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
