  This paper is a contribution to the exploration of the parametric Kalman
filter (PKF), which is an approximation of the Kalman filter, where the error
covariances are approximated by a covariance model. Here we focus on the
covariance model parameterized from the variance and the anisotropy of the
local correlations, and whose parameters dynamics provides a proxy for the full
error-covariance dynamics. For this covariance model, we aim to provide the
boundary condition to specify in the prediction of PKF for bounded domains,
focusing on Dirichlet and Neumann conditions when they are prescribed for the
physical dynamics. An ensemble validation is proposed for the transport
equation and for the heterogeneous diffusion equation over a bounded 1D domain.
This ensemble validation requires to specify the auto-correlation time-scale
needed to populate boundary perturbation that leads to prescribed uncertainty
characteristics. The numerical simulations show that the PKF is able to
reproduce the uncertainty diagnosed from the ensemble of forecast appropriately
perturbed on the boundaries, which show the ability of the PKF to handle
boundaries in the prediction of the uncertainties. It results that Dirichlet
condition on the physical dynamics implies Dirichlet condition on the variance
and on the anisotropy. Codes are available at
https://github.com/opannekoucke/pkf-boundary
