Discrete Ordinates Method

In the theory of radiative transfer, of either thermal or neutron radiation, a position and direction-dependent intensity function is usually sought for the description of the radiation field. The intensity field can in principle be solved from the integrodifferential radiative transfer equation (RTE), but an exact solution is usually impossible and even in the case of geometrically simple systems can contain unusual special functions such as the Chandrasekhar's H-function and Chandrasekhar's X- and Y-functions. The method of discrete ordinates, or the Sn method, is one way to approximately solve the RTE by discretizing both the xyz-domain and the angular variables that specify the direction of radiation. The methods were developed by Subrahmanyan Chandrasekhar when he was working on radiative transfer. In the theory of radiative transfer, of either thermal or neutron radiation, a position and direction-dependent intensity function is usually sought for the description of the radiation field. The intensity field can in principle be solved from the integrodifferential radiative transfer equation (RTE), but an exact solution is usually impossible and even in the case of geometrically simple systems can contain unusual special functions such as the Chandrasekhar's H-function and Chandrasekhar's X- and Y-functions. The method of discrete ordinates, or the Sn method, is one way to approximately solve the RTE by discretizing both the xyz-domain and the angular variables that specify the direction of radiation. The methods were developed by Subrahmanyan Chandrasekhar when he was working on radiative transfer. In the case of time-independent monochromatic radiation in an elastically scattering medium, the RTE is s ⋅ ∇ I ( r , s ) = κ I b ( r ) − β I ( r , s ) + σ 4 π ∫ 4 π I ( r , s ) Φ ( s , s ′ ) d Ω ′ {displaystyle mathbf {s} cdot abla I(mathbf {r} ,mathbf {s} )=kappa I_{b}(mathbf {r} )-eta I(mathbf {r} ,mathbf {s} )+{frac {sigma }{4pi }}int _{4pi }I(mathbf {r} ,mathbf {s} )Phi (mathbf {s} ,mathbf {s'} )dOmega '} where the first term on the RHS is the contribution of emission, the second term the contribution of absorption and the last term is the contribution from scattering in the medium. The variable s {displaystyle mathbf {s} } is a unit vector that specifies the direction of radiation and the variable s ′ {displaystyle mathbf {s'} } is a dummy integration variable for the calculation of scattering from direction s {displaystyle mathbf {s} } to direction s ′ {displaystyle mathbf {s'} } . In the discrete ordinates method, the full solid angle of 4 π {displaystyle 4pi } is divided to some number of discrete angular intervals, and the continuous direction variable s {displaystyle mathbf {s} } is replaced by a discrete set of direction vectors s j {displaystyle mathbf {s_{j}} } . Then the scattering integral in the RTE, which makes the solution problematic, becomes a sum ∑ j = 1 n w j I ( r , s j ) Φ ( s j , s i ) {displaystyle sum _{j=1}^{n}w_{j}I(mathbf {r} ,mathbf {s_{j}} )Phi (mathbf {s_{j}} ,mathbf {s_{i}} )} where the numbers w j {displaystyle w_{j}} are weighting coefficients for the different direction vectors. With this the RTE becomes a linear system of equations for a multi-index object, the number of indices depending on the dimensionality and symmetry properties of the problem. It is possible to solve the resulting linear system directly with Gauss-Jordan elimination, but this is problematic due to the large memory requirement for storing the matrix of the linear system. Another way is to use iterative methods, where the required number of iterations for a given degree of accuracy depends on the strength of scattering. The discrete ordinates method, or some variation of it, is applied for solving radiation intensities in several physics and engineering simulation programs, such as COMSOL Multiphysics or the Fire Dynamics Simulator.