求解刚性燃烧化学反应系统的Krylov子空间中的指数积分法
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摘要
为了提高求解大时间步长的刚性燃烧化学反应方程组的效率,本文发展了一种使用Krylov子空间中的指数积分(EIKS)求解的方法。该方法基于多时间尺度法(MTS),使用指数积分格式求解线性化后的化学反应方程组,并使用Krylov子空间方法进行矩阵降维。本文使用MTS-EIKS方法模拟了H_2/CO/空气和CH4/空气的自着火过程,并引入自相关动态自适应化学(CoDAC)机理简化。结果表明,CoDAC-MTS-EIKS的联合简化加速方法可以准确、快速地模拟时间步长为10~(-6)s的自着火问题,在计算大步长刚性燃烧化学反应方程组方面有很好的应用前景。
In order to improve the efficiency in solving stiff combustion chemical reaction equations with large time steps, an exponential integrator in Krylov subspace(EIKS) was developed. The method is based on the Multi-timescale(MTS) method. It employs exponential scheme to solve linearized stiff chemical reaction equations, and to reduce matrix dimension with Krylov subspace method. The MTS-EIKS was used to simulate the ignition delays of mixtures of H2/CO/air and CH4/air. Additionally, the Correlated dynamic adaptive chemistry(CoDAC) method was introduced.The results show that the combined reduction method CoDAC-MTS-EIKS method can predict the ignition delays accurately and efficiently with a time steps of 10~(-6)s, which demonstrates that the method has a good application prospect in solving stiff chemistry problems with large time steps.
In order to improve the efficiency in solving stiff combustion chemical reaction equations with large time steps, an exponential integrator in Krylov subspace(EIKS) was developed. The method is based on the Multi-timescale(MTS) method. It employs exponential scheme to solve linearized stiff chemical reaction equations, and to reduce matrix dimension with Krylov subspace method. The MTS-EIKS was used to simulate the ignition delays of mixtures of H2/CO/air and CH4/air. Additionally, the Correlated dynamic adaptive chemistry(CoDAC) method was introduced.The results show that the combined reduction method CoDAC-MTS-EIKS method can predict the ignition delays accurately and efficiently with a time steps of 10~(-6)s, which demonstrates that the method has a good application prospect in solving stiff chemistry problems with large time steps.
引文
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[2]Chen J H.Petascale Direct Numerical Simulation of Turbulent Combustion-Fundamental Insights Towards Predictive Models[J].Proceedings of the Combustion Institute,2011,33(1):99-123
[3]GOU Xiaolong,SUN Wenting,CHEN Zheng.Numerical Methods for Complicated Chemical Mechanism Involved in Combustion Simulation(in Chinese)[J].SCIENTIA SINICA Physica,Mechanica&Astronomica,2017,47(7):070006
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[6]Lu T F,Law C K.A Directed Relation Graph Method for Mechanism Reduction[J].Proceedings of the Combustion Institute,2005,30:1333-1341
[7]Lu T F,Law C K.On the Applicability of Directed Relation Graphs to the Reduction of Reaction Mechanisms[J].Combustion and Flame,2006,146(3):472-483
[8]LUO Zhaoyu,LU Tianfeng,Maciaszek M J,et al.A Reduced Mechanism for High-temperature Oxidation of Biodiesel Surrogates[J].Energy and Fuels,2010,24(12):6283-6293
[9]Zheng X L,Lu T F,Law C K.Experimental Counterflow Ignition Temperatures and Reaction Mechanisms of 1,3-butadiene[J].Proceedings of the Combustion Institute,2007,31(1):367-375
[10]Pepiot-Desjardins P,Pitsch H.An Efficient Errorpropagation-based Reduction Method for Large Chemical Kinetic Mechanisms[J].Combustion and Flame,2008,154(1):67-81
[11]Sun Wenting,Chen Zheng,Gou Xiaolong,et al.A Path Flux Analysis Method for the Reduction of Detailed Chemical Kinetic Mechanisms[J].Combustion and Flame,2010,157(7):1298-1307
[12]Liang L,Stevens J G,Farrell J T.A Dynamic Adaptive Chemistry Scheme for Reactive Flow Computations[J].Proceedings of the Combustion Institute,2009,32(1):527-534
