  We study the approximation and stability properties of a recently popularized
discretization strategy for the speed variable in kinetic equations, based on
pseudo spectral collocation on a grid defined by the zeros of a non-standard
family of orthogonal polynomials called Maxwell polynomials. Taking a
one-dimensional equation describing energy diffusion due to Fokker-Planck
collisions with a Maxwell-Boltzmann background distribution as the test bench
for the performance of the scheme, we find that Maxwell based discretizations
outperform other commonly used schemes in most situations, often by orders of
magnitude. This provides a strong motivation for their use in high-dimensional
gyrokinetic simulations. However, we also show that Maxwell based schemes are
subject to a non-modal time stepping instability in their most straightforward
implementation, so that special care must be given to the discrete
representation of the linear operators in order to benefit from the advantages
provided by Maxwell polynomials.
