  In this paper, we study the numerical stability of reduced order models for
convection-dominated stochastic systems in a relatively simple setting: a
stochastic Burgers equation with linear multiplicative noise. Our preliminary
results suggest that, in a convection-dominated regime, standard reduced order
models yield inaccurate results in the form of spurious numerical oscillations.
To alleviate these oscillations, we use the Leray reduced order model, which
increases the numerical stability of the standard model by smoothing
(regularizing) the convective term with an explicit spatial filter. The Leray
reduced order model yields significantly better results than the standard
reduced order model and is more robust with respect to changes in the strength
of the noise.
