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Fourier decomposition of sharply peaked phase functions: Legendre expansions versus trapezoidal rule

Most numerical methods used in radiation transfer problems decompose the radiation field in Fourier series and solve a reduced radiative transfer equation for each Fourier mode. Classically, Legendre polynomials expansions provide the kernels of these reduced transfer equations. For highly peaked phase functions, the Legendre series expansion converges very slowly. We show in this paper that this expansion can advantageously be replaced by a direct numerical evaluation using the trapezoidal rule. The improvement afforded by this direct evaluation yields highly accurate results with orders of magnitude fewer arithmetic operations than the Legendre series, avoids the very slow convergence of the Legendre series and exploits instead the rapid decay of the Fourier coefficients for exponential convergence, and finally bypasses the need for phase function truncations.