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We present an adaptive spectral method for solving the Landau/Fokker-Planckequation for electron-ion systems. The heart of the algorithm is an expansionin Laguerre polynomials, which has several advantages, including automaticconservation of both energy and particles without the need for any specialdiscretization or time-stepping schemes. One drawback is the $O(N^3)$ memoryrequirement, where $N$ is the number of polynomials used. This can impose aninconvenient limit in cases of practical interest, such as when two particlespecies have widely separated temperatures. The algorithm we describe hereaddresses this problem by periodically re-projecting the solution onto ajudicious choice of new basis functions that are still Laguerre polynomials buthave arguments adapted to the current physical conditions. This results in areduction in the number of polynomials needed, at the expense of increasedsolution time. Because the equations are solved with little difficulty, thisadded time is not of much concern compared to the savings in memory. Todemonstrate the algorithm, we solve several relaxation problems that could notbe computed with the spectral method without re-projection. Another majoradvantage of this method is that it can be used for collision operators morecomplicated than that of the Landau equation, and we demonstrate this here byusing it to solve the non-degenerate quantum Lenard-Balescu equation for ahydrogen plasma.

Adaptive spectral solution method for the Landau and Lenard-Balescu equations