  We examine the merits of using a family of polynomials that are orthogonal
with respect to a non-classical weight function to discretize the speed
variable in continuum kinetic calculations. We consider a model one-dimensional
partial differential equation describing energy diffusion in velocity space due
to Fokker-Planck collisions. This relatively simple case allows us to compare
the results of the projected dynamics with an expensive but highly accurate
spectral transform approach. It also allows us to integrate in time exactly,
and to focus entirely on the effectiveness of the discretization of the speed
variable. We show that for a fixed number of modes or grid points, the
non-classical polynomials can be many orders of magnitude more accurate than
classical Hermite polynomials or finite-difference solvers for kinetic
equations in plasma physics. We provide a detailed analysis of the difference
in behavior and accuracy of the two families of polynomials. For the
non-classical polynomials, if the initial condition is not smooth at the origin
when interpreted as a three-dimensional radial function, the exact solution
leaves the polynomial subspace for a time, but returns (up to roundoff
accuracy) to the same point evolved to by the projected dynamics in that time.
By contrast, using classical polynomials, the exact solution differs
significantly from the projected dynamics solution when it returns to the
subspace. We also explore the connection between eigenfunctions of the
projected evolution operator and (non-normalizable) eigenfunctions of the full
evolution operator, as well as the effect of truncating the computational
domain.
