Symbolic implicit Monte Carlo radiation transport in the difference formulation: a piecewise constant discretization
The paper deals with equations of radiation transport for thermal photons. In particular, the slab geometry is considered. In this case the transport equation for the specific intensity \(I(x,t;\nu;\mu)\) reads \[ \frac{1}{c} \frac{\partial I}{\partial t} + \mu \frac{\partial I}{\partial x} = - \sigma'_{a}(\nu,T(x,t)) [I - B(\nu,T(x,t))] \;, \] where \(t\) is time, \(x\) is coordinate in the direction perpendicular to the slab, \(c\) is the speed of light, \(\nu\) is the radiation frequency, \(\mu = \cos \theta\), where \(\theta\) is the propogation angle, \(\sigma'_{a}\) is the absorption coefficient, and \(B(\nu,T(x,t))\) is the thermal (Planck) distribution at the material temperature \(T(x,t)\). The solution of such transport equations in thick media is a notoriously difficult problem. To overcome this difficulty, the authors use the difference formulation proposed in an earlier work. In this scheme, the transport equation is transformed by considering a new field \(D(x,t;\nu;\mu) = I(x,t;\nu;\mu) - B(\nu,T(x,t))\) that is the deviation of the local intensity from the local equilibrium intensity. It yields the transport equation \[ \frac{1}{c} \frac{\partial D}{\partial t} + \mu \frac{\partial D}{\partial x} = - \sigma'_{a} D - \frac{1}{c} \frac{\partial B}{\partial t} - \mu \frac{\partial B}{\partial x} \] for the new variable \(D\), which is solved by means of the symbolic implicit Monte Carlo method using a piecewise constant treatment of the material temperature. Several test problems are considered and comparisons are made between the standard formulation and the difference formulation. It is shown that the difference formulation leads to significant reduction of the noise and has a great computational advantage. The results also well agree with the analytical solution available for a linearised problem.
scientific article; zbMATH DE number 2175591
scientific article