用于心电学双域模型的多分辨率时空自适应格式
A multiresolution space-time adaptive scheme for the bidomain model in electrocardiology
AI总结：
提出一种结合全自适应多分辨率方法和时间自适应策略的有限体积格式，高效求解心肌组织电活动的单域和双域模型。
摘要：
本文研究心肌组织电活动的单域和双域模型的数值求解。双域模型是一个系统，由可能退化的抛物型偏微分方程与椭圆型偏微分方程耦合而成，分别描述跨膜电位和细胞外电位。这两个标量PDE系统补充了一个时间依赖的ODE，用于模拟所谓的门控变量的演化。在单域模型的更简单子情形中，椭圆型PDE简化为一个代数方程。考虑了两种简单的膜和离子电流模型：Mitchell-Schaeffer模型和更简单的FitzHugh-Nagumo模型。由于双域和单域模型的典型解呈现具有陡峭梯度的波前，我们提出一种由全自适应多分辨率方法增强的有限体积格式，其基本目的是将计算精力集中在解的强变化区域。时间自适应通过两种替代方案实现，即局部变时间步长和Runge-Kutta-Fehlberg型自适应时间积分。一系列数值例子表明，这些方法足够高效和精确，能够以可承受的计算量模拟心肌组织中的电活动。此外，推导了在多分辨率表示中丢弃非显著信息的最优阈值，并通过CPU时间加速比、内存压缩和不同范数下的误差来衡量方法的数值效率和精度。
英文摘要：
This work deals with the numerical solution of the monodomain and bidomain models of electrical activity of myocardial tissue. The bidomain model is a system consisting of a possibly degenerate parabolic PDE coupled with an elliptic PDE for the transmembrane and extracellular potentials, respectively. This system of two scalar PDEs is supplemented by a time-dependent ODE modeling the evolution of the so-called gating variable. In the simpler sub-case of the monodomain model, the elliptic PDE reduces to an algebraic equation. Two simple models for the membrane and ionic currents are considered, the Mitchell-Schaeffer model and the simpler FitzHugh-Nagumo model. Since typical solutions of the bidomain and monodomain models exhibit wavefronts with steep gradients, we propose a finite volume scheme enriched by a fully adaptive multiresolution method, whose basic purpose is to concentrate computational effort on zones of strong variation of the solution. Time adaptivity is achieved by two alternative devices, namely locally varying time stepping and a Runge-Kutta-Fehlberg-type adaptive time integration. A series of numerical examples demonstrates thatthese methods are efficient and sufficiently accurate to simulate the electrical activity in myocardial tissue with affordable effort. In addition, an optimalthreshold for discarding non-significant information in the multiresolution representation of the solution is derived, and the numerical efficiency and accuracy of the method is measured in terms of CPU time speed-up, memory compression, and errors in different norms.
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