Variable Time-stepping Exponential Integrators for Chemical Reactors with Analytical Jacobians

Jared Stewart; Mayya Tokman; Valentin Dallerit; Fabrizio Bisetti; Oscar Diaz-Ibarra

doi:doi:10.11648/j.acm.20241302.11

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Variable Time-stepping Exponential Integrators for Chemical Reactors with Analytical Jacobians

Jared Stewart^*, Mayya Tokman, Valentin Dallerit, Fabrizio Bisetti, Oscar Diaz-Ibarra

Published in Applied and Computational Mathematics (Volume 13, Issue 2)

Received: 18 February 2024 Accepted: 19 March 2024 Published: 7 April 2024

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Abstract

Chemical combustion problems are known to be stiff and therefore difficult to efficiently integrate in time when numerically simulated. Implicit methods, such as backwards differentiation formula (BDF), are widely considered to be the state-of-the-art methods owing their capability of taking relatively large time-steps while maintaining accurate combustion characteristics. Exponential time integration methods have recently demonstrated the ability to accurately and efficiently solve large scale systems of ordinary differential equations. This study introduces a novel adaptive time stepping exponential integrator named EPI3V. Its performance is measured on spatially homogeneous isobaric reactive mixtures involving three hydrocarbon fuel mechanisms. The full combustion process is simulated using gas compositions with sufficient temperature to obtain auto-ignition. Simulations are run until the steady state is obtained, then a comparison of the computational efficiency and accuracy between a BDF and EPI3V method is made. The novel EPI3V method exhibits comparable computational efficiency to a well-established implementation of the variable time-stepping BDF implicit methods for two of the mechanisms investigated. In certain situations it even demonstrates a slight advantage over the implicit solver. However, in one specific case, the EPI3V shows relative performance degradation compared to the implicit method, but it still converges for this case. These results indicate that exponential time integration methods may be applicable to a larger variety of combustion problems.

Published in	Applied and Computational Mathematics (Volume 13, Issue 2)
DOI	10.11648/j.acm.20241302.11
Page(s)	29-37
Creative Commons	This is an Open Access article, distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution and reproduction in any medium or format, provided the original work is properly cited.
Copyright	Copyright © The Author(s), 2024. Published by Science Publishing Group

Keywords

Exponential Time Integrators, Chemical Reactors, Time Integrator, Analytic Jacobian, Numerical Methods

