Rudolph Manfred
Chemische Fakultät, Am Steiger 3, Friedrich-Schiller-Universität, D-07743 Jena, Germany.
J Comput Chem. 2005 Apr 30;26(6):619-32. doi: 10.1002/jcc.20200.
It is a well-known phenomenon called superconvergence in the mathematical literature that the error level of an integral quantity can be much smaller than the magnitude of the local errors involved in the computation of this quantity. When discretizing an integrated form of Fick's second law of diffusion the local errors reflect the accuracy of individual concentration points while the integral quantity has the physical meaning of the flux. This article demonstrates how an extraordinary fast exponential convergence towards zero can be achieved for the simulated flux error on the basis of finite-difference approximations that are only second-order (Box 2 method) or fourth-order (Box 4 method) accurate as far as the level of local errors is concerned.
在数学文献中，有一个被称为超收敛的著名现象，即一个积分量的误差水平可能比计算该量时所涉及的局部误差的大小小得多。在离散菲克第二扩散定律的积分形式时，局部误差反映了各个浓度点的精度，而积分量具有通量的物理意义。本文展示了，就局部误差水平而言，基于仅为二阶精度的有限差分近似（盒式2方法）或四阶精度的有限差分近似（盒式4方法），如何能使模拟通量误差实现向零的异常快速指数收敛。
