Semi-Implicit Time-Discretization Schemes for the Bidomain Model

The bidomain model is a system of partial differential equations used to model the propagation of electrical potential waves in the myocardium. It is composed of coupled parabolic and elliptic partial differential equations, as well as at least one ordinary differential equation to model the ion activity through the cardiac cell membranes. The purpose of this paper is to propose and analyze several implicit, semi-implicit, and explicit time-stepping methods to solve that model, in particular to avoid the expensive resolution of a nonlinear system through the Newton-Raphson method. We identify necessary stability conditions on the time step $\Delta t$ for the proposed methods through a theoretical analysis based on energy estimates. We next compare the methods for one- and two-dimensional test cases, in terms of both stability and accuracy of the numerical solutions. The theoretical stability conditions are seen to be consistent with those observed in practice. Our analysis allows us to recommend using either the Crank-Nicolson/Adams-Bashforth method or the second order semi-implicit backward differention method. These semi-implicit methods produce a good numerical solution; unlike the explicit methods, their stability does not depend on the spatial grid size; and unlike the implicit methods, they do not require the resolution of a system of nonlinear equations.