Multi-scale methods for the solution of the radiative transfer equation

- Multi-scale models are applied to 3D transient problems with collimated radiation.
- Two multi-scale models are used: micro-macro and hybrid transport-diffusion.
- Their performance is compared with that of the radiative transfer equation.
- Performance depends on order of accuracy of spatial/temporal discretization schemes.
- Performance depends also on the radiative properties of the medium.

Various methods have been developed and tested over the years to solve the radiative transfer equation (RTE) with different results and trade-offs. Although the RTE is extensively used, the approximate diffusion equation is sometimes preferred, particularly in optically thick media, due to the lower computational requirements. Recently, multi-scale models, namely the domain decomposition methods, the micro-macro model and the hybrid transport-diffusion model, have been proposed as an alternative to the RTE. In domain decomposition methods, the domain is split into two subdomains, namely a mesoscopic subdomain where the RTE is solved and a macroscopic subdomain where the diffusion equation is solved. In the micro-macro and hybrid transport-diffusion models, the radiation intensity is decomposed into a macroscopic component and a mesoscopic one. In both cases, the aim is to reduce the computational requirements, while maintaining the accuracy, or to improve the accuracy for similar computational requirements. In this paper, these multi-scale methods are described, and the application of the micro-macro and hybrid transport-diffusion models to three-dimensional transient problems is reported. It is shown that when the diffusion approximation is accurate, but not over the entire domain, the multi-scale methods may improve the solution accuracy in comparison with the solution of the RTE. The order of accuracy of the numerical schemes and the radiative properties of the medium play a key role in the performance of the multi-scale methods.