The activity and dynamics of excitable cells are fundamentally regulated and moderated by extracellular and intracellular ion concentrations and their electric potentials. The increasing availability of dense reconstructions of excitable tissue at extreme geometric detail pose a new and clear scientific computing challenge for computational modelling of ion dynamics and transport. In this paper, we design, develop and evaluate a scalable numerical algorithm for solving the time-dependent and nonlinear KNP-EMI equations describing ionic electrodiffusion for excitable cells with an explicit geometric representation of intracellular and extracellular compartments and interior interfaces. We also introduce and specify a set of model scenarios of increasing complexity suitable for benchmarking. Our solution strategy is based on an implicit-explicit discretization and linearization in time, a mixed finite element discretization of ion concentrations and electric potentials in intracellular and extracellular domains, and an algebraic multigrid-based, inexact block-diagonal preconditioner for GMRES. Numerical experiments with up to $10^8$ unknowns per time step and up to 256 cores demonstrate that this solution strategy is robust and scalable with respect to the problem size, time discretization and number of cores.

翻译：可兴奋细胞的活动和动力学从根本上受到细胞外和细胞内离子浓度及其电位的调节与调控。随着具有极端几何细节的可兴奋组织密集重建数据的日益增多，这为离子动力学和传输的计算建模带来了新的明确科学计算挑战。本文设计、开发并评估了一种可扩展数值算法，用于求解描述可兴奋细胞离子电扩散的时变非线性KNP-EMI方程，该方程明确表示了细胞内和细胞外隔室及其内部界面。我们还引入并指定了一套复杂度递增、适用于基准测试的模型场景。我们的求解策略基于时间上的隐式-显式离散化与线性化、细胞内和细胞外区域中离子浓度与电位的混合有限元离散化，以及基于代数多重网格的非精确块对角预条件器（用于GMRES）。针对每时间步多达$10^8$个未知量、最多256个核心的数值实验表明，该求解策略在问题规模、时间离散化和核心数量方面具有鲁棒性和可扩展性。