Schémas de discrétisation en temps pour le modèle bidomaine

Abstract

The bidomain model is a system of partial differential equations frequently used to model the propagation of electrical potential waves in the heart muscle. It contains a coupled parabolic and elliptic equation, as well as at least one ordinary differential equation to model the ion activity of the myocardium. The nature of this system makes it especially hard to solve. We will consider here several implicit, semi-implicit and explicit time-stepping methods applied to the solution of this model. We will show that they remain stable under certain conditions for the time step Deltat, and we will compare the numerical solutions produced by these methods for two test cases: one in 1D, the propagation of a potential wave across the domain, and another in 2D, the formation of a spiral wave. We recommend using either the Crank-Nicolson-Adams-Brashforth method or the second order SBDF method. These semi-implicit methods produce a good numerical solution, unlike, the explicit methods, their stability does not depend on how fine the spatial mesh is, and unlike the implicit methods, they do not require the resolution of a system of nonlinear equations.