Adding and doubling solution to the 1D Fokker-Planck Equation

Abstract

The 1D Fokker-Planck equation (FPE) plays a major role in the propagation of light in the universe. It specifically describes small angle scattering of photons (and electrons) as they travel in participating media. In particular, the differential scattering term representing the phase function scattering law enables the small angle scattering. This term also makes the FPE a challenge to solve in the discrete ordinate sense. Our approach utilizes adding and doubling, which has been successfully applied since the 1960s to solve the linear Boltzmann equation. With the help of Morel’s discrete ordinate equivalence of the angular Laplacian, the FPE becomes similar to the discrete ordinates equation of linear transport theory. We then take advantage of the similarity through adding and doubling for its solution.