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This manuscript deals with the analysis of numerical methods for the fulldiscretization (in time and space) of the linear heat equation with Neumannboundary conditions, and it provides the reader with error estimates that areuniform in time. First, we consider the homogeneous equation with homogeneousNeumann boundary conditions over a finite interval. Using finite differences inspace and the Euler method in time, we prove that our method is of order 1 inspace, uniformly in time, under a classical CFL condition, and despite its lackof consistency at the boundaries. Second, we consider the nonhomogeneousequation with nonhomogeneous Neumann boundary conditions over a finiteinterval. Using a tailored similar scheme, we prove that our method is also oforder 1 in space, uniformly in time, under a classical CFL condition. Weindicate how this numerical method allows for a new way to compute steadystates of such equations when they exist. We conclude by several numericalexperiments to illustrate the sharpness and relevance of our theoreticalresults, as well as to examine situations that do not meet the hypotheses ofour theoretical results, and to illustrate how our results extend to higherdimensions.

Uniform estimates for a fully discrete scheme integrating the linear heat equation on a bounded interval with pure Neumann boundary conditions