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Analysis of time-stepping methods for the monodomain model

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Abstract

To a large extent, the stiffness of the bidomain and monodomain models depends on the choice of the ionic model, which varies in terms of complexity and realism. In this paper, we compare and analyze a variety of time-stepping methods: explicit or semi-implicit, operator splitting, exponential, and deferred correction methods. We compare these methods for solving the monodomain model coupled with three ionic models of varying complexity and stiffness: the phenomenological Mitchell–Schaeffer model, the more realistic Beeler–Reuter model, and the stiff and very complex ten Tuscher–Noble–Noble-Panfilov (TNNP) model. For each method, we derive absolute stability criteria of the spatially discretized monodomain model and verify that the theoretical critical time steps obtained closely match the ones in numerical experiments. We also verify that the numerical methods achieve an optimal order of convergence on the model variables and derived quantities (such as speed of the wave, depolarization time), and this in spite of the local non-differentiability of some of the ionic models. The efficiency of the different methods is also considered by comparing computational times for similar accuracy. Conclusions are drawn on the methods to be used to solve the monodomain model based on the model stiffness and complexity, measured, respectively, by the eigenvalues of the model’s Jacobian and the number of variables, and based on strict stability and accuracy criteria.

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Acknowledgements

The authors would like to thank the University of Ottawa for graduate scholarships to the first author, and the Natural Science and Engineering Research Council (NSERC) of Canada for a research grant to the second author. Research exchanges between France and Canada were funded by a grant from the Agence Nationale de la Recherche of France (ANR project HR-CEM no. 13-MONU-0004-01).

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Authors and Affiliations

  1. Mathematical Institute, University of Oxford, Oxford, UK

    Thomas Roy

  2. Department of Mathematics and Statistics, University of Ottawa, Ottawa, ON, Canada

    Yves Bourgault

  3. Laboratoires de Mathématiques et de leurs Applications de Pau, UMR CNRS 5142, Université de Pau et des Pays de l’Adour, Pau, France

    Charles Pierre

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  1. Thomas Roy
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  2. Yves Bourgault
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  3. Charles Pierre
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Correspondence to Yves Bourgault.

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Communicated by Jose Alberto Cuminato.

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Roy, T., Bourgault, Y. & Pierre, C. Analysis of time-stepping methods for the monodomain model. Comp. Appl. Math. 39, 230 (2020). https://doi.org/10.1007/s40314-020-01254-z

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  • DOI: https://doi.org/10.1007/s40314-020-01254-z

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