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

The development of efficient solvers in cardiac electrophysiology requires high order (semi) explicit and stable time stepping methods. In this paper are introduced two new exponential integrators of orders 3 and 4. They generalize the order 2 Rush Larsen scheme derived by Perego and Veneziani [24] in 2009. They have been named Rush Larsen of order k, shortly RL k. The RL k schemes are explicit exponential multistep integrators. They display a simple general formulation and an easy implementation. The RL k schemes are shown to be stable under perturbation (or 0-stable) and con-vergent 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.