An operator splitting method for solving the bidomain equations coupled to a volume conductor model for the torso

Sundnes, Joakim; Lines, Glenn Terje; Tveito, Aslak

doi:10.1016/j.mbs.2005.01.001

Cited by 136 publications

(111 citation statements)

References 14 publications

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“…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].…”

Section: Time Discretizationmentioning

confidence: 99%

“…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.…”

Section: Introductionmentioning

confidence: 99%

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

2013

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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.Mathematics Subject Classification. 65N55, 65M55, 65F10, 92C30.

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Section: Time Discretizationmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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

2013

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“…where the reaction and diffusion part of the parabolic PDE is split by employing a Strang or a Gudunov scheme (Qu & Garfinkel, 1999;Sundnes et al, 2005). The parabolic portion (12) is solved either by choosing θ = 0.5, which results in a Crank-Nicholson scheme, or θ = 0.0, which results in an explicit forward Euler scheme.…”

Section: Numerical Schemesmentioning

confidence: 99%

Modeling Defibrillation

Plank¹,

Trayanov²

2011

Cardiac Defibrillation - Mechanisms, Challenges and Implications

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“…Furthermore, Quan et al [16] present a domain decomposition approach using an alternating direction implicit (ADI) method. Sundnes et al [17] introduce an operator splitting method to solve a fully coupled discretization of three PDEs modelling the interaction between the myocardium and the torso surrounding the heart. With respect to time adaptivity, Skouibine et al [18] present a predictor-corrector time stepping strategy to accelerate a given finite difference scheme for the bidomain equations using active membrane kinetics (Luo-Rudy phase II).…”

Section: B Related Workmentioning

confidence: 99%

A multiresolution space‐time adaptive scheme for the bidomain model in electrocardiology

Bendahmane¹,

Bürger²,

Ruiz–Baier³

2009

Numerical Methods Partial

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The bidomain model of electrical activity of myocardial tissue consists of a possibly degenerate parabolic PDE coupled with an elliptic PDE for the transmembrane and extracellular potentials, respectively. This system of two scalar PDEs is supplemented by a time-dependent ODE modeling the evolution of the gating variable. In the simpler subcase of the monodomain model, the elliptic PDE reduces to an algebraic equation. Since typical solutions of the bidomain and monodomain models exhibit wavefronts with steep gradients, we propose a finite volume scheme enriched by a fully adaptive multiresolution method, whose basic purpose is to concentrate computational effort on zones of strong variation of the solution. Time adaptivity is achieved by two alternative devices, namely locally varying time stepping and a Runge-Kutta-Fehlberg-type adaptive time integration. A series of numerical examples demonstrates that these methods are efficient and sufficiently accurate to simulate the electrical activity in myocardial tissue with affordable effort. In addition, the optimal choice of the threshold for discarding nonsignificant information in the multiresolution representation of the solution is addressed, and the numerical efficiency and accuracy of the method is measured in terms of CPU time speed-up, memory compression, and errors in different norms.

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An operator splitting method for solving the bidomain equations coupled to a volume conductor model for the torso

Cited by 136 publications

References 14 publications

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

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

Modeling Defibrillation

A multiresolution space‐time adaptive scheme for the bidomain model in electrocardiology

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