Analysis of time-stepping methods for the monodomain model
- PDF / 1,585,593 Bytes
- 32 Pages / 439.37 x 666.142 pts Page_size
- 18 Downloads / 146 Views
Analysis of time-stepping methods for the monodomain model Thomas Roy1 · Yves Bourgault2 · Charles Pierre3 Received: 8 November 2017 / Revised: 17 May 2020 / Accepted: 7 July 2020 © SBMAC - Sociedade Brasileira de Matemática Aplicada e Computacional 2020
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. Keywords Cardiac electrophysiology · Monodomain model · Stiff problems · Time-stepping methods · Absolute stability
Communicated by Jose Alberto Cuminato.
B
Yves Bourgault [email protected] Thomas Roy [email protected] Charles Pierre [email protected]
1
Mathematical Institute, University of Oxford, Oxford, UK
2
Department of Mathematics and Statistics, University of Ottawa, Ottawa, ON, Canada
3
Laboratoires de Mathématiques et de leurs Applications de Pau, UMR CNRS 5142, Université de Pau et des Pays de l’Adour, Pau, France 0123456789().: V,-vol
123
230
Page 2 of 32
T. Roy et al.
Mathematics Subject Classification 65M12 · 65L04 · 65L06 · 35K57 · 35Q92
1 Introduction The modelling of the electrical activity of the heart offers an interesting perspective on the understanding of cardiac pathologies and its treatments. This subject has great potential in biomedical sciences, as experiments on living hearts require considerable resources and provide only a partial picture of the electrical activity of the heart. For instance, realistically simulating the behaviour of the heart reduces the necessity for these kinds of experiments. Considering that heart diseases are a leading cause of death in Western countries (Canada 2009), modelling in cardiac electrophysiology has the potenti
Data Loading...
