  We present an adaptive spectral method for solving the Landau/Fokker-Planck
equation for electron-ion systems. The heart of the algorithm is an expansion
in Laguerre polynomials, which has several advantages, including automatic
conservation of both energy and particles without the need for any special
discretization or time-stepping schemes. One drawback is the $O(N^3)$ memory
requirement, where $N$ is the number of polynomials used. This can impose an
inconvenient limit in cases of practical interest, such as when two particle
species have widely separated temperatures. The algorithm we describe here
addresses this problem by periodically re-projecting the solution onto a
judicious choice of new basis functions that are still Laguerre polynomials but
have arguments adapted to the current physical conditions. This results in a
reduction in the number of polynomials needed, at the expense of increased
solution time. Because the equations are solved with little difficulty, this
added time is not of much concern compared to the savings in memory. To
demonstrate the algorithm, we solve several relaxation problems that could not
be computed with the spectral method without re-projection. Another major
advantage of this method is that it can be used for collision operators more
complicated than that of the Landau equation, and we demonstrate this here by
using it to solve the non-degenerate quantum Lenard-Balescu equation for a
hydrogen plasma.
