Radiative Transfer Verification Solutions

Physical processes involving fluids in which radiation is strong enough to convey significant energy require some radiative heat transfer (RT) description to model. It is standard to assume that photons may be described as particles and that they travel in straight lines between collisions. This allows the radiation to be modeled by the linear Boltzmann transport equation, often called simply ``the transport equation,'' or ``transport.'' The transport equation has seven independent variables, one for time, three for direction, two for angle, and one for energy. Until recently, even advanced computers did not have the computational power to solve the full transport equation. This compelled simplifications. One of these, the P1 approximation, involves taking angular moments of the transport equation with spherical harmonics and solving the resulting angular-independent coupled system. Monte Carlo methods were developed to circumvent the dimensionality question, but they converge slowly. Now, however, advances in computing allow ambitious codes that simulate the full transport equation in all of the independent variables to be coupled with complex physics simulations and run on state of the art machines. The advancement of simulation code makes the issue of verification even more pertinent. Analytic solution methods have historically been the focus of verification in RT and from these methods have come many invaluable solutions. However, it can be argued that much of the low hanging fruit on the analytic solution tree has been picked. A modification in approach therefore is in order. This approach, presented here, is a discretized benchmark generator which is able to achieve the same levels of accuracy as the semi-analytic benchmarks in the field. To solve the one-dimensional, energy-independent transport equation, which is still a challenging integro-differential equation, the angular dependence is discretized by discrete ordinates (SN). Spatial variation is resolved by a spectral elements method with moving mesh edges. The integration of the resulting ordinary differential equations in time is accomplished with a Runge-Kutta integrator. In addition to this, the uncollided part of the solution is separated out and solved analytically. This method is tested on existing analytic solutions and novel solutions developed specifically for this purpose and extended to provide new benchmarks for a nonlinear RT problem and an uncoupled blast wave imaging problem on one dimensional Cartesian coordinates, also called ``slab geometry''. Derivation of a one dimensional spherical geometry version of the method is also proven on known solutions. Additionally, new asymptotic analysis is used to verify that the method is asymptotic preserving in the early time initial layer and late time diffusive regimes.