Recently, there has been a growing interest in the extracellular-membrane-intracellular (EMI) cardiac model, which accurately represents the discrete cellular structure of the myocardium. This model consists of piecewise electrostatic equations coupled through a nonstandard time-dependent transmission condition on the membrane. Given the presence of the jumps and fluxes on the interfaces, it seems that finite volume methods are of real interest. In this work, we study the convergence of the two-point flux approximation (TPFA) finite volume method, as defined in [1], with Backward Differentiation Formula and Rush-Larsen [2] methods for time integration. The convergence analysis using the Implicit-Explicit time-stepping approaches, was carried out on the electrical potential in the intra- and extra-cellular subdomains and the transmembrane voltage on the membranes, using a discrete H^1 -like semi norm, L^2 and L^∞ norms in space, along with L^2 and L^∞ norms in time. Errors were computed with respect to manufactured and reference solutions. The main challenge encountered lies in constructing admissible meshes. The extension of the TPFA to higher dimensions proves inherently challenging. Consequently, exploring more flexible schemes, such as the diamond scheme adapted to more general meshes, emerges as a genuine interest. This work was part of the MICROCARD project. This project has received funding from the European High-Performance Computing Joint Undertaking EuroHPC (JU) under the grant agreement No 955495. The JU receives support from the European Union’s Horizon 2020 research and innovation program and France, Italy, Germany, Austria, Norway, Switzerland.