[13]Liang L,Stevens J G,Farrell J T.A Dynamic Multi-Zone Partitioning Scheme for Solving Detailed Chemical Kinetics in Reactive Flow Computations[J].Combustion Science and Technology,2009,181(11):1345-1371
[14]Liang L,Stevens J G,Raman S,et al.The Use of Dynamic Adaptive Chemistry in Combustion Simulation of Gasoline Surrogate Fuels[J].Combustion and Flame,2009,156(7):1493-1502
[15]Gou Xiaolong,Chen Zheng,Sun Wenting,et al.A Dynamic Adaptive Chemistry Scheme with Error Control for Combustion Modeling with a Large Detailed Mechanism[J].Combustion and Flame,2013,160(2):225-231
[16]Sun Weiqi,Gou Xiaolong,El-Asrag H A,et al.Multitimescale and Correlated Dynamic Adaptive Chemistry Modeling of Ignition and Flame Propagation Using a Real Jet Fuel Surrogate Model[J].Combustion and Flame,2015,162(4):1530-1539
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[18]刘再刚,杨帆,孔文俊.基于CO-DAC的合成气反应极限分析[J].燃烧科学与技术,2017,23(1):61-67LIU Zaigang,YANG Fan,KONG Wenjun.Reaction Limit Analysis of Syngas Based on CO-DAC Method[J].Journal of Combustion Science and Technology,2017,23(1):61-67
[19]Pope S B.Computationally Efficient Implementation of Combustion Chemistry Using in Situ Adaptive Tabulation[J].Combustion Theory and Modelling,1997,1:41-63
[20]Yang B,Pope S B Treating Chemistry in Combustion with Detailed Mechanisms-In Situ Adaptive Tabulation in Principal Directions Premixed Combustion[J].Combustion and Flame,1998,112(1/2):85-112
[21]Contino F,Jeanmart H,Lucchini T,et al.Coupling of in Situ Adaptive Tabulation and Dynamic Adaptive Chemistry:an Effective Method for Solving Combustion in Engine Simulations[J].Proceedings of the Combustion Institute,2011,33:3057-3064
[22]SUN Wenting,WON S H,JU Yiguang.In Situ Plasma Activated Low Temperature Chemistry and The S-curve Transition in DME/Oxygen/Helium Mixture[J].Combustion and Flame,2014,161(8):2054-2063
[23]Lefkowitz J K,Uddi M,Windom B C,et al.In Situ Species Diagnostics and Kinetic Study of Plasma Activated Ethylene Dissociation and Oxidation In a Low Temperature Flow Reactor[J].Proceedings of the Combustion Institute,2015,35(3):3505-3512
[24]Katta V R,Roquemore W M.Calculation of Multidimensional Flames Using Large Chemical Kinetics[J].AIAA Journal,2008,46(7):1640-1650
[25]Yang S,Ranjan R,Yang V,et al.Parallel On-The-Fly Adaptive Kinetics in Direct Numerical Simulation of Turbulent Premixed Flame[J].Proceedings of the Combustion Institute,2017,36(2):2025-2032
[26]Lu Tianfeng,Law C K,Yoo C S,et al.Dynamic Stiffness Removal for Direct Numerical Simulations[J].Combustion and Flame,2009,156(8):1542-1551
[27]Gou Xiaolong,Sun Wenting,Chen Zheng,et al.A Dynamic Multi-Timescale Method for Combustion Modeling with Detailed and Reduced Chemical Kinetic Mechanisms[J].Combustion and Flame,2010,157(6):1111-1121
[28]GAO Yang,LIU Yufeng,REN Zhuyin,et al.A Dynamic Adaptive Method for Hybrid Integration of Stiff Chemistry[J].Combustion and Flame,2015,162(2):287-295
[29]Hochbruck M,Ostermann A.Exponential Integrators[J].ACTA Numerica,2010,19:209-286
[30]Tangman D Y,Gopaul A,Bhuruth M.Exponential Time Integration and Chebychev Discretisation Schemes for Fast Pricing of Options[J].Applied Numerical Mathematics,2008,58(9):1309-1319
[31]Niesen J,Wright W M.A Krylov Subspace Method for Option Pricing[EB/OL].[2017-08-19].https://ssrn.com/abstract-1799124
[32]Hochbruck M,Honig M,Ostermann A.Regularization of Nonlinear Ill-Posed Problems by Exponential Integrators[J].Esaim-Mathematical Modelling and Numerical Analysis-Modelisation Mathematique Et Analyse Numerique,2009,43(4):709-720
[33]Tokman M,Loffeld J.Efficient Design of ExponentialKrylov Integrators for Large Scale Computing[J].Iccs2010-International Conference on Computational Science,Proceedings,2010,1(1):229-237
