The foundations of the algorithmic scale-bridging ideas we will build on reside in statistical mechanics, kinetic theory, and transport theory. These ideas have matured in steady-state neutron transport for nuclear reactors. In steady-state neutron transport, using this algorithm, the transport (Boltzmann) equation would be referred to as the high order (HO) problem and is six-dimensional (6-D). The low order (LO) problem is constructed from combining the 0th and 1st phase-space moments of the transport equation. This LO equation is only three-dimensional (3-D), and thus is a reduced space representation of the HO problem. In the appropriate limit, the LO problem asymptotes to the neutron diffusion equation. This LO (moment-based) problem is used to accelerate the convergence of a brute force transport source iteration via bridging the transport and diffusion scales. This algorithm has characteristics of a multigrid method. The LO problem lives in a reduced dimensional space (3-D). It is an efficient way to relax the system-scale “diffusion” features that naturally reside in the HO problem. The more expensive HO problem (which lives in 6-D space) is only required to relax the fine-scale transport features. The HO and LO problem are connected with restriction and prolongation processes related to the moments. Additionally, the LO problem is frequently on a coarser configuration space mesh as compared to the HO problem. While this algorithmic idea is frequently referred to as an accelerator for the transport (HO) solver, it can also be viewed as a powerful physics-based scale-bridging algorithm.
Development and use of similar moment-based acceleration methods can be found in a variety of application areas such as neutron transport, thermal radiation (photon) transport, kinetic plasma simulation, rarified gas dynamics, and material science. We have ongoing efforts to advance these algorithms in the areas of thermal radiation transport, kinetic plasma simulation and neutron transport. In this talk we will draw the majority of our example results from photon and neutron transport.
While much is known about these algorithms, and they are highly utilized by some, further algorithm development and analysis remains. In this talk we will provide an algorithmic overview with some application specific details. Next we will discuss two important algorithmic issues. The first issue is enforcing discrete consistency between the HO solver and the LO solver. The second issue is related to challenges that arise when the HO solver is stochastic, such as the use of Monte Carlo as the HO solver for photon transport. At that point we have a hybrid algorithm with both deterministic features and stochastic features. Finally, this algorithm has a number of characteristics that will be beneficial in moving to exascale computing. This general moment-based HO/LO algorithm has natural heterogeneity and concurrency. We will discuss our ongoing computation co-design efforts on emerging architectures to maximize the impact of these algorithms for future multiphysics, multiscale simulation.