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This paper derives and analyzes new diffusion synthetic acceleration (DSA) preconditioners for the SN transport equation when discretized with a high-order (HO) discontinuous Galerkin (DG) discretization. DSA preconditioners address the need to accelerate solving SN transport when the mean free path $\varepsilon$ of particles is small and the condition number of the SN transport equation scales like $O(\varepsilon^{-2})$. By expanding the SN transport operator in $\varepsilon$ and employing a singular matrix perturbation analysis, we derive a DSA matrix that reduces to the symmetric interior penalty (SIP) DG discretization of diffusion when the mesh is first-order and the total opacity is constant. We prove that preconditioning HO DG SN transport with the SIP DSA matrix results in an $O(\varepsilon)$ perturbation of the identity, and fixed-point iteration therefore converges rapidly for optically thick problems. However, the SIP DSA matrix is conditioned like $O(\varepsilon^{-1})$, making it difficult to invert for small $\varepsilon$. We further derive a new two-part, additive DSA preconditioner based on a continuous discretization of diffusion-reaction, which has a condition number independent of $\varepsilon$, and prove that this DSA variant has the same theoretical efficiency as the SIP DSA preconditioner in the optically thick limit. The analysis is extended to the case of HO (curved) meshes, where so-called mesh cycles can result from elements both being upwind of each other. In particular, we prove that performing two additional transport sweeps between DSA steps yields the same theoretical conditioning of fixed-point iterations as in the cycle-free case. Theoretical results are validated by numerical experiments on HO, highly curved meshes generated from an ALE hydrodynamics code, where the additional inner sweeps between DSA steps offer up to a 4x reduction in total sweeps., Comment: 26 pages, Accepted to SISC

We propose a graph-based sweep algorithm for solving the steady state, mono-energetic discrete ordinates on meshes of high-order curved mesh elements. Our spatial discretization consists of arbitrarily high-order discontinuous Galerkin finite elements using upwinding at mesh element faces. To determine mesh element sweep ordering