A Fully Implicit Parallel Algorithm for Simulating the Nonlinear Electrical Activation of the Heart

Maria Murillo

Xiao-Chuan Cai

Campus Box 430

Boulder, CO 80309

Abstract

In this research we developed and tested a fully implicit and highly parallel Newton-Krylov-Schwarz method for solving the bidomain equations representing the excitation process of the heart. Newton-Krylov-Schwarz method has been used successfully for many nonlinear problems, but this is the first attempt to use this method for the bidomain system which consists of time dependent partial differential equations of mixed types. Our experiments on parallel computers show that the method is scalable and robust with respect to many of the parameters in the bidomain system. In the outer layer of the algorithm, we use a nonlinearly implicit backward Euler method to discretize the time derivative, and the resulting systems of large sparse nonlinear equations are solved using an inexact Newton method. The Jacobian system required solving in each Newton iteration is solved with a preconditioned GMRES method. The efficiency and robustness of the overall method depends heavily on the preconditioning step of the linear solver. By comparing several preconditioners, we found the restricted

additive Schwarz method offers the best performance. Our parallel software is developed using the PETSc package of Argonne National Laboratory, and our numerical results were obtained on the IBM-SP of the San Diego Supercomputer Center.