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.

翻译：完全显式稳定多速率（mRKC）方法因其优越的稳定性特性，非常适用于求解大型多尺度刚性常微分方程组。为证明该方法在求解刚性、多尺度、非线性抛物型偏微分方程方面的效率，我们将mRKC方法应用于心脏电生理学中的单域方程。在此基础上，我们提出了一种改进版本，该版本专门针对单域模型进行优化，从而形成了显式指数多速率稳定（emRKC）方法。通过多项数值实验，我们评估了mRKC和emRKC方法的效率，同时考虑了不同的有限元网格（结构化与非结构化）及实际的离子模型。新的emRKC方法通常优于心脏电生理学中标准的隐式-显式基线方法。代码性能分析和强可扩展性结果进一步表明，emRKC在不牺牲精度的前提下速度更快且具有内在并行性。