A new 3D finite volume formulation and discretization is presented to simulate the electrocardiograms. It allows for the full coupling relations between the heart and the torso to be computed. Additionally, realistic ionic models are used, with very stiff dynamics. Hence, thousands of iterations are needed, each of which requires to solve a large and sparse linear system. The system is proved to be well-posed. Numerical results are presented, that rely on the choice of a good preconditioning technique.