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

TL;DR

This paper investigates exponential Adams Bashforth integrators for stiff ODEs in cardiac electrophysiology, demonstrating their stability and efficiency comparable to implicit methods through theoretical analysis and numerical experiments.

Contribution

It introduces a new stability analysis framework for EAB integrators with a variable stabilizer, showing their A(alpha)-stability under certain conditions.

Findings

EAB methods are as stable as implicit solvers.

EAB methods are computationally cheaper at similar accuracy.

Numerical experiments confirm the effectiveness of EAB in cardiac models.

Abstract

Models in cardiac electrophysiology are coupled systems of reaction diffusion PDE and of ODE. The ODE system displays a very stiff behavior. It is non linear and its upgrade at each time step is a preponderant load in the computational cost. The issue is to develop high order explicit and stable methods to cope with this situation.In this article, is 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. It 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 classes of stiff models in cardiac electrophysiology. They include performances comparisons with several classical methods. The EAB method is observed to be as stable as implicit solvers and cheaper at equal level of accuracy.