References

[1]	Z. X. Chen, I. Langella, and N. Swaminathan, The Role of CFD in Modern Jet Engine Combustor Design, Environmental Impact of Aviation and Sustainable Solutions. IntechOpen. 2020 https://doi.org/10.5772/intechopen.88267 .
[2]	R. Reitz, Directions in internal combustion engine research, Combustion and Flame, Volume 160, Issue 1, 2013, Pages 1-8, ISSN 0010-2180. https://doi.org/10.1016/j.combustflame.2012.11.002
[3]	A. D Gomez, N. Deak, F. Bisetti, Jacobian- free Newton-Krylov method for the simulation of non-thermal plasma discharges with high-order time integration and physics-based preconditioning, Journal of Computational Physics, Vol 480, 2023. https://doi.org/10.1016/j.jcp.2023.112007
[4]	P. N. Brown, G. D. Byrne, and A. C. Hindmarsh, VODE: A variable coefficient ODE solver, SIAM J. Sci. Statist. Comput. 10(5) (1989), pp. 1038-1051. https://doi.org/10.1137/0910062
[5]	G. D. Byrne and A. M. Dean, The numerical solution of some kinetics models with VODE and CHEMKIN II, Comput. Chem. 17(3) (1993), pp. 297-302. https://doi.org/10.1016/0097-8485(93)80011-2
[6]	F. Bisetti, High-order methods for the simulation of unsteady counterflow flames subject to stochastic forcing of large amplitude, Combustion Theory and Modeling, 2023, https://doi.org/10.1080/13647830.2023.2218621
[7]	Sandia National Laboratories. Available from https://github.com/sandialabs/TChem, [Accessed 12 March 2024]
[8]	D. G Goodwin, H. K. Moffat, and R. L. Speth, Cantera: An Object-oriented Software Toolkit for Chemical Kinetics, Thermodynamics, and Transport Processes. Version 2.2.0, 2015.
[9]	R. J. Kee, F. M. Rupley, and J. A. Miller, Chemkin-II: A Fortran chemical kinetics package for the analysis of gas- phase chemical kinetics, Sandia National Laboratories, 1989, SAND-89-8009.
[10]	M. J. McNenly, R. A. Whitesides, D. L. Flowers Faster solvers for large kinetic mechanisms using adaptive preconditioners, Proceedings of the combustion institute, Volume 35, Issue 1, 2015. https://doi.org/10.1016/j.proci.2014.05.113
[11]	M. Tokman, A new class of exponential propagation iterative methods of Runge-Kutta type (EPIRK), Journal of Computational Physics Volume 230, Issue 24, 1 October 2011, Pages 8762-8778. https://doi.org/10.1016/j.jcp.2011.08.023
[12]	M. Hochbruck, A. Ostermann, Exponential Runge- Kutta methods for parabolic problems, Applied Numerical Mathematics, Volume 53, Issues 2-4, 2005, Pages 323-339, ISSN 0168-9274. https://doi.org/10.1016/j.apnum.2004.08.005 .
[13]	F. Bisetti, Integration of large chemical kinetic mechanisms via exponential methods with Krylov approximations to Jacobian matrix functions, Combustion theory and modeling, Vol 16, Number 3, 2012. https://doi.org/org/10.1080/13647830.2011.631032
[14]	W. Dabelstein, A. Reglitzky, A. Schütze and K. Reders Automotive Fuels, Ullmann’s Encyclopedia of Industrial Chemistry 2007, Wiley-VCH, Weinheim. https://doi.org/10.1002/14356007.a16 719.pub2
[15]	D. Kim, J. Martz, A. Violi, A surrogate for emulating the physical and chemical properties of conventional jet fuel, Combustion and Flame Volume 161, Issue 6, June 2014, Pages 1489-1498. https://doi.org/10.1016/j.combustflame.2013.12.015
[16]	T. Kathrotia, P. Oßwald, C. Naumann, S. Richter, M. Köhler, Combustion kinetics of alternative jet fuels, Part-II: Reaction model for fuel surrogate, Fuel, Volume 302, 2021, 120736, ISSN 0016-2361, https://doi.org/10.1016/j.fuel.2021.120736 .
[17]	G. P. Smith, D. M. Golden, M. Frenklach, N. W. Moriarty, B. Eiteneer, M. Goldenberg, C. T. Bowman, R. K. Hanson, S. Song, W. C. Gardiner Jr., V. V. Lissianski, and Z. Qin http://www.me.berkeley.edu/gri mech/
[18]	N. M Marinov, W. J. Pitz, C. K. Westbrook, A. M. Vincitore, M. J. Castaldi, S. M. Senkan, Aromatic and Polycyclic Aromatic Hydrocarbon Formation in a Laminar Premixed n-Butane Flame, Combustion and Flame 114 192-213 (1998). https://doi.org/10.1016/S0010-2180(97)00275-7
[19]	Z. Luo., S. Som , S. M. Sarathy, M. Plomer, W. J. Pitz, D. E. Longman, T. F. Lu , Development and Validation of an n-Dodecane Skeletal Mechanism for Diesel Spray- Combustion Applications, Combust. Theory Model., https://doi.org/10.1080/13647830.2013.872807 , 2014
[20]	S. C. Moldoveanu, Pyrolysis of Organic Molecules (Second Edition), Elsevier, 2019, ISBN 9780444640000.
[21]	K. Kim, O. Diaz-Ibarra, H. Najm, J. Zádor, and C. Safta. TChem: A performance portable parallel software toolkit for complex kinetic mechanisms, Computer Physics Communications, Vol 285, 2023, https://doi.org/10.1016/j.cpc.2022.108628 .
[22]	S. Gaudreault, G. Rainwater, M. Tokman, KIOPS: A fast adaptive Krylov subspace solver for exponential integrators, Journal of Computational Physics, Volume 372, 2018, Pages 236-255, ISSN 0021-9991, https://doi.org/10.1016/j.jcp.2018.06.026 .
[23]	N. J. Higham. The Scaling and Squaring Method for the Matrix Exponential Revisited. SIAM J. Matrix Anal. Appl. 26, 4 (2005), 1179-1193. https://doi.org/10.1137/04061101X
[24]	J. Loffeld, and M. Tokman. Implementation of parallel adaptive-Krylov exponential solvers for large scale stiff problems. SIAM Journal on Scientific Computing, Vol 36, Iss 5, 2014. https://doi.org/10.1137/13094462X
[25]	G. Wanner, E. Hairer, S. P. Norsett, Solving Ordinary Differential Equations I: Nonstiff Problems., Germany: Springer Berlin Heidelberg, 2013. https://doi.org/10.1007/978-3-540-78862-1
[26]	D. J. Gardner, D. R. Reynolds, C. S. Woodward, and C. J. Balos. Enabling new flexibility in the SUNDIALS suite of nonlinear and differential/algebraic equation solvers., ACM Transactions on Mathematical Software (TOMS) 48, no. 3 (2022): 1-24. https://doi.org/10.1145/3539801
[27]	J. Niesen, W. M. Wright Algorithm 919: A Krylov subspace algorithm for evaluating the ϕ-functions appearing in exponential integrators., ACM Transactions on Mathematical Software (TOMS), 38(3), 1-19, 2012. https://doi.org/10.1145/2168773.2168781
[28]	A. Kassam, L. N. Trefethen, Fourth-Order Time- Stepping for Stiff PDEs, SIAM Journal on Scientific Computing, 26(4), pages 1214-1233, 2005. https://doi.org/10.1137/S1064827502410633