[34]Gaudreault S,Pudykiewicz J A.An Efficient Exponential Time Integration Method for the Numerical Solution of the Shallow Water Equations on the Sphere[J].Journal of Computational Physics,2016,322:827-848
[35]Perini F,Galligani E,Reitz R.D.An Analytical Jacobian Approach to Sparse Reaction Kinetics for Computationally Efficient Combustion Modeling with Large Reaction Mechanisms[J].Energy&Fuels,2012,26(8):4804-4822
[36]Moler C,Van Loan C.Nineteen Dubious Ways to Compute the Exponential of a Matrix,Twenty-Five Years Later[J].Siam Review,2003,45(1):3-49
[37]Al-Mohy A H,Higham N J.A New Scaling and Squaring Algorithm for the Matrix Exponential[J].Siam Journal on Matrix Analysis and Applications,2009,31(3):970-989
[38]Saad Y.Analysis of Some Krylov Subspace Approximations to the Matrix Exponential Operator[J].Siam Journal On Numerical Analysis,1992,29(1):209-228
[39]Sidje R B.Expokit:a Software Package for Computing Matrix Exponentials[J].ACM Transactions on Mathematical Software,1998,24(1):130-156
[40]Niesen J,Wright W M.Algorithm 919:a Krylov Subspace Algorithm for Evaluating the phi-Functions Appearing in Exponential Integrators[J].ACM Transactions on Mathematical Software,2012,38(3):22:1-22:19
[41]Lord G J,Stone D.New Efficient Substepping Methods for Exponential Timestepping[J].Applied Mathematics and Computation,2017,307:342-365
[42]Vo H D,Sidje R B.Implementation of Variable Parameters in the Krylov-Based Finite State Projection for Solving the Chemical Master Equation[J].Applied Mathematics and Computation,2017,293:334-344
[43]Bisetti F:Integration of Large Chemical Kinetic Mechanisms Via Exponential Methods with Krylov Approximations to Jacobian Matrix Functions[J].Combustion Theory and Modelling,2012,16:387-418
[44]Lam S H.Model Reductions with Special CSP Data[J].Combustion and Flame,2013,160(12):2707-2711
[45]Li J,Zhao Z W,Kazakov A,et al.A Comprehensive Kinetic Mechanism for CO,CH_2O,and CH_3OH Combustion[J].International Journal of Chemical Kinetics,2007,39(3):109-136
[46]Smith Gregory P,Golden David M,Frenklach Michael,et al.GRI-Mech 3.0[EB/OL].[2017-08-20].http://www.me.b erkeley.ed u/gr i_mech/
[47]杨帆,林博颖,隋春杰,等.湍流强旋流预混火焰大涡模拟研究[J].工程热物理学报,2015,36(4):895-900Yang Fan,Lin Boying,Sui Chunjie,et al.Large Eddy Simulation of Turbulent Strongly Swirling Premixed Flame[J].Journal of Engineering Thermophysics,2015,36(4):895-900
[48]Kim J,Moin P.Application of a Fractional-Step Method to Incompressible Navier-Stokes Equations[J].Journal of Computational Physics,1985,59(2):308-323
[49]Desjardins O,Blanquart G,Balarac G,et al.High Order Conservative Finite Difference Scheme for Variable Density Low Mach Number Turbulent Flows[J].Journal of Computational Physics,2008,227(15):7125-7159
[2]Chen J H.Petascale Direct Numerical Simulation of Turbulent Combustion-Fundamental Insights Towards Predictive Models[J].Proceedings of the Combustion Institute,2011,33(1):99-123
[3]GOU Xiaolong,SUN Wenting,CHEN Zheng.Numerical Methods for Complicated Chemical Mechanism Involved in Combustion Simulation(in Chinese)[J].SCIENTIA SINICA Physica,Mechanica&Astronomica,2017,47(7):070006
[4]Brown P N,Byrne G D,Hindmarsh A C.VODE,a Variable-Coefficient ODE Solver[J].Siam Journal on Scientific and Statistical Computing,1989,10(5):1038-1051
[5]Petzold L R.A Description of DASSL:a Differential/algebraic System Solver[R].Sandia National Laboratories,SAND82-8637,1982
[6]Lu T F,Law C K.A Directed Relation Graph Method for Mechanism Reduction[J].Proceedings of the Combustion Institute,2005,30:1333-1341
[7]Lu T F,Law C K.On the Applicability of Directed Relation Graphs to the Reduction of Reaction Mechanisms[J].Combustion and Flame,2006,146(3):472-483
