A comparison of coupled and uncoupled solvers for the cardiac Bidomain model^∗,∗∗

¹ Dipartimento di Matematica, Università di Pavia, Via Ferrata, 27100 Pavia, Italy.
colli@imati.cnr.it
² Dipartimento di Matematica, Università di Milano, Via Saldini 50, 20133 Milano, Italy.
luca.pavarino@unimi.it; simone.scacchi@unimi.it

Received: 20 December 2011
Revised: 13 May 2012

Abstract

The aim of this work is to compare a new uncoupled solver for the cardiac Bidomain model with a usual coupled solver. The Bidomain model describes the bioelectric activity of the cardiac tissue and consists of a system of a non-linear parabolic reaction-diffusion partial differential equation (PDE) and an elliptic linear PDE. This system models at macroscopic level the evolution of the transmembrane and extracellular electric potentials of the anisotropic cardiac tissue. The evolution equation is coupled through the non-linear reaction term with a stiff system of ordinary differential equations (ODEs), the so-called membrane model, describing the ionic currents through the cellular membrane. A novel uncoupled solver for the Bidomain system is here introduced, based on solving twice the parabolic PDE and once the elliptic PDE at each time step, and it is compared with a usual coupled solver. Three-dimensional numerical tests have been performed in order to show that the proposed uncoupled method has the same accuracy of the coupled strategy. Parallel numerical tests on structured meshes have also shown that the uncoupled technique is as scalable as the coupled one. Moreover, the conjugate gradient method preconditioned by Multilevel Hybrid Schwarz preconditioners converges faster for the linear systems deriving from the uncoupled method than from the coupled one. Finally, in all parallel numerical tests considered, the uncoupled technique proposed is always about two or three times faster than the coupled approach.