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We present high-order, finite element-based Second Moment Methods (SMMs) for solving radiation transport problems in two spatial dimensions. We leverage the close connection between the Variable Eddington Factor (VEF) method and SMM to convert existing discretizations of the VEF moment system to discretizations of the SMM moment system. The moment discretizations are coupled to a high-order Discontinuous Galerkin discretization of the Discrete Ordinates transport equations. We show that the resulting methods achieve high-order accuracy on high-order (curved) meshes, preserve the thick diffusion limit, and are effective on a challenging multi-material problem both in outer fixed-point iterations and in inner preconditioned iterative solver iterations for the discrete moment systems. We also present parallel scaling results and provide direct comparisons to the VEF algorithms the SMM algorithms were derived from.

We apply high-order mixed finite element discretization techniques and their associated preconditioned iterative solvers to the Variable Eddington Factor (VEF) equations in two spatial dimensions. The mixed finite element VEF discretizations are coupled to a high-order Discontinuous Galerkin (DG) discretization of the Discrete Ordinates transport equation to form effective linear transport algorithms that are compatible with high-order (curved) meshes. This combination of VEF and transport discretizations is motivated by the use of high-order mixed finite element methods in hydrodynamics calculations at the Lawrence Livermore National Laboratory. Due to the mathematical structure of the VEF equations, the standard Raviart Thomas (RT) mixed finite elements cannot be used to approximate the vector variable in the VEF equations. Instead, we investigate three alternatives based on the use of continuous finite elements for each vector component, a non-conforming RT approach where DG-like techniques are used, and a hybridized RT method. We present numerical results that demonstrate high-order accuracy, compatibility with curved meshes, and robust and efficient convergence in iteratively solving the coupled transport-VEF system and in the preconditioned linear solvers used to invert the discretized VEF equations.

We present a family of discretizations for the Variable Eddington Factor (VEF) equations that have high-order accuracy on curved meshes and efficient preconditioned iterative solvers. The VEF discretizations are combined with a high-order Discontinuous Galerkin transport discretization to form an effective high-order, linear transport method. The VEF discretizations are derived by extending the unified analysis of Discontinuous Galerkin methods for elliptic problems to the VEF equations. This framework is used to define analogs of the interior penalty, second method of Bassi and Rebay, minimal dissipation local Discontinuous Galerkin, and continuous finite element methods. The analysis of subspace correction preconditioners, which use a continuous operator to iteratively precondition the discontinuous discretization, is extended to the case of the non-symmetric VEF system. Numerical results demonstrate that the VEF discretizations have arbitrary-order accuracy on curved meshes, preserve the thick diffusion limit, and are effective on a proxy problem from thermal radiative transfer in both outer transport iterations and inner preconditioned linear solver iterations. In addition, a parallel weak scaling study of the interior penalty VEF discretization demonstrates the scalability of the method out to 1152 processors.

We present a new approach for solving high-order thermal radiative transfer (TRT) using the Variable Eddington Factor (VEF) method (also known as quasidiffusion). Our approach leverages the VEF equations, which consist of the first and second moments of the $S_N$ transport equation, to more efficiently compute the TRT solution for each time step. The scheme consists of two loops - an outer loop to converge the Eddington tensor and an inner loop to converge the iteration between the temperature equation and the VEF system. By converging the outer iteration, one obtains the fully implicit TRT solution for the given time step with a relatively low number of transport sweeps. However, one could choose to perform exactly one outer iteration (and therefore exactly one sweep) per time step, resulting in a semi-implicit scheme that is both highly efficient and robust. Our results indicate that the error between the one-sweep and fully implicit variants of our scheme may be small enough for consideration in many problems of interest., Comment: Submitted to the American Nuclear Society Mathematics & Computation 2021 Conference (The International Conference on Mathematics and Computational Methods Applied to Nuclear Science and Engineering, Raleigh, North Carolina, October 3-7, 2021)

