Computational uncertainty and optimal grid size and time step of the Lax–Friedrichs scheme for the 1D advection equation

Abstract

This paper examines truncation and round-off errors in the numerical solution of the 1D advection equation with the Lax–Friedrichs scheme, and accumulation of the errors as they are propagated to high temporal layers. The authors obtain a new theoretical approximation formula for the upper bound of the total error of the numerical solution, as well as theoretical formulae for the optimal grid size and time step. The reliability of the obtained formulae is demonstrated with numerical experimental examples. Next, the ratio of the optimal time steps under two different machine precisions is found to satisfy a universal relation that depends only on the machine precision involved. Finally, theoretical verification suggests that this problem satisfies the computational uncertainty principle when the grid ratio is fixed, demonstrating the inevitable existence of an optimal time step size under a finite machine precision.

摘要

本文对于应用Lax- Friedrichs 格式数值求解一维平流方程, 研究数值求解过程中产生的截断误差与舍入误差, 以及两种误差逐层向高时间层传播的累积, 得到新的数值解总误差上界的理论近似公式, 以及最优格距和最优时间步长的理论公式. 通过数值算例验证了所得公式的可靠性. 然后, 发现了两种不同机器精度下最优时间步长之比满足的一个仅与机器精度有关的普适关系. 最后, 理论验证了在网格比固定的情况下, 此问题满足数值计算的不确定性原理, 以及在机器有限精度下最优时间步长的必然存在.