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

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 preci-sion 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.

Automated summary

This paper investigates truncation and round-off errors in the numerical solution of the 1D advection equation using the Lax-Friedrichs scheme, and their accumulation in high temporal layers. The authors derive a new theoretical approximation formula for the upper bound of the total error, as well as theoretical formulas for the optimal grid size and time step. The reliability of the formulas is demonstrated through numerical experiments. Furthermore, the paper explores the relationship between the optimal time steps under different machine precisions and presents theoretical verification of the computational uncertainty principle under a fixed grid ratio.