One of the most popular model describing the electrical propagation in the heart tissue is the monodomain model. It is a parabolic PDE, coupled with systems of ODEs called ionic models. The monodomain model belongs to the class of stiff reaction diffusion equations, where the PDE and ODE both are responsible of the stiffness. Because of the nonlinearity (in particular of the ODE) and the stiffness of these equations, their numerical resolution is very challenging.
The PDE displays a diffusion term that is usually solved implicitly by a scheme of order 1 or 2. The reaction term coupled with the ODE system is strongly nonlinear and is usually solved explicitly with a scheme of order 1 or 2.
In this talk, we will introduce the Rush Larsen methods of higher order (order 3 and 4). These are exponential multistep methods and will make possible to integrate efficiently the monodomain model with higher order of accuracy. This will be done by solving the PDE by the well known SBDF method of order k, and the system of ODEs by the Rush Larsen method of order k. A motivation of using high order methods for the numerical integration of the monodomain model will be investigated on some spiral waves.