Efficient fully implicit time integration methods for modeling cardiac dynamics.

Area of Science:

• Computational biology
• Mathematical modeling
• Cardiac electrophysiology

Background:

• Implicit methods offer superior stability for stiff systems but are often avoided due to computational complexity.
• Modeling cardiac electrical activity involves stiff systems, necessitating stable and efficient numerical methods.

Purpose of the Study:

• To apply and evaluate the backward Euler (BE) method and a second-order composite backward differentiation formula (C-BDF2) for the monodomain equations.
• To address the computational complexity associated with implicit methods in cardiac modeling.

Main Methods:

• Application of the L-stable C-BDF2 implicit time integration scheme, built upon the forward Euler and BE methods.
• Solution of the nonlinear system from the BE method using a Jacobian-free Newton-Krylov solver with a nonlinear elimination technique.
• Reduction of the global system Jacobian size, enhancing symmetry and positive definiteness for efficient solving.

Main Results:

• The C-BDF2 scheme provides accurate results for cardiac electrical activity modeling.
• The implemented nonlinear elimination method significantly reduces the computational burden of implicit solvers.
• The proposed methods achieve accurate results with reduced CPU times compared to explicit methods.

Conclusions:

• The C-BDF2 scheme is a stable, accurate, and easily implementable implicit method for cardiac electrophysiology modeling.
• The nonlinear elimination approach effectively tackles the computational challenges of implicit methods for the monodomain equations.
• This work demonstrates the practical viability of implicit methods for efficient and accurate cardiac modeling.