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APA Style

Stewart, J., Tokman, M., Dallerit, V., Bisetti, F., Diaz-Ibarra, O. (2024). Variable Time-stepping Exponential Integrators for Chemical Reactors with Analytical Jacobians. Applied and Computational Mathematics, 13(2), 29-37. https://doi.org/10.11648/j.acm.20241302.11

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ACS Style

Stewart, J.; Tokman, M.; Dallerit, V.; Bisetti, F.; Diaz-Ibarra, O. Variable Time-stepping Exponential Integrators for Chemical Reactors with Analytical Jacobians. Appl. Comput. Math. 2024, 13(2), 29-37. doi: 10.11648/j.acm.20241302.11

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AMA Style

Stewart J, Tokman M, Dallerit V, Bisetti F, Diaz-Ibarra O. Variable Time-stepping Exponential Integrators for Chemical Reactors with Analytical Jacobians. Appl Comput Math. 2024;13(2):29-37. doi: 10.11648/j.acm.20241302.11

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@article{10.11648/j.acm.20241302.11,
  author = {Jared Stewart and Mayya Tokman and Valentin Dallerit and Fabrizio Bisetti and Oscar Diaz-Ibarra},
  title = {Variable Time-stepping Exponential Integrators for Chemical Reactors with Analytical Jacobians},
  journal = {Applied and Computational Mathematics},
  volume = {13},
  number = {2},
  pages = {29-37},
  doi = {10.11648/j.acm.20241302.11},
  url = {https://doi.org/10.11648/j.acm.20241302.11},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.acm.20241302.11},
  abstract = {Chemical combustion problems are known to be stiff and therefore difficult to efficiently integrate in time when numerically simulated. Implicit methods, such as backwards differentiation formula (BDF), are widely considered to be the state-of-the-art methods owing their capability of taking relatively large time-steps while maintaining accurate combustion characteristics. Exponential time integration methods have recently demonstrated the ability to accurately and efficiently solve large scale systems of ordinary differential equations. This study introduces a novel adaptive time stepping exponential integrator named EPI3V. Its performance is measured on spatially homogeneous isobaric reactive mixtures involving three hydrocarbon fuel mechanisms. The full combustion process is simulated using gas compositions with sufficient temperature to obtain auto-ignition. Simulations are run until the steady state is obtained, then a comparison of the computational efficiency and accuracy between a BDF and EPI3V method is made. The novel EPI3V method exhibits comparable computational efficiency to a well-established implementation of the variable time-stepping BDF implicit methods for two of the mechanisms investigated. In certain situations it even demonstrates a slight advantage over the implicit solver. However, in one specific case, the EPI3V shows relative performance degradation compared to the implicit method, but it still converges for this case. These results indicate that exponential time integration methods may be applicable to a larger variety of combustion problems.},
 year = {2024}
}

