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High-Order Operator-Splitting Methods for the Bidomain and Monodomain Models

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Mathematical and Numerical Modeling of the Cardiovascular System and Applications

Part of the book series: SEMA SIMAI Springer Series ((SEMA SIMAI,volume 16))

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Abstract

The bidomain and monodomain models are among the most widely used mathematical models to describe cardiac electrophysiology. They take the form of multi-scale reaction-diffusion partial differential equations that couple the dynamic behaviour on the cellular scale with that on the tissue scale. The systems of differential equations associated with these models are large and strongly non-linear, but they also have a distinct structure due to their multi-scale nature. For these reasons, numerical solutions to these systems are often found via operator-splitting methods. In this chapter, we provide a survey of operator-splitting methods for the numerical solution of differential equations. In particular, we focus on splitting methods with order higher than two that, according to the Sheng–Suzuki theorem, require backward time integration and historically have been considered unstable for solving deterministic parabolic systems. We demonstrate the stability of operator-splitting methods of up to order four to solve the bidomain and monodomain models on several examples arising in the field of cardiovascular modeling.

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Author information

Authors and Affiliations

  1. Department of Mathematics and Statistics, University of Saskatchewan, SK, Canada

    Jessica Cervi

  2. Department of Computer Science, University of Saskatchewan, SK, Canada

    Raymond J. Spiteri

Authors
  1. Jessica Cervi
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  2. Raymond J. Spiteri
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Correspondence to Jessica Cervi .

Editor information

Editors and Affiliations

  1. Department of Mathematics “F. Casorati”, University of Pavia, Pavia, Italy

    Daniele Boffi

  2. Department of Mathematics “F. Casorati”, University of Pavia, Pavia, Italy

    Luca F. Pavarino

  3. SISSA mathLab, Mathematics Area, International School for Advanced Studies, Trieste, Italy

    Gianluigi Rozza

  4. Department of Mathematics, University of Milan, Milano, Italy

    Simone Scacchi

  5. Department of Mathematics, Politecnico di Milano, Milano, Italy

    Christian Vergara

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Cervi, J., Spiteri, R.J. (2018). High-Order Operator-Splitting Methods for the Bidomain and Monodomain Models. In: Boffi, D., Pavarino, L., Rozza, G., Scacchi, S., Vergara, C. (eds) Mathematical and Numerical Modeling of the Cardiovascular System and Applications. SEMA SIMAI Springer Series, vol 16. Springer, Cham. https://doi.org/10.1007/978-3-319-96649-6_2

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