This manuscript deals with the analysis of numerical methods for the full discretization (in time and space) of the linear heat equation with Neumann boundary conditions, and it provides the reader with error estimates that are uniform in time. First, we consider the homogeneous equation with homogeneous Neumann boundary conditions over a finite interval. Using finite differences in space and the Euler method in time, we prove that our method is of order 1 in space, uniformly in time, under a classical CFL condition, and despite its lack of consistency at the boundaries. Second, we consider the nonhomogeneous equation with nonhomogeneous Neumann boundary conditions over a finite interval. Using a tailored similar scheme, we prove that our method is also of order 1 in space, uniformly in time, under a classical CFL condition. We indicate how this numerical method allows for a new way to compute steady states of such equations when they exist. We conclude by several numerical experiments to illustrate the sharpness and relevance of our theoretical results, as well as to examine situations that do not meet the hypotheses of our theoretical results, and to illustrate how our results extend to higher dimensions.

翻译：本文研究带Neumann边界条件的线性热方程全离散（时间与空间）数值方法分析，并给出时间一致误差估计。首先，考虑有限区间上具齐次Neumann边界条件的齐次方程。通过空间有限差分与时间欧拉方法，我们证明在经典CFL条件下，尽管边界处存在非相容性，该方法在空间上仍达到一阶精度且时间一致。其次，考虑有限区间上具非齐次Neumann边界条件的非齐次方程。采用类似定制格式，我们证明在经典CFL条件下，该方法同样在空间上达到一阶精度且时间一致。我们指出该数值方法为计算此类方程稳态解（若存在）提供了新途径。最后通过多个数值实验验证理论结果的精确性与适用性，考察理论假设不成立的情形，并展示该方法向高维空间的推广。