Chebyshev spectral-SN method for the neutron transport equation

We study convergence of a combined spectral and (SN) discrete ordinates approximation for a multidimensional, steady state, linear transport problem with isotropic scattering. The procedure is based on expansion of the angular flux in a truncated series of Chebyshev polynomials in spatial variables that results in the transformation of the multidimensional problems into a set of one-dimensional problems. The convergence of this approach is studied in the context of the discrete-ordinates equations based on a special quadrature rule for the scattering integral. The discrete-ordinates and quadrature errors are expanded in truncated series of Chebyshev polynomials of degree ≤ L, and the convergence is derived assuming L ≤ σt - 4πσs, where σt and σs are total- and scattering cross-sections, respectively.