Efficient Fully Implicit Time Integration Methods for Modeling Cardiac Dynamics

Implicit methods are well known to have greater stability than explicit methods for stiff systems, but they often are not used in practice due to perceived computational complexity. This paper applies the backward Euler (BE) method and a second-order one-step two-stage composite backward differentiation formula (C-BDF2) for the monodomain equations arising from mathematically modeling the electrical activity of the heart. The C-BDF2 scheme is an L-stable implicit time integration method and easily implementable. It uses the simplest forward Euler and BE methods as fundamental building blocks. The nonlinear system resulting from application of the BE method for the monodomain equations is solved for the first time by a nonlinear elimination method, which eliminates local and nonsymmetric components by using a Jacobian-free Newton solver, called Newton--Krylov solver. Unlike other fully implicit methods proposed for the monodomain equations in the literature, the Jacobian of the global system after the nonlinear elimination has much smaller size, is symmetric and possibly positive definite, which can be solved efficiently by standard optimal solvers. Numerical results are presented demonstrating that the C-BDF2 scheme can yield accurate results with less CPU times than explicit methods for both a single patch and spatially extended domains.