Whiteley Jonathan P
Computing Laboratory, University of Oxford, Wolfson Bldg., Parks Road, Oxford OX1 3QD, UK.
IEEE Trans Biomed Eng. 2006 Nov;53(11):2139-47. doi: 10.1109/TBME.2006.879425.
Most numerical schemes for solving the monodomain or bidomain equations use a forward approximation to some or all of the time derivatives. This approach, however, constrains the maximum timestep that may be used by stability considerations as well as accuracy considerations. Stability may be ensured by using a backward approximation to all time derivatives, although this approach requires the solution of a very large system of nonlinear equations at each timestep which is computationally prohibitive. In this paper we propose a semi-implicit algorithm that ensures stability. A linear system is solved on each timestep to update the transmembrane potential and, if the bidomain equations are being used, the extracellular potential. The remainder of the equations to be solved uncouple into small systems of ordinary differential equations. The backward Euler method may be used to solve these systems and guarantee numerical stability: as these systems are small, only the solution of small nonlinear systems are required. Simulations are carried out to show that the use of this algorithm allows much larger timesteps to be used with only a minimal loss of accuracy. As a result of using these longer timesteps the computation time may be reduced substantially.
大多数用于求解单域或双域方程的数值格式对部分或所有时间导数采用向前近似。然而，这种方法在稳定性和精度方面都限制了可使用的最大时间步长。通过对所有时间导数采用向后近似可以确保稳定性，不过这种方法需要在每个时间步求解一个非常大的非线性方程组，这在计算上是难以实现的。在本文中，我们提出了一种确保稳定性的半隐式算法。在每个时间步求解一个线性系统来更新跨膜电位，如果使用双域方程，则更新细胞外电位。其余待求解的方程可分解为常微分方程的小系统。可以使用向后欧拉方法来求解这些系统并保证数值稳定性：由于这些系统规模较小，只需要求解小型非线性系统。进行的模拟表明，使用该算法允许使用大得多的时间步长，而精度损失极小。由于使用了这些更长的时间步长，计算时间可以大幅减少。
