; Moriarty, N. W.; Eiteneer, B.; Goldenberg, M.; Bowman, C. T.; Hanson, R. K.; Song, S.; Gardiner, W. C.; Lissianski, V. V.; Qin, Z., GRI-mech 3.0, available at:
[83] Blanquart, G., Caltechmech v2.1, available at:
[84] Wang, W.; Luo, K.; Fan, J., Direct numerical simulation and conditional statistics of hydrogen/air turbulent premixed flames, Energy Fuels, 27, 549-560, (2013)
[85] Hawkes, E.; Chatakonda, O.; Kolla, H.; Kerstein, A.; Chen, J., A petascale direct numerical simulation study of the modelling of flame wrinkling for large-eddy simulations in intense turbulence, Combust. Flame, 159, 2690-2703, (2012)
[86] Day, M.; Bell, J.; Bremer, P.-T.; Pascucci, V.; Beckner, V.; Lijewski, M., Turbulence effects on cellular burning structures in Lean premixed hydrogen flames, Combust. Flame, 156, 1035-1045, (2009)
[87] Aspden, A.; Day, M.; Bell, J., Turbulence-chemistry interaction in Lean premixed hydrogen combustion, Proc. Combust. Inst., 35, 2, 1321-1329, (2015)
[88] Peters, N., Local quenching due to flame stretch and non-premixed turbulent combustion, Combust. Sci. Technol., 30, 1, 1-17, (1983)
[89] Kailasanathan, R. K.A.; Yelverton, T.; Fang, T.; Roberts, W., Effect of diluents on soot precursor formation and temperature in ethylene laminar diffusion flames, Combust. Flame, 160, 3, 656-670, (2013)
[90] Kailasanathan, R. K.A.; Book, E.; Fang, T.; Roberts, W., Hydrocarbon species concentrations in nitrogen diluted ethylene-air laminar jet diffusion flames at elevated pressures, Proc. Combust. Inst., 34, 1, 1035-1043, (2013)
[91] Ju, Y.; Sun, W.; Burke, M.; Gou, X.; Chen, Z., Multi-timescale modeling of ignition and flame regimes of n-heptane-air mixtures near spark assisted homogeneous charge compression ignition conditions, Proc. Combust. Inst., 33, 1, 1245-1251, (2011)
[92] Goyal, G.; Paul, P.; Mukunda, H.; Deshpande, S., Time dependent operator-split and unsplit schemes for one dimensional premixed flames, Combust. Sci. Technol., 60, 1-3, 167-189, (1988)
[93] Zhong, X., Additive semi-implicit Runge-Kutta methods for computing high-speed nonequilibrium reactive flows, J. Comput. Phys., 128, 1, 19-31, (1996) · Zbl 0861.76057
[94] Hiremath, V.; Lantz, S.; Wang, H.; Pope, S., Large-scale parallel simulations of turbulent combustion using combined dimension reduction and tabulation of chemistry, Proc. Combust. Inst., 34, 205-215, (2013)
[95] Lu, T.; Law, C. K., Strategies for mechanism reduction for large hydrocarbons: n-heptane, Combust. Flame, 154, 1-2, 153-163, (2008)
[96] Gruber, A.; Sankaran, E. R.; Hawkes, E. R.; Chen, J., Turbulent flame-wall interaction: a direct numerical simulation study, J. Fluid Mech., 658, 5-32, (2010) · Zbl 1205.76288
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