[8]LUO Zhaoyu,LU Tianfeng,Maciaszek M J,et al.A Reduced Mechanism for High-temperature Oxidation of Biodiesel Surrogates[J].Energy and Fuels,2010,24(12):6283-6293
[9]Zheng X L,Lu T F,Law C K.Experimental Counterflow Ignition Temperatures and Reaction Mechanisms of 1,3-butadiene[J].Proceedings of the Combustion Institute,2007,31(1):367-375
[10]Pepiot-Desjardins P,Pitsch H.An Efficient Errorpropagation-based Reduction Method for Large Chemical Kinetic Mechanisms[J].Combustion and Flame,2008,154(1):67-81
[11]Sun Wenting,Chen Zheng,Gou Xiaolong,et al.A Path Flux Analysis Method for the Reduction of Detailed Chemical Kinetic Mechanisms[J].Combustion and Flame,2010,157(7):1298-1307
[12]Liang L,Stevens J G,Farrell J T.A Dynamic Adaptive Chemistry Scheme for Reactive Flow Computations[J].Proceedings of the Combustion Institute,2009,32(1):527-534
[13]Liang L,Stevens J G,Farrell J T.A Dynamic Multi-Zone Partitioning Scheme for Solving Detailed Chemical Kinetics in Reactive Flow Computations[J].Combustion Science and Technology,2009,181(11):1345-1371
[14]Liang L,Stevens J G,Raman S,et al.The Use of Dynamic Adaptive Chemistry in Combustion Simulation of Gasoline Surrogate Fuels[J].Combustion and Flame,2009,156(7):1493-1502
[15]Gou Xiaolong,Chen Zheng,Sun Wenting,et al.A Dynamic Adaptive Chemistry Scheme with Error Control for Combustion Modeling with a Large Detailed Mechanism[J].Combustion and Flame,2013,160(2):225-231
[16]Sun Weiqi,Gou Xiaolong,El-Asrag H A,et al.Multitimescale and Correlated Dynamic Adaptive Chemistry Modeling of Ignition and Flame Propagation Using a Real Jet Fuel Surrogate Model[J].Combustion and Flame,2015,162(4):1530-1539
[17]Sun Weiqi,Won S H,Gou Xiaolong,et al.Multi-scale Modeling of Dynamics and Ignition to Flame Transitions of High Pressure Stratified N-heptane/toluene Mixtures[J].Proceedings of the Combustion Institute,2015,35(1):1049-1056
[18]刘再刚,杨帆,孔文俊.基于CO-DAC的合成气反应极限分析[J].燃烧科学与技术,2017,23(1):61-67LIU Zaigang,YANG Fan,KONG Wenjun.Reaction Limit Analysis of Syngas Based on CO-DAC Method[J].Journal of Combustion Science and Technology,2017,23(1):61-67
[19]Pope S B.Computationally Efficient Implementation of Combustion Chemistry Using in Situ Adaptive Tabulation[J].Combustion Theory and Modelling,1997,1:41-63
[20]Yang B,Pope S B Treating Chemistry in Combustion with Detailed Mechanisms-In Situ Adaptive Tabulation in Principal Directions Premixed Combustion[J].Combustion and Flame,1998,112(1/2):85-112
[21]Contino F,Jeanmart H,Lucchini T,et al.Coupling of in Situ Adaptive Tabulation and Dynamic Adaptive Chemistry:an Effective Method for Solving Combustion in Engine Simulations[J].Proceedings of the Combustion Institute,2011,33:3057-3064
[22]SUN Wenting,WON S H,JU Yiguang.In Situ Plasma Activated Low Temperature Chemistry and The S-curve Transition in DME/Oxygen/Helium Mixture[J].Combustion and Flame,2014,161(8):2054-2063
[23]Lefkowitz J K,Uddi M,Windom B C,et al.In Situ Species Diagnostics and Kinetic Study of Plasma Activated Ethylene Dissociation and Oxidation In a Low Temperature Flow Reactor[J].Proceedings of the Combustion Institute,2015,35(3):3505-3512
[24]Katta V R,Roquemore W M.Calculation of Multidimensional Flames Using Large Chemical Kinetics[J].AIAA Journal,2008,46(7):1640-1650
[25]Yang S,Ranjan R,Yang V,et al.Parallel On-The-Fly Adaptive Kinetics in Direct Numerical Simulation of Turbulent Premixed Flame[J].Proceedings of the Combustion Institute,2017,36(2):2025-2032
[26]Lu Tianfeng,Law C K,Yoo C S,et al.Dynamic Stiffness Removal for Direct Numerical Simulations[J].Combustion and Flame,2009,156(8):1542-1551
[27]Gou Xiaolong,Sun Wenting,Chen Zheng,et al.A Dynamic Multi-Timescale Method for Combustion Modeling with Detailed and Reduced Chemical Kinetic Mechanisms[J].Combustion and Flame,2010,157(6):1111-1121
