Numerical solutions of matrix Riccati equations for radiative transfer in a plane-parallel geometry

In this paper, we conduct numerical experiments with matrix Riccati equations (MREs) which describe the reflection (R) and transmission CT) matrices of the specific intensities in a layer containing randomly distributed scattering particles. The theoretical formulation of MREs is discussed in our previous paper where we show that R and T for a thick layer can be efficiently computed by successively doubling R and T matrices for a thin layer (with small optical thickness tau(Delta)). We can compute R(tau(Delta)) and T(tau(Delta)) very accurately using either a fourth-order Runge-Kutta scheme or the fourth-order iterative solution. The differences between these results and those computed by the eigenmode expansion technique (EMET) are very small (< 0.1%). Although the MRE formulation cannot be extended to handle the inhomogeneous term (source term) in the differential equation, we show that the force term can be reformulated as an equivalent boundary condition which is consistent with MRE methods. MRE methods offer an alternative way of solving plane-parallel radiative transport problems. For large problems that do not fit into computer memory, the MRE method provides a significant reduction in computer memory and computational time.