Convergence Analysis of a Finite Volume Scheme for Periodic Numerical Solutions of the Monodomain Model in Cardiac Electrophysiology

Résumé fourni par la source

Cardiac electrophysiology is a scientific field that studies the propagation of electrical signals through myocardial tissue. From a mathematical standpoint, this electrical activity can be represented by the monodomain model, a simplification of the well-known bidomain model. It consists of a highly nonlinear parabolic partial differential equation (PDE), coupled with an ordinary differential equations (ODE). Understanding the periodic behavior of the solution is essential for analyzing cardiac pathologies and to better capture the underlying electrophysiological processes. Recent works have provided existence and uniqueness results of time-periodic solutions at the continuous level. In this work, we consider the monodomain model and construct a numerical scheme preserving the periodicity of the solution. We employ the cell-centered finite volume method for the space discretization and the implicit Euler scheme for the time discretization. We prove that our periodic numerical solution converges to the weak periodic solution of the continuous problem. We emphasize here on the fact that the periodicity is ensured by the appropriate choice of an initial condition, solution to some nonlinear problem. This constitutes the main originality of our work. Finally, we provide numerical experiments showing the strength of the proposed approach.

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