  In this work, we prove rigorous convergence properties for a semi-discrete,
moment-based approximation of a model kinetic equation in one dimension. This
approximation is equivalent to a standard spectral method in the velocity
variable of the kinetic distribution and, as such, is accompanied by standard
algebraic estimates of the form $N^{-q}$, where $N$ is the number of modes and
$q>0$ depends on the regularity of the solution. However, in the multiscale
setting, the error estimate can be expressed in terms of the scaling parameter
$\epsilon$, which measures the ratio of the mean-free-path to the
characteristic domain length. We show that, for isotropic initial conditions,
the error in the spectral approximation is $\mathcal{O}(\epsilon^{N+1})$. More
surprisingly, the coefficients of the expansion satisfy super convergence
properties. In particular, the error of the $\ell^{th}$ coefficient of the
expansion scales like $\mathcal{O}(\epsilon^{2N})$ when $\ell =0$ and
$\mathcal{O}(\epsilon^{2N+2-\ell})$ for all $1\leq \ell \leq N$. This result is
significant, because the low-order coefficients correspond to physically
relevant quantities of the underlying system. All the above estimates involve
constants depending on $N$, the time $t$, and the initial condition. We
investigate specifically the dependence on $N$, in order to assess whether
increasing $N$ actually yields an additional factor of $\epsilon$ in the error.
Numerical tests will also be presented to support the theoretical results.
