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Spectral methods, thanks to the high accuracy and the possibility of usingfast algorithms, represent an effective way to approximate collisional kineticequations in kinetic theory. On the other hand, the loss of some localinvariants can lead to the wrong long time behavior of the numerical solution.We introduce in this paper a novel Fourier-Galerkin spectral method thatimproves the classical spectral method by making it conservative on the momentsof the approximated distribution, without sacrificing its spectral accuracy orthe possibility of using fast algorithms. The method is derived directly usinga constrained best approximation in the space of trigonometric polynomials andcan be applied to a wide class of problems where preservation of moments isessential. We then apply the new spectral method to the evaluation of theBoltzmann collision term, and prove spectral consistency and stability of theresulting Fourier-Galerkin approximation scheme. Various numerical experimentsillustrate the theoretical findings.

Moment preserving Fourier-Galerkin spectral methods and application to the Boltzmann equation