High-order discontinuous Galerkin methods for the monodomain and bidomain models

Federica Botta; Matteo Calafà; Pasquale C. Africa; Christian Vergara; Paola F. Antonietti; Federica Botta; Matteo Calafà; Pasquale C. Africa; Christian Vergara; Paola F. Antonietti

doi:10.3934/mine.2024028

Mathematics in Engineering

2024, Volume 6, Issue 6: 726-741. doi: 10.3934/mine.2024028

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Research article

High-order discontinuous Galerkin methods for the monodomain and bidomain models

1.
Independent researcher; federica.botta@mail.polimi.it
2.
Department of Mechanical and Production Engineering, Aarhus University, Katrinebjergvej 89F, Aarhus 8200, Denmark
3.
mathLab, SISSA, via Bonomea, 265, Trieste 34136, Italy
4.
LABS, Dipartimento di Chimica, Materiali e Ingegneria Chimica, Politecnico di Milano, Piazza Leonardo da Vinci 32, Milano 20133, Italy
5.
MOX, Dipartimento di Matematica, Politecnico di Milano, Piazza Leonardo da Vinci 32, Milano 20133, Italy
^† These authors equally contributed to this work.

Received: 01 June 2024 Accepted: 24 December 2024 Published: 31 December 2024

This work aims at presenting a discontinuous Galerkin (DG) formulation employing a spectral basis for two important models employed in cardiac electrophysiology, namely the monodomain and bidomain models. The use of DG methods is motivated by the characteristic of the mathematical solution of such equations which often corresponds to a highly steep wavefront. Hence, the built-in flexibility of discontinuous methods in developing adaptive approaches, combined with the high-order accuracy, can well represent the underlying physics. The choice of a semi-implicit time integration allows for a fast solution at each time step. The article includes some numerical tests to verify the convergence properties and the physiological behaviour of the numerical solution. Also, a pseudo-realistic simulation turns out to fully reconstruct the propagation of the electric potential, comprising the phases of depolarization and repolarization, by overcoming the typical issues related to the steepness of the wave front.
- discontinuous Galerkin methods,
- electrophysiology,
- spectral methods,
- cardiac modelling,
- reaction-diffusion equations
Citation: Federica Botta, Matteo Calafà, Pasquale C. Africa, Christian Vergara, Paola F. Antonietti. High-order discontinuous Galerkin methods for the monodomain and bidomain models[J]. Mathematics in Engineering, 2024, 6(6): 726-741. doi: 10.3934/mine.2024028

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Abstract

This work aims at presenting a discontinuous Galerkin (DG) formulation employing a spectral basis for two important models employed in cardiac electrophysiology, namely the monodomain and bidomain models. The use of DG methods is motivated by the characteristic of the mathematical solution of such equations which often corresponds to a highly steep wavefront. Hence, the built-in flexibility of discontinuous methods in developing adaptive approaches, combined with the high-order accuracy, can well represent the underlying physics. The choice of a semi-implicit time integration allows for a fast solution at each time step. The article includes some numerical tests to verify the convergence properties and the physiological behaviour of the numerical solution. Also, a pseudo-realistic simulation turns out to fully reconstruct the propagation of the electric potential, comprising the phases of depolarization and repolarization, by overcoming the typical issues related to the steepness of the wave front.

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