Abstract
We investigate spectral deferred correction (SDC) methods for time stepping and their interplay with spatio-temporal adaptivity, applied to the solution of the cardiac electro-mechanical coupling model. This model consists of the Monodomain equations, a reaction-diffusion system modeling the cardiac bioelectrical activity, coupled with a quasi-static mechanical model describing the contraction and relaxation of the cardiac muscle. The numerical approximation of the cardiac electro-mechanical coupling is a challenging multiphysics problem, because it exhibits very different spatial and temporal scales. Therefore, spatio-temporal adaptivity is a promising approach to reduce the computational complexity. SDC methods are simple iterative methods for solving collocation systems. We exploit their flexibility for combining them in various ways with spatio-temporal adaptivity. The accuracy and computational complexity of the resulting methods are studied on some numerical examples.
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References
Aliev, R.R., Panfilov, A.V.: A simple two-variable model of cardiac excitation. Chaos Solitons Fractals 7, 293–301 (1996)
Bendahmane, M., Buerger, R., Ruiz-Baier, R.: A multiresolution space-time adaptive scheme for the bidomain model in electrocardiology. Numer. Methods Partial Differ. Equ. 26, 1377–1404 (2010)
Colli Franzone, P., Deuflhard, P., Erdmann, B., Lang, J., Pavarino, L.F.: Adaptivity in space and time for reaction-diffusion systems in electrocardiology. SIAM J. Sci. Comput. 28, 942–962 (2006)
Dutt, A., Greengard, L., Rokhlin, V.: Spectral deferred correction methods for ordinary differential equations. BIT Numer. Math. 40(2), 241–266 (2000)
G\(\ddot{\mbox{ o}}\) ktepe, S., Kuhl, E.: Electromechanics of the heart – a unified approach to the strongly coupled excitation-contraction problem. Comput. Mech. 80, 227–243 (2010)
Humphrey, J.D.: Cardiovascular Solid Mechanics, Cells, Tissues and Organs. Springer, New York (2001)
Kerckhoffs, R.C.P., Bovendeerd, P.H.M., Kotte, J.C.S., Prinzen, F.W., Smits, K., Arts, T.: Homogeneity of cardiac contraction despite physiological asynchrony of depolarization: a model study. Ann. Biomed. Eng. 31, 536–547 (2003)
Lang, J.: Adaptive multilevel solution of nonlinear parabolic PDE systems. Lecture Notes in Computational Science and Engineering, vol. 16. Springer, New York (2001)
Nash, M.P., Panfilov, A.V.: Electromechanical model of excitable tissue to study reentrant cardiac arrhythmias. Prog. Biophys. Mol. Biol. 85, 501–522 (2004)
Pathmanathan, P.J., Whiteley, J.P.: A numerical method for cardiac mechanoelectric simulations. Ann. Biomed. Eng. 37, 860–873 (2009)
Vetter, F.J., McCulloch, A.D.: Three-dimensional stress and strain in passive rabbit left ventricle: a model study. Ann. Biomed. Eng. 28, 781–792 (2000)
Whiteley, J.P., Bishop, M.J., Gavaghan, D.J.: Soft tissue modelling of cardiac fibres for use in coupled mechano-electric simulations. Bull. Math. Biol. 69, 2199–2225 (2007)
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Weiser, M., Scacchi, S. (2016). Spectral Deferred Correction Methods for Adaptive Electro-Mechanical Coupling in Cardiac Simulation. In: Russo, G., Capasso, V., Nicosia, G., Romano, V. (eds) Progress in Industrial Mathematics at ECMI 2014. ECMI 2014. Mathematics in Industry(), vol 22. Springer, Cham. https://doi.org/10.1007/978-3-319-23413-7_42
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DOI: https://doi.org/10.1007/978-3-319-23413-7_42
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