Small angle approximation and total solution to the transport equation in the mesh algorithm of the discrete ordinate method

The method to solve the direct problem for the transport equation in a slab layer illuminated by the monodirectional beam under a peaked-forward scattering phase function is presented. The solution is decomposed into the singular component (small angle approximation) and the regular one. The new method to the small angle approximation and source in the problem for the regular component is suggested.
The small angle approximation is defined for directions near the direction of the incident beam. It is obtained via the mesh equations and solved by the iteration method. The mesh equations are find via the characteristics method. The regular component is defined for all directions. It is found by solving the transport equation with the source defined via the small angle approximation; this problem is solved by the mesh scheme of the discrete ordinate method by the code Радуга-6.2.
Reflectance and transmittance for sea water and cloud layers, obtained by the new method and the direct calculations are presented. The forward peak of the sea water phase functions is more than the back peak by 5-7 orders. This value is about 3 orders for the cloud phase function.
One shows the new method permits use sparse angular meshs in high accurate calculations.