Exponential Adams Bashforth integrators for stiff ODEs, application to cardiac electrophysiology

Yves Coudière (1) , Charlie Douanla Lontsi (1) , Charles Pierre (2)

1 CARMEN - Modélisation et calculs pour l'électrophysiologie cardiaque
2 LMAP - Laboratoire de Mathématiques et de leurs Applications [Pau]

Models in cardiac electrophysiology are coupled systems of reaction diffusion PDE and of ODE. The ODE system displays a very stiff behaviour. It is non linear and its upgrade at each time step is a preponderant load in the total amount of computational cost. The issue is to develop high order explicit and stable methods to cope with that situation. In this article is analyzed the resort to exponential Adams Bashforth (\EAB) integrators in cardiac electrophysiology. The method is presented in the framework of a general and varying stabilizer, that is well suited in this context. Stability under perturbation (or 0-stability) is proven. This provides a new approach for the convergence analysis of the method. The Dahlquist stability properties of the method is performed. It is presented in a new framework that incorporates the discrepancy between the stabilizer and the system Jacobian matrix. Provided this discrepancy is small enough, the method is shown to be $A(\alpha)$-stable. This result is interesting for an explicit time stepping method. Numerical experiments are presented for two class of stiff models in electrophysiology, including performances comparisons with several classical methods. The \EAB method is observed to be as stable as implicit solvers and cheaper at equal accuracy level.

Mots clés: Dahlquist stability .; stability and convergence; stiff equations; explicit high-order multistep methods; exponential integrators of Adams type

Domaines: Analyse numérique [math.NA]

Yves Coudière, Charlie Douanla Lontsi, Charles Pierre. Exponential Adams Bashforth integrators for stiff ODEs, application to cardiac electrophysiology. 2017. ⟨hal-01394036v2⟩