A Petrov-Galerkin spectral method for the inelastic Boltzmann equation using mapped Chebyshev functions

We develop in this paper a Petrov-Galerkin spectral method for the inelastic Boltzmann equation in one dimension. Solutions to such equations typically exhibit heavy tails in the velocity space so that domain truncation or Fourier approximation would suffer from large truncation errors. Our method is based on the mapped Chebyshev functions on unbounded domains, hence requires no domain truncation. Furthermore, the test and trial function spaces are carefully chosen to obtain desired convergence and conservation properties. Through a series of examples, we demonstrate that the proposed method performs better than the Fourier spectral method and yields highly accurate results.