A comparative study of fully implicit staggered and monolithic solution methods. Part II: Coupled excitation–contraction equations of cardiac electromechanics

A comparative study of fully implicit staggered and monolithic solution algorithms for the coupled bidomain equations of cardiac electrophysiology was discussed in Part I (JCAM 407:114021, 2022). In this folow-up study, we further employ the similar approach and present an extensive comparative analysis between the staggered and monolithic solution techniques for the coupled partial differential equations (PDEs) of cardiac electromechanics both in the monodomain and bidomain setting. For the monolithic solution technique, we adopt the work of Göktepe et al. (CM 45: 227-243, 2010) and Dal et al. (CMAME 253: 353-336, 2013), where the governing PDEs of the excitation-contraction problem are solved simultaneously and all rate equations are treated with an implicit time integration scheme. For the staggered scheme, however, we simply suggest a decoupled solution of the governing PDEs while keeping other aspects of the monolithic algorithm the same, such as utilization of the implicit time integration scheme and coupling of ordinary differential equations, which describe the state variables of cardiac electrophysiology, to the parabolic PDE of the electrophysiology. Both solution algorithms are applied to several problems, where we simulate regular planar wave propagations and scroll waves for different spatial and temporal resolution and material parameters. The comparison between the solution schemes is performed in terms of accuracy, efficiency and stability. We reveal that for regular wave propagation the results of different solution algorithms are identical. The only deviation is observed during scroll wave propagation, if the parameters controlling the mutual interaction between the mechanical and electrical field are relatively large and at the same time if the time increment is big, e.g., Δt=2.0ms. Nevertheless, the suggested staggered solution scheme yields almost identical results even for high coupling if smaller time increments are used. Besides, the staggered scheme provides an enormous gain in computational efficiency, particularly, as the number of degrees of freedom is increased and we do not encounter any stability issue arising specifically from the decoupled solution of the governing PDEs. Similar to the conclusion that we made for cardiac electrophysiology in Part I, we report that the suggested fully implicit staggered solution algorithm has a tremendous application potential for computer simulations of exciation-contraction coupling of the heart both in the monodomain and bidomain setting.