A positive cell vertex Godunov scheme for a Beeler–Reuter based model of cardiac electrical activity. Issue 1 (26th August 2020)

• Record Type: Journal Article
• Authors: Bendahmane, Mostafa; Mroue, Fatima; Saad, Mazen
• Is Part Of: Numerical methods for partial differential equations. Volume 37:Issue 1(2021)
• Page Start: 262
• Page End: 301
• Publication Date: 2020-08-26
• Subjects: convergence -- finite element -- finite volume -- Godunov scheme -- maximum principle -- monodomain model; Differential equations, Partial -- Numerical solutions -- Periodicals; 515.353
• DOI: 10.1002/num.22528
• Languages: English
• ISSNs: 0749-159X

Abstract

Abstract: The monodomain model is a widely used model in electrocardiology to simulate the propagation of electrical potential in the myocardium. In this paper, we investigate a positive nonlinear control volume finite element scheme, based on Godunov's flux approximation of the diffusion term, for the monodomain model coupled to a physiological ionic model (the Beeler–Reuter model) and using an anisotropic diffusion tensor. In this scheme, degrees of freedom are assigned to vertices of a primal triangular mesh, as in conforming finite element methods. The diffusion term which involves an anisotropic tensor is discretized on a dual mesh using the diffusion fluxes provided by the conforming finite element reconstruction on the primal mesh and the other terms are discretized by means of an upwind finite volume method on the dual mesh. The scheme ensures the validity of the discrete maximum principle without any restriction on the transmissibility coefficients. By using a compactness argument, we obtain the convergence of the discrete solution and as a consequence, we get the existence of a weak solution of the original model. Finally, we illustrate by numerical simulations that the proposed scheme successfully removes nonphysical oscillations in the propagation of the wavefront and maintains conduction velocity close to physiological values.