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We present a spectral Petrov--Galerkin method for the Boltzmann collision operator. We expand the density distribution $f$ to high order orthogonal polynomials multiplied by a Maxwellian. By that choice, we can approximate on the whole momentum domain $mathbb{R}^3$ resulting in high accuracy at the evaluation of the collision operator. Additionally, the special choice of the test space naturally ensures conservation of mass, momentum, and energy. By numerical examples we demonstrate the convergence (w.r.t. time) to the exact stationary solution. For efficiency we transfer between nodal and Maxwellian weighted spherical harmonics which are orthogonal w.r.t. the innermost integrals of the collision operator. Combined with efficient transformations between the bases and the calculation of the outer integrals this gives an algorithm of complexity $mathcal{O}(N^7)$ and a storage requirement $mathcal{O}(N^4)$ for the evaluation of the nonlinear Boltzmann collision operator. The presented method is applicable...

A Polynomial Spectral Method for the Spatially Homogeneous Boltzmann Equation