The purpose of this note is to investigate the coupling of Dirichlet and Neumann numerical boundary conditions for the transport equation set on an interval. When one starts with a stable finite difference scheme on the lattice $\mathbb{Z}$ and each numerical boundary condition is taken separately with the Neumann extrapolation condition at the outflow boundary, the corresponding numerical semigroup on a half-line is known to be bounded. It is also known that the coupling of such numerical boundary conditions on a compact interval yields a stable approximation, even though large time exponentially growing modes may occur. We review the different stability estimates associated with these numerical boundary conditions and give explicit examples of such exponential growth phenomena for finite difference schemes with ''small'' stencils. This provides numerical evidence for the optimality of some stability estimates on the interval.

翻译：本笔记旨在研究定义在区间上的输运方程中Dirichlet与Neumann数值边界条件的耦合问题。当从格点$\mathbb{Z}$上的稳定有限差分格式出发，并分别在出流边界处采用Neumann外推条件处理各数值边界条件时，已知对应半直线上的数值半群是有界的。此外，尽管可能出现长时间指数增长模态，但已知此类数值边界条件在紧区间上的耦合仍能产生稳定近似。我们回顾了与这些数值边界条件相关的不同稳定性估计，并针对具有"小"模板的有限差分格式给出了此类指数增长现象的具体实例。这为区间上某些稳定性估计的最优性提供了数值证据。