Consistent Second Moment Methods with Scalable Linear Solvers for Radiation Transport

Abstract

Second Moment Methods (SMMs) are developed that are consistent with the Discontinuous Galerkin (DG) spatial discretization of the discrete ordinates (or \Sn) transport equations. The low-order (LO) diffusion system of equations is discretized with fully consistent \Pone, Local Discontinuous Galerkin (LDG), and Interior Penalty (IP) methods. A discrete residual approach is used to derive SMM correction terms that make each of the LO systems consistent with the high-order (HO) discretization. We show that the consistent methods are more accurate and have better solution quality than independently discretized LO systems, that they preserve the diffusion limit, and that the LDG and IP consistent SMMs can be scalably solved in parallel on a challenging, multi-material benchmark problem.

Figures (10)

• Figure 1: Sparsity plots depicting DG discretizations of the Poisson equation in first-order form. The degrees of freedom are ordered such that the first block row corresponds to the current and the second to the scalar flux, as in \ref{['eq:block']}. Plot (a) shows the sparsity pattern for a P$_1$-like numerical flux. Observe that the top diagonal block is coupled to its neighbors, preventing the efficient elimination of the current. Plot (b) shows the sparsity pattern associated with the approach taken by the LDG and IP discretizations where the coupling in the normal component of the current is severed and replaced with alternate stabilization terms, allowing the current to be efficiently eliminated on each element. Plot (c) shows the sparsity pattern that results after eliminating the current with block Gaussian elimination.
• Figure 1: Errors on the MMS problem. The P$_1$, LDG, and IP methods were equivalent to the iterative tolerance of $10^{-10}$.
• Figure 2: Consistency between the HO and LO solution variables on the MMS problem. The P$_1$, LDG, and IP methods were equivalent to below the iterative tolerance of $10^{-10}$.
• Figure 3: Lineouts of the scalar flux along $y=0.5\cm$ for the consistent SMMs as $\epsilon \rightarrow 0$.
• Figure 4: Depiction of the geometry of the crooked pipe benchmark problem. The optically thin pipe and optically thick wall are depicted as gray and black, respectively.