Spectral computation of low probability tails for the homogeneous Boltzmann equation

We apply the spectral-Lagrangian method of Gamba and Tharkabhushanam for solving homogeneous Boltzmann equation to compute low probability tails velocity distribution function, f, a particle species. This is based on truncation, Qtr(f,f), collision operator, Q(f,f), whose Fourier transform given by weighted convolution. The truncated operator models situation in which two colliding particles ignore each other if their relative speed exceeds threshold, gtr. demonstrate that choice truncation parameter plays critical role accuracy numerical computation Q. Significantly, gtr too large, then accurate convolution integral not feasible, since degree oscillation weighting function increases as increases. derive an upper bound pointwise error between Q Qtr, assuming both operators are computed exactly. provides some additional theoretical justification method, can be used guide computations. how choose discretization parameters so good approximation tails. Finally, several different initial conditions, we feasibility accurately computing time evolution pdf down density levels ranging from 10−5 10−9.