Radiative Transfer

The equations of radiative transfer describe the evolution of radiative intensity, which depends on time, space, direction of flight (of photons, electrons, etc.), and possibly frequency/energy. Absorption and scattering properties of the background material (think of particles flying through the human body in radiotherapy, or through concrete in a protective hull around a radiation source) are described by kernels in integral terms in the governing equations. Due to the high-dimensional phase space, the direct numerical simulation of the radiative transfer equations is in general very costly, even for simple geometries.

Monte-Carlo methods are one class of approaches to approximate the equations of radiative transfer. As commonly encountered, these types of methods tend to yield good quality approximations even with relatively straightforward sampling strategies. Important weaknesses of Monte-Carlo methods are: a) a certain level of statistical noise is always present in the approximate solution; and b) they provide a black-box solution method, whose inversion or adjunction (e.g., as needed for optimization and control) is not easily possible. Because optimization and control is important in many applications, it is important to study deterministic methods, that provide direct approximations to the partial (integro-)differential equation that describes the radiative transfer problem.

Moment methods: Moment models expand the radiative intensity as a Fourier series in the direction variable, whose coefficients are direction-independent. Thus, the radiative transfer equation is transformed into an infinite system of macroscopic evolution equations for the moments. The lowest couple of moment are typically the quantities of interest. This infinite moment system is then truncated (see research on the moment closure problem in radiative transfer) after the N-th moment, by modeling the (N+1)-st moment in terms of the lower moments. Examples considered in my research are:

• the P_N closure, leading to spherical Harmonics methods;
• the diffusion correction closure, D_N, that possesses a diffusive term in the highest moment;
• crescendo-diffusion and crescendo-diffusion correction closures;
• the simplified P_N closure, leading to the SP_N equations, as well as the reordered P_N, or RP_N equations;
• nonlinear closures, most prominently the maximum entropy closure, leading to the M_N; and
• Kershaw closures, that lead to the K_N equations, which approximate the M_N equations.

The P_N equations, as well as the time-dependent SP_N equations, are (for arbitrary moment order ) efficiently implemented in the StaRMAP software project.
For work on the D_N equations, please consult the work on the moment closure problem via optimal prediction, and the paper in [Kinet. Relat. Models 4(3):717-733, 2011].
The and R_N equations and crescendo-diffusion are introduced in the paper [Continuum Mech. Thermodyn. 21(6):511-527, 2009].
The time-dependent SP_N equations are studied in detail in the paper [J. Comput. Phys. 238:315-336, 2013].
Discontinuous Galerkin methods for maximum entropy methods, and Kershaw closures, are the subject of current research.

Related Publications

• B. Seibold, M. Frank, StaRMAP - A second order staggered grid method for spherical harmonics moment equations of radiative transfer
• E. Olbrant, E. W. Larsen, M. Frank, B. Seibold, Asymptotic derivation and numerical investigation of time-dependent simplified P_N equations, J. Comput. Phys., Vol. 238, 2013, pp. 315-336.
• M. Frank, B. Seibold, Optimal prediction for radiative transfer: A new perspective on moment closure, Kinet. Relat. Models, Vol. 4, No. 3, 2011, pp. 717-733.
• B. Seibold, M. Frank, Optimal prediction for moment models: Crescendo diffusion and reordered equations, Continuum Mech. Thermodyn., Vol. 21, No. 6, 2009, pp. 511-527.