[28]GAO Yang,LIU Yufeng,REN Zhuyin,et al.A Dynamic Adaptive Method for Hybrid Integration of Stiff Chemistry[J].Combustion and Flame,2015,162(2):287-295
[29]Hochbruck M,Ostermann A.Exponential Integrators[J].ACTA Numerica,2010,19:209-286
[30]Tangman D Y,Gopaul A,Bhuruth M.Exponential Time Integration and Chebychev Discretisation Schemes for Fast Pricing of Options[J].Applied Numerical Mathematics,2008,58(9):1309-1319
[31]Niesen J,Wright W M.A Krylov Subspace Method for Option Pricing[EB/OL].[2017-08-19].https://ssrn.com/abstract-1799124
[32]Hochbruck M,Honig M,Ostermann A.Regularization of Nonlinear Ill-Posed Problems by Exponential Integrators[J].Esaim-Mathematical Modelling and Numerical Analysis-Modelisation Mathematique Et Analyse Numerique,2009,43(4):709-720
[33]Tokman M,Loffeld J.Efficient Design of ExponentialKrylov Integrators for Large Scale Computing[J].Iccs2010-International Conference on Computational Science,Proceedings,2010,1(1):229-237
[34]Gaudreault S,Pudykiewicz J A.An Efficient Exponential Time Integration Method for the Numerical Solution of the Shallow Water Equations on the Sphere[J].Journal of Computational Physics,2016,322:827-848
[35]Perini F,Galligani E,Reitz R.D.An Analytical Jacobian Approach to Sparse Reaction Kinetics for Computationally Efficient Combustion Modeling with Large Reaction Mechanisms[J].Energy&Fuels,2012,26(8):4804-4822
[36]Moler C,Van Loan C.Nineteen Dubious Ways to Compute the Exponential of a Matrix,Twenty-Five Years Later[J].Siam Review,2003,45(1):3-49
[37]Al-Mohy A H,Higham N J.A New Scaling and Squaring Algorithm for the Matrix Exponential[J].Siam Journal on Matrix Analysis and Applications,2009,31(3):970-989
[38]Saad Y.Analysis of Some Krylov Subspace Approximations to the Matrix Exponential Operator[J].Siam Journal On Numerical Analysis,1992,29(1):209-228
[39]Sidje R B.Expokit:a Software Package for Computing Matrix Exponentials[J].ACM Transactions on Mathematical Software,1998,24(1):130-156
[40]Niesen J,Wright W M.Algorithm 919:a Krylov Subspace Algorithm for Evaluating the phi-Functions Appearing in Exponential Integrators[J].ACM Transactions on Mathematical Software,2012,38(3):22:1-22:19
[41]Lord G J,Stone D.New Efficient Substepping Methods for Exponential Timestepping[J].Applied Mathematics and Computation,2017,307:342-365
[42]Vo H D,Sidje R B.Implementation of Variable Parameters in the Krylov-Based Finite State Projection for Solving the Chemical Master Equation[J].Applied Mathematics and Computation,2017,293:334-344
[43]Bisetti F:Integration of Large Chemical Kinetic Mechanisms Via Exponential Methods with Krylov Approximations to Jacobian Matrix Functions[J].Combustion Theory and Modelling,2012,16:387-418
[44]Lam S H.Model Reductions with Special CSP Data[J].Combustion and Flame,2013,160(12):2707-2711
[45]Li J,Zhao Z W,Kazakov A,et al.A Comprehensive Kinetic Mechanism for CO,CH_2O,and CH_3OH Combustion[J].International Journal of Chemical Kinetics,2007,39(3):109-136
[46]Smith Gregory P,Golden David M,Frenklach Michael,et al.GRI-Mech 3.0[EB/OL].[2017-08-20].http://www.me.b erkeley.ed u/gr i_mech/
[47]杨帆,林博颖,隋春杰,等.湍流强旋流预混火焰大涡模拟研究[J].工程热物理学报,2015,36(4):895-900Yang Fan,Lin Boying,Sui Chunjie,et al.Large Eddy Simulation of Turbulent Strongly Swirling Premixed Flame[J].Journal of Engineering Thermophysics,2015,36(4):895-900
[48]Kim J,Moin P.Application of a Fractional-Step Method to Incompressible Navier-Stokes Equations[J].Journal of Computational Physics,1985,59(2):308-323
[49]Desjardins O,Blanquart G,Balarac G,et al.High Order Conservative Finite Difference Scheme for Variable Density Low Mach Number Turbulent Flows[J].Journal of Computational Physics,2008,227(15):7125-7159