Elliptic formulation of the Simplified Spherical Harmonics Method in radiative heat transfer

The SimplifiedPN-Approximation was introduced some 50 years ago, as an intuitive three-dimensional extension to the one-dimensional slab PN-formulation. The set of simplified-PN or SPN equations was formulated, such that they reduced to the standard PN-approximation for a one-dimensional slab, resulting in 2(N+1) simultaneous first order differential equations, subject to a modified set of Marshak boundary conditions. In this article the method is developed further for radiative heat transfer applications by reducing the governing equations to a small set of simultaneous second-order, elliptic PDEs, similar to the simple P1 equation. Each of the equations is subject to P1-type boundary conditions. Several two-dimensional examples are presented, including scattering, strongly inhomogeneous temperatures and absorption coefficients.