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TY  - JOUR
T1  - Variable Time-stepping Exponential Integrators for Chemical Reactors with Analytical Jacobians
AU  - Jared Stewart
AU  - Mayya Tokman
AU  - Valentin Dallerit
AU  - Fabrizio Bisetti
AU  - Oscar Diaz-Ibarra
Y1  - 2024/04/07
PY  - 2024
N1  - https://doi.org/10.11648/j.acm.20241302.11
DO  - 10.11648/j.acm.20241302.11
T2  - Applied and Computational Mathematics
JF  - Applied and Computational Mathematics
JO  - Applied and Computational Mathematics
SP  - 29
EP  - 37
PB  - Science Publishing Group
SN  - 2328-5613
UR  - https://doi.org/10.11648/j.acm.20241302.11
AB  - Chemical combustion problems are known to be stiff and therefore difficult to efficiently integrate in time when numerically simulated. Implicit methods, such as backwards differentiation formula (BDF), are widely considered to be the state-of-the-art methods owing their capability of taking relatively large time-steps while maintaining accurate combustion characteristics. Exponential time integration methods have recently demonstrated the ability to accurately and efficiently solve large scale systems of ordinary differential equations. This study introduces a novel adaptive time stepping exponential integrator named EPI3V. Its performance is measured on spatially homogeneous isobaric reactive mixtures involving three hydrocarbon fuel mechanisms. The full combustion process is simulated using gas compositions with sufficient temperature to obtain auto-ignition. Simulations are run until the steady state is obtained, then a comparison of the computational efficiency and accuracy between a BDF and EPI3V method is made. The novel EPI3V method exhibits comparable computational efficiency to a well-established implementation of the variable time-stepping BDF implicit methods for two of the mechanisms investigated. In certain situations it even demonstrates a slight advantage over the implicit solver. However, in one specific case, the EPI3V shows relative performance degradation compared to the implicit method, but it still converges for this case. These results indicate that exponential time integration methods may be applicable to a larger variety of combustion problems.
VL  - 13
IS  - 2
ER  -

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Author Information

Jared Stewart

Applied Math Department, University of California Merced, Merced, USA

Contact Email
Mayya Tokman

Applied Math Department, University of California Merced, Merced, USA
Valentin Dallerit

Applied Math Department, University of California Merced, Merced, USA
Fabrizio Bisetti

Aerospace Engineering & Engineering Mechanics, The University of Texas at Austin, Austin, USA
Oscar Diaz-Ibarra

Center for Computing Research, Sandia National Laboratories, Albuquerque, USA

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APA Style

Stewart, J., Tokman, M., Dallerit, V., Bisetti, F., Diaz-Ibarra, O. (2024). Variable Time-stepping Exponential Integrators for Chemical Reactors with Analytical Jacobians. Applied and Computational Mathematics, 13(2), 29-37. https://doi.org/10.11648/j.acm.20241302.11

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ACS Style

Stewart, J.; Tokman, M.; Dallerit, V.; Bisetti, F.; Diaz-Ibarra, O. Variable Time-stepping Exponential Integrators for Chemical Reactors with Analytical Jacobians. Appl. Comput. Math. 2024, 13(2), 29-37. doi: 10.11648/j.acm.20241302.11

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AMA Style

Stewart J, Tokman M, Dallerit V, Bisetti F, Diaz-Ibarra O. Variable Time-stepping Exponential Integrators for Chemical Reactors with Analytical Jacobians. Appl Comput Math. 2024;13(2):29-37. doi: 10.11648/j.acm.20241302.11

