Cellwise Block Iteration as a Multigrid Smoother for Discrete-Ordinates Radiation-Transport Calculations

Abstract

We improve the convergence properties of cellwise block iteration for discrete-ordinates radiation-transport calculations by adapting it for use as a smoother within a multigrid method. Cellwise block iteration by itself converges very slowly for optically thin spatial cells. However, multigrid methods involve a sequence of increasingly coarser grids such that cells on the coarsest grid should be optically thick, for which cellwise block iteration converges quickly. This fast convergence on the coarsest grid should enable fast convergence overall. A novel aspect of this paper is that, along with the usual first-order form of the transport equation, we also consider the Self-Adjoint Angular Flux (SAAF) form. We present numerical results generated using several multigrid methods based on cellwise block iteration for smoothing that demonstrate our approach yields robust convergence regardless of cell optical thickness as specified by the finest grid as well as for heterogeneous media. In addition, we find that the multigrid methods for the SAAF form of the transport equation have superior convergence properties as compared to those for the firstorder form.

AIR multigrid with GMRES polynomials (AIRG) and additive preconditioners for Boltzmann transport

“…This section details the iterative method we use to solve (11). For simplicity we begin by writing the unseparated form (9) with our sparsified D−1 and introduce a preconditioner M −1 on the right to give…”

“…These methods do not use GS/sweep smoothers however and do not scale well in the streaming limit where σ s tends to zero. Similarly, most multigrid methods in the literature that achieve good performance for the BTE have relied on either blockbased smoothers, often on a cell/element, which do not scale with increasing angle size but which perform well in the scattering limit [6,7,8,9], or GS/sweeps as smoothers which perform well in the streaming limit [10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25] but as mentioned can be difficult to parallelise on unstructured grids. Developing multigrid methods which perform well with local smoothers is difficult, particularly as hyperbolic problems have long proved difficult to solve with multigrid due to the lack of developed theory for non-SPD matrices.…”

A comparison of element agglomeration algorithms for unstructured geometric multigrid

“…Traditionally in radiation transport a common smoother choice is a Gauss-Seidel (known as a "sweep"), as the use of structured grids with DG-FEM in space and a discrete ordinates discretisation (known as S n ) in angle results in the streaming/removal part of the discretised transport equation (i.e., the LHS in (1)) having lower triangular structure. This pure-advection problem can therefore be solved exactly in one Gauss-Seidel iteration, making a powerful smoother for (1) (e.g., see [22,23,24,25,26]).…”