Fast conservative and entropic numerical methods for the Boson Boltzmann equation

In this paper we derive accurate numerical methods for the quan­ tum Boltzmann equation for a gas of interacting bosons. The schemes preserve the main physical features of the continuous problem, namely conservation of mass and energy, the entropy inequality and general­ ized Bose­Einstein distributions as steady states. These properties are essential in order to develop schemes that are able to capture the energy concentration behavior of bosons. In addition we develop fast algorithms for the numerical evaluation of the resulting quadra­ ture formulas which allow the final schemes to be computed only in O(N^2 log 2 N) operations instead of O(N^3 ).