Copy | Download

@article{10.11648/j.acm.20241302.11,
  author = {Jared Stewart and Mayya Tokman and Valentin Dallerit and Fabrizio Bisetti and Oscar Diaz-Ibarra},
  title = {Variable Time-stepping Exponential Integrators for Chemical Reactors with Analytical Jacobians},
  journal = {Applied and Computational Mathematics},
  volume = {13},
  number = {2},
  pages = {29-37},
  doi = {10.11648/j.acm.20241302.11},
  url = {https://doi.org/10.11648/j.acm.20241302.11},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.acm.20241302.11},
  abstract = {Chemical combustion problems are known to be stiff and therefore difficult to efficiently integrate in time when numerically simulated. Implicit methods, such as backwards differentiation formula (BDF), are widely considered to be the state-of-the-art methods owing their capability of taking relatively large time-steps while maintaining accurate combustion characteristics. Exponential time integration methods have recently demonstrated the ability to accurately and efficiently solve large scale systems of ordinary differential equations. This study introduces a novel adaptive time stepping exponential integrator named EPI3V. Its performance is measured on spatially homogeneous isobaric reactive mixtures involving three hydrocarbon fuel mechanisms. The full combustion process is simulated using gas compositions with sufficient temperature to obtain auto-ignition. Simulations are run until the steady state is obtained, then a comparison of the computational efficiency and accuracy between a BDF and EPI3V method is made. The novel EPI3V method exhibits comparable computational efficiency to a well-established implementation of the variable time-stepping BDF implicit methods for two of the mechanisms investigated. In certain situations it even demonstrates a slight advantage over the implicit solver. However, in one specific case, the EPI3V shows relative performance degradation compared to the implicit method, but it still converges for this case. These results indicate that exponential time integration methods may be applicable to a larger variety of combustion problems.},
 year = {2024}
}

Copy | Download

TY  - JOUR
T1  - Variable Time-stepping Exponential Integrators for Chemical Reactors with Analytical Jacobians
AU  - Jared Stewart
AU  - Mayya Tokman
AU  - Valentin Dallerit
AU  - Fabrizio Bisetti
AU  - Oscar Diaz-Ibarra
Y1  - 2024/04/07
PY  - 2024
N1  - https://doi.org/10.11648/j.acm.20241302.11
DO  - 10.11648/j.acm.20241302.11
T2  - Applied and Computational Mathematics
JF  - Applied and Computational Mathematics
JO  - Applied and Computational Mathematics
SP  - 29
EP  - 37
PB  - Science Publishing Group
SN  - 2328-5613
UR  - https://doi.org/10.11648/j.acm.20241302.11
AB  - Chemical combustion problems are known to be stiff and therefore difficult to efficiently integrate in time when numerically simulated. Implicit methods, such as backwards differentiation formula (BDF), are widely considered to be the state-of-the-art methods owing their capability of taking relatively large time-steps while maintaining accurate combustion characteristics. Exponential time integration methods have recently demonstrated the ability to accurately and efficiently solve large scale systems of ordinary differential equations. This study introduces a novel adaptive time stepping exponential integrator named EPI3V. Its performance is measured on spatially homogeneous isobaric reactive mixtures involving three hydrocarbon fuel mechanisms. The full combustion process is simulated using gas compositions with sufficient temperature to obtain auto-ignition. Simulations are run until the steady state is obtained, then a comparison of the computational efficiency and accuracy between a BDF and EPI3V method is made. The novel EPI3V method exhibits comparable computational efficiency to a well-established implementation of the variable time-stepping BDF implicit methods for two of the mechanisms investigated. In certain situations it even demonstrates a slight advantage over the implicit solver. However, in one specific case, the EPI3V shows relative performance degradation compared to the implicit method, but it still converges for this case. These results indicate that exponential time integration methods may be applicable to a larger variety of combustion problems.
VL  - 13
IS  - 2
ER  -

Copy | Download