Fourth-order compact schemes with adaptive time step for monodomain reaction–diffusion equations

Abstract

Multigrid applied to fourth-order compact schemes for monodomain reaction-diffusion equations in two dimensions has been developed. The scheme accounts for the anisotropy of the medium, allows for any cellular activation model to be used, and incorporates an adaptive time step algorithm. Numerical simulations show up to a 40% reduction in computational time for complex cellular models as compared to second-order schemes for the same solution error. These results point to high-order schemes as valid alternatives for the efficient solution of the cardiac electrophysiology problem when complex cellular activation models are used.

Mathematical Cardiac Electrophysiology

“…Finite difference methods have been studied in [66,223,368,399,413,449,529,568,569,591]. Finite element methods have been widely used, see e.g.…”

Compact schemes for anisotropic reaction–diffusion equations with adaptive time step

“…The order of the method has been demonstrated on an analytical solution with Neumann boundary conditions. The results also demonstrate the ability of the method in handling fully anisotropic electrophysiology problems (previous high-order schemes have been proposed for isotropic or orthotropic media with constant coefficients [21,23]). Anisotropic square samples of normal and ischemic cardiac tissue have been simulated by means of the monodomain model with the reactive term defined by the LuoRudy II dynamics.…”

“…In this regard, high-order integration methods based on compact finite difference schemes [21][22][23] offer a valid alternative for solving this problem. These schemes are in general operator specific.…”

“…Figure 1 shows the action potential for the LR2 model and the associated total ionic current, I ion . The solution of problem (5) with boundary conditions (6) is usually performed by means of finite differences or finite elements for the spatial discretization and an explicit or semi-implicit method for the temporal part [13,14,18,23]. By semi-implicit it is understood, implicit for the diffusive term and explicit for the reactive term.…”

Boosting the accuracy of finite difference schemes via optimal time step selection and non-iterative defect correction

“…High-order numerical methods for partial differential equations (PDEs) will always be valuable for increasing the computational efficiency of numerical simulations. Thus, it is not at all surprising that a great deal of effort in numerical PDEs continues to be focused on the development of high-order numerical schemes [ 1,2,3,4,5,6,7]. Typically, high-order accuracy is achieved by constructing schemes that are formally high-order accurate.…”