The time-dependent radiation transport equation describes the dynamics of radiation traveling through and interacting with a background medium. These dynamics are important in a diversity of fields including nuclear reactor kinetics, stellar evolution, and inertial confinement fusion. Except for trivial problems, the transport equation must be solved numerically. This research is concerned with developing a new deterministic time discretization for numerical solutions of the radiation transport equation.
To preserve maximal parallelism, a deterministic transport method must maintain locality, meaning that the solution at a point in space is dependent only upon information that is locally available. Furthermore, computational efficiency requires that a method be unconditionally stable, meaning that it provides positive, physically permissible solutions for time steps of any length.
Existing unconditionally stable radiation transport methods require mesh sweeps, which make the methods non-local and inhibit their parallelism, thereby reducing their efficiency on large supercomputers. We present a new Staggered-Block Jacobi (SBJ) method, which produces unconditionally stable numerical solutions while maintaining locality.
The SBJ time discretization operates by forming blocks of cells. In one dimension, a block is composed of two cells. The incident information into the block is evaluated at the beginning of the time step. This decouples every block, and allows the solution in the blocks to be computed in parallel.
We apply the SBJ method to the linear diffusion and transport equations, as well as the linearized thermal radiation transport equations. We find that the SBJ time discretization, applied to the linear diffusion and transport equations, produces methods that are accurate and efficient when the particle wave advances about 20% of a cell per time step, i.e., where the time steps are small or the problem is optically thick. In the case of the thermal radiation transport equations, we find that the SBJ method is accurate and efficient whenever a time step length is chosen such that the error resulting from the linearization is small. The SBJ method should be more efficient than sweep-based methods for many problems of interest on massively parallel computers.