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

The stability and accuracy of numerical methods for reaction-diffusion equations still need improvements, which prompts the development of high-order and stable time-stepping methods. This is particularly true in the context of cardiac electrophysiology, where reaction-diffusion equations are coupled with stiff systems of ordinary differential equations. So as to address these issues, much research on implicit-explicit methods and exponential integrators has been carried out during the past 15 years. In 2009, Perego and Veneziani [Electron. Trans. Numer. Anal., 35 (2009), pp. 234–256] proposed an innovative time-stepping scheme of order 2. In this paper we present an extension of this scheme to the orders 3 and 4, which we call Rush-Larsen schemes of order k. These new schemes are explicit multistep methods, which belong to the classical class of exponential integrators. Their general formulation is simple and easy to implement. We prove that they are stable under perturbation and convergent of order k. We analyze their Dahlquist stability and show that they have a very large stability domain provided that the stabilizer associated with the method captures well enough the stiff modes of the problem. We study their application to a system of equations that models the action potential in cardiac electrophysiology.