Stability of Two-Integrators for the Aliev-Panfilov System

Abstract

We propose a second-order accurate method for computing the solutions to the Aliev-Panfilov model of cardiac excitation. This two-variable reaction-diffusion system is due to its simplicity a popular choice for modeling important problems in electrocardiology; e.g. cardiac arrhythmias. The solutions might be very complicated in structure, and hence highly resolved numerical simulations are called for to capture the fine details. Usually the forward Euler time-integrator is applied in these computations; it is very simple to implement and can be effective for coarse grids. For fine-scale simulations, however, the forward Euler method suffers from a severe time-step restriction, rendering it less efficient for simulations where high resolution and accuracy are important.
We analyze the stability of the proposed second-order method and the forward Euler scheme when applied to the Aliev-Panfilov model. Compared to the Euler method the suggested scheme has a much weaker time-step restriction, and promises to be more efficient for computations on finer meshes.