Rush-Larsen time-stepping methods of high order for stiff problems in cardiac electrophysiology

TL;DR

This paper introduces high-order explicit multistep exponential integrators called Rush-Larsen schemes for stiff reaction-diffusion equations in cardiac electrophysiology, demonstrating their stability, convergence, and large stability domains.

Contribution

The paper extends the Rush-Larsen method to orders 3 and 4, providing a simple, stable, and convergent high-order explicit integrator for stiff cardiac models.

Findings

RL_k schemes are stable under perturbation.

RL_k schemes are convergent of order k.

They have large stability domains.

Abstract

To address the issues of stability and accuracy for reaction-diffusion equations, the development of high order and stable time-stepping methods is necessary. This is particularly true in the context of cardiac electrophysiology, where reaction-diffusion equations are coupled with stiff ODE systems. Many research have been led in that way in the past 15 years concerning implicit-explicit methods and exponential integrators. In 2009, Perego and Veneziani proposed an innovative time-stepping method of order 2. In this paper we present the extension of this method to the orders 3 and 4 and introduce the Rush-Larsen schemes of order k (shortly denoted RL\_k). The RL\_k schemes are explicit multistep exponential integrators. They display a simple general formulation and an easy implementation. The RL\_k schemes are shown to be stable under perturbation and convergent of order k. Their Dahlquist stability analysis is performed. They have a very large stability domain provided that the stabilizer associated with the method captures well enough the stiff modes of the problem. The RL\_k method is numerically studied as applied to the membrane equation in cardiac electrophysiology. The RL k schemes are shown to be stable under perturbation and convergent oforder k. Their Dahlquist stability analysis is performed. They have a very large stability domain provided that the stabilizer associated with the method captures well enough the stiff modes of the problem. The RL k method is numerically studied as applied to the membrane equation in cardiac electrophysiology.