Numerical experiments on the Helmholtz equation derived from the solar radiation transfer equation in three-dimensional space

A numerical scheme using the finite-difference approach to solve the modified Helmholtz partial differential equation derived from the solar radiative transfer equation is developed and tested along with the method of evaluating the slant-path optical depth. For overhead solar incidence, we obtain good agreement between the finite-difference approach and the semianalytical solution in terms of the local intensity, local flux, average intensity, and average flux. For face-parallel oblique solar incidence in which the semianalytical method is not applicable, we compare the results with those of previous studies utilizing the Monte Carlo method and the approximate semianalytical method. We show that the present numerical scheme can be applied to any incident solar angle which the approximate semianalytical method is incapable of. Comparisons with results from Monte Carlo method reveal reasonable agreement for the averaged intensity and flux density.