We present a method for accelerating discrete ordinates radiative transfer calculations for radiative transfer. Our method works with nonlinear positivity fixes, in contrast to most acceleration schemes. The method is based on the dynamic mode decomposition (DMD) and using a sequence of rank-one updates to compute the singular value decomposition needed for DMD. Using a sequential method allows us to automatically determine the number of solution vectors to include in the DMD acceleration. We present results for slab geometry discrete ordinates calculations with the standard temperature linearization. Compared with positive source iteration, our results demonstrate that our acceleration method reduces the number of transport sweeps required to solve the problem by a factor of about 3 on a standard diffusive Marshak wave problem, a factor of several thousand on a cooling problem where the effective scattering ratio approaches unity, and a factor of 20 improvement in a realistic, multimaterial radiating shock problem., Comment: 29 pages, 8 figures

A standard approach to solving the S$_N$ transport equations is to use source iteration with diffusion synthetic acceleration (DSA). Although this approach is widely used and effective on many problems, there remain some practical issues with DSA preconditioning, particularly on highly heterogeneous domains. For large-scale parallel simulation, it is critical that both (i) preconditioned source iteration converges rapidly, and (ii) the action of the DSA preconditioner can be applied using fast, scalable solvers, such as algebraic multigrid (AMG). For heterogeneous domains, these two interests can be at odds. In particular, there exist DSA diffusion discretizations that can be solved rapidly using AMG, but they do not always yield robust/fast convergence of the larger source iteration. Conversely, there exist robust DSA discretizations where source iteration converges rapidly on difficult heterogeneous problems, but fast parallel solvers like AMG tend to struggle applying the action of such operators. Moreover, very few current methods for the solution of deterministic transport are compatible with voids. This paper develops a new heterogeneous DSA preconditioner based on only preconditioning the optically thick subdomains. The resulting method proves robust on a variety of heterogeneous transport problems, including a linearized hohlraum mesh related to inertial confinement fusion. Moreover, the action of the preconditioner is easily computed using $\mathcal{O}(1)$ AMG iterations, {convergence of the transport iteration typically requires $2-5\times$ less iterations than current state-of-the-art ``full DSA,'' and the proposed method is} trivially compatible with voids. On the hohlraum problem, rapid convergence is obtained by preconditioning less than 3\% of the mesh elements with $5-10$ AMG iterations., Comment: 23 pages, accepted to NSE

We present a new flux-fixup approach for arbitrarily high-order discontinuous Galerkin discretizations of the SN transport equation. This approach is sweep-compatible: as the transport sweep is performed, a local quadratic programming (QP) problem is solved in each spatial cell to ensure that the solution satisfies certain physical constraints, including local particle balance. The constraints can be chosen in two ways, leading to two variants of the method: QP Zero (QPZ) and QP Maximum Principle (QPMP). The coefficients of the solution are constrained to be nonnegative in QPZ, and they are constrained by an approximate discrete maximum principle in QPMP. There are two primary takeaways in this paper. First, it is shown that the QPMP method, when used with the positive Bernstein basis, eliminates negativities, preserves high-order accuracy for smooth problems, and significantly dampens unphysical oscillations in the solution. The second takeaway is that the Variable Eddington Factor (VEF) method can be used to accelerate the convergence of source iteration with fixup for problems with optically thick regions. Regardless of whether a fixup is applied, source iteration converges slowly when optically thick regions are present and acceleration is needed. When VEF is combined with fixed-up transport sweeps, the result is a scheme that produces a nonnegative solution, converges independently of the mean free path, and, in the case of the QPMP fixup, adheres to an approximate discrete maximum principle., Comment: Revised draft submitted to Journal of Computational Physics

We present a novel acceleration technique for improving the convergence of source iteration for discrete ordinates transport calculations. Our approach uses the idea of the dynamic mode decomposition (DMD) to estimate the slowly decaying modes from source iteration and remove them from the solution. The memory cost of our acceleration technique is that the scalar flux for a number of iterations must be stored; the computational cost is a singular value decomposition of a matrix comprised of those stored scalar fluxes.On 1-D slab geometry problems we observe an order of magnitude reduction in the number of transport sweeps required compared to source iteration and that the number of sweeps required is independent of the scattering ratio in the problem. These observations hold for an extremely heterogeneous problem and a 2-D problem. In 2-D we do observe that the effectiveness of the approach slowly degrades as the mesh is refined, but is still about one order of magnitude faster than source iteration., Comment: 4 pages