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

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
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[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.
[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,
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[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.
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Cite This Article
  • 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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    AU  - Jared Stewart
    AU  - Mayya Tokman
    AU  - Valentin Dallerit
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    DO  - 10.11648/j.acm.20241302.11
    T2  - Applied and Computational Mathematics
    JF  - Applied and Computational Mathematics
    JO  - Applied and Computational Mathematics
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    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
  • Applied Math Department, University of California Merced, Merced, USA

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

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

  • Aerospace Engineering & Engineering Mechanics, The University of Texas at Austin, Austin, USA

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

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