Solving the Coupled System Improves Computational Efficiency of the Bidomain Equations

Abstract

The bidomain equations are frequently used to model the propagation of cardiac action potentials across cardiac tissue. At the whole organ level the size of the computational mesh required makes their solution a significant computational challenge. As the accuracy of the numerical solution cannot be compromised, efficiency of the solution technique is important to ensure that the results of the simulation can be obtained in a reasonable time whilst still encapsulating the complexities of the system.In an attempt to increase efficiency of the solver, the bidomain equations are often decoupled into one parabolic equation that is computationally very cheap to solve and an elliptic equation that is much more expensive to solve. In this study the performance of this uncoupled solution method is compared with an alternative strategy in which the bidomain equations are solved as a coupled system. This seems counter-intuitive as the alternative method requires the solution of a much larger linear system at each time step. However, in tests on two 3-D rabbit ventricle benchmarks it is shown that the coupled method is up to 80% faster than the conventional uncoupled methodand that parallel performance is better for the larger coupled problem.

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

“…For what concerns efficiency, all the parallel numerical tests considered have shown that the uncoupled technique is about 2.5-3 times faster than the coupled approach, employing the same PCG relative tolerances for both the coupled and uncoupled systems. A recent study [45] has shown that solving the coupled system can be more efficient than the uncoupled one when considering different uncoupled methods without the last parabolic correction step, that requires different PCG relative tolerances in order to achieve the same level of accuracy for the coupled and uncoupled approaches. We have also performed some tests with a standard uncoupled method without the last parabolic correction step and we have found that also this variant is 2-3 times faster than the coupled approach, although less accurate than our uncoupled method.…”

“…Previous works have considered several variants of these splitting and/or uncoupling techniques. In particular, the uncoupled approach has been considered in [1,22,35,36,45,47,[51][52][53][54]. The splitting of reaction and diffusion terms has been considered in [35,53,54], while a three steps Strang splitting, with reaction -diffusion -reaction steps, has been considered in [22,47,51].…”

“…In the standard uncoupled method, the first parabolic correction can be solved explicitly as e.g. in [36,45] or by a semi-implicit method as e.g. in [1,45].…”

“…Alternatively, most previous works have considered IMEX time discretizations and/or operator splitting schemes, where the reaction and diffusion terms are treated separately, see e.g. [4,6,8,14,20,22,28,35,38,45,47,[53][54][55]. The advantage of IMEX and operator splitting schemes is that they only require the solution of a linear system for the parabolic and elliptic PDEs at each time step.…”

BPX preconditioners for the Bidomain model of electrocardiology

“…Alternatively, most previous works have considered semi-implicit (IMEX) time discretizations and/or operator splitting schemes, where the reaction and diffusion terms are treated separately, see e.g. [5,6,7,8,9,10,11,12,13,14,15]. The advantage of IMEX and operator splitting schemes is that they only require the solution of linear systems at each time step.…”