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To address the issues of stability and accuracy for reaction-diffusionequations, the development of high order and stable time-stepping methods isnecessary. This is particularly true in the context of cardiacelectrophysiology, where reaction-diffusion equations are coupled with stiffODE systems. Many research have been led in that way in the past 15 yearsconcerning implicit-explicit methods and exponential integrators. In 2009,Perego and Veneziani proposed an innovative time-stepping method of order 2. Inthis paper we present the extension of this method to the orders 3 and 4 andintroduce the Rush-Larsen schemes of order k (shortly denoted RL\_k). The RL\_kschemes are explicit multistep exponential integrators. They display a simplegeneral formulation and an easy implementation. The RL\_k schemes are shown tobe stable under perturbation and convergent of order k. Their Dahlquiststability analysis is performed. They have a very large stability domainprovided that the stabilizer associated with the method captures well enoughthe stiff modes of the problem. The RL\_k method is numerically studied asapplied to the membrane equation in cardiac electrophysiology. The RL k schemesare shown to be stable under perturbation and convergent oforder k. TheirDahlquist stability analysis is performed. They have a very large stabilitydomain provided that the stabilizer associated with the method captures wellenough the stiff modes of the problem. The RL k method is numerically studiedas applied to the membrane equation in cardiac electrophysiology.

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