数值积分过程中截断误差和舍入误差的分离方法及其效果检验

Separation of Truncation Error and Round-off Error in the Numerical Integration and Its Validation

本文讨论数值积分过程中截断误差和舍入误差的分离方法和理论,解析地给出某些数值计算方法的理论截断误差,并以此来分离计算结果中的误差。然后引入参考解的办法,用来分离更为一般的微分方程求解过程中的截断误差和舍入误差。以参考解算法为基础,对一个偏微分方程的数值解进行计算,所得结果与采用理论截断误差得到的结果进行了对比,发现:(1)当使用迎风差和中央差格式时,理论截断误差和近似截断误差在数值上高度一致,说明了参考解方法的正确性;(2)对于一阶的波动方程,迎风差和中央差格式的理论截断误差在形式上也具有波动的周期特征,振幅的大小与计算参数有关;(3)理论截断误差可以适用于任意t时刻,而近似截断误差的适用时间范围为一个有限的时间段,不过它可以很容易的获取一般微分方程的截断误差,而不需要复杂的理论推导。

The authors propose a method to separate the truncation error and the round-off error from the numerical solution.The analytical truncation error formulas of a partial differential equation are given for the upstream scheme and the centered difference scheme,respectively.The reference solution method is then introduced to separate these two types of errors for more general equations.A scheme based on the reference solution is used to obtain the approximate truncation error.Comparing the results for the upstream scheme and the centered difference scheme,the authors find that：1） the approximate truncation error is highly consistent with the analytical one.2） The truncation errors of 1-D wave equations for the two schemes both show wavy periodicities with amplitudes being related to the parameters of computation.3） The analytical error is suitable for the analysis of any slice of t,while the approximate one is only suitable for the analysis of a certain time range.However,the approximate error can be more easily obtained for general differential equations without a complex theoretical deduction.