A fully implicit parallel algorithm for simulating the non‐linear electrical activity of the heart

Abstract In this paper, we study a fully implicit parallel Newton–Krylov–Schwarz method (NKS) for solving the bidomain equations describing the electrical excitation process of the heart. NKS has been used successfully for many non‐linear problems, but this is the first attempt to use this method for the bidomain model which consists of a system of time dependent partial differential equations of mixed type. Our experiments on parallel computers show that the method is scalable and robust with respect to many of the parameters in the bidomain model. In the outer layer of the algorithm, we use a non‐linearly implicit backward Euler method to discretize the time derivative, and the resulting systems of large sparse non‐linear equations are solved using an inexact Newton method. The Jacobian system required to solve in each Newton iteration is solved with a GMRES method preconditioned by a new component‐wise restricted additive Schwarz preconditioner. The efficiency and robustness of the overall method depend heavily on what preconditioner we use. By comparing several preconditioners, we found our new restricted additive Schwarz method offers the best performance. Our parallel software is developed using the PETSc package of Argonne National Laboratory. Numerical results obtained on an IBM SP will be reported. Copyright © 2004 John Wiley & Sons, Ltd.