Efficient high order schemes for stiff ODEs in cardiac electrophysiology

TL;DR

This paper evaluates high order exponential solvers for stiff ODEs in cardiac electrophysiology, demonstrating their efficiency and accuracy at large time steps compared to classical methods.

Contribution

It introduces and benchmarks high order exponential Adams-Bashforth and Rush-Larsen schemes for cardiac models, showing their advantages over traditional explicit and implicit methods.

Findings

Exponential solvers enable large time steps despite stiffness.

Significant cost savings for a given accuracy with exponential methods.

High order exponential schemes perform well in cardiac electrophysiology simulations.

Abstract

In this work we analyze the resort to high order exponential solvers for stiff ODEs in the context of cardiac electrophysiology modeling. The exponential Adams-Bashforth and the Rush-Larsen schemes will be considered up to order 4. These methods are explicit multistep schemes.The accuracy and the cost of these methods are numerically analyzed in this paper and benchmarked with several classical explicit and implicit schemes at various orders. This analysis has been led considering data of high particular interest in cardiac electrophysiology : the activation time ($t\_a$ ), the recovery time ($t\_r $) and the action potential duration ($APD$). The Beeler Reuter ionic model, especially designed for cardiac ventricular cells, has been used for this study. It is shown that, in spite of the stiffness of the considered model, exponential solvers allow computation at large time steps, as large as for implicit methods. Moreover, in terms of cost for a given accuracy, a significant gain is achieved with exponential solvers. We conclude that accurate computations at large time step are possible with explicit high order methods. This is a quite important feature when considering stiff non linear ODEs.