Explicit stabilized multirate methods for the monodomain model in cardiac electrophysiology

TL;DR

This work addresses the computational challenge of solving stiff, multiscale ODE systems in cardiac electrophysiology by developing explicit stabilized multirate solvers for the monodomain model. It introduces mRKC and a tailored emRKC that leverages explicit exponential multirate stabilization to couple diffusion and ionic dynamics, and benchmark them against the IMEX-RL baseline across 2D/3D meshes, realistic left atrial geometries, and multiple ionic models including fibrosis. The results show that emRKC typically offers superior efficiency and stability, with first-order time convergence and strong parallel scalability, achieving substantial speedups over IMEX-RL in many settings. Overall, the paper provides a fast, scalable, open-source explicit time-integration framework for large-scale, physics-based cardiac simulations with realistic tissue structure and heterogeneity.

Abstract

Fully explicit stabilized multirate (mRKC) methods are well-suited for the numerical solution of large multiscale systems of stiff ordinary differential equations thanks to their improved stability properties. To demonstrate their efficiency for the numerical solution of stiff, multiscale, nonlinear parabolic PDE's, we apply mRKC methods to the monodomain equation from cardiac electrophysiology. In doing so, we propose an improved version, specifically tailored to the monodomain model, which leads to the explicit exponential multirate stabilized (emRKC) method. Several numerical experiments are conducted to evaluate the efficiency of both mRKC and emRKC, while taking into account different finite element meshes (structured and unstructured) and realistic ionic models. The new emRKC method typically outperforms a standard implicit-explicit baseline method for cardiac electrophysiology. Code profiling and strong scalability results further demonstrate that emRKC is faster and inherently parallel without sacrificing accuracy.

Figures (15)

• Figure 1: Effective stability. Relative norm of the numerical solution $\mathbf{V}_N$ and number of stages for increasing step size $\Delta t$. Recall that $s$, $m$ are the number of stages used by the outer and inner RKC iterations, respectively.
• Figure 2: Two dimensional rectangle. Convergence experiments with different mesh sizes $\Delta x$ and ionic models.
• Figure 3: Two dimensional rectangle. Number of stages taken by the mRKC and emRKC methods, with different mesh sizes $\Delta x$ and ionic models. Recall that $s$, $m$ are the number of stages taken by the outer and inner RKC iterations, respectively.
• Figure 4: Two dimensional rectangle. Efficiency experiments with different mesh sizes $\Delta x$ and ionic models.
• Figure 5: Three dimensional cuboid. Efficiency experiments with different mesh sizes $\Delta x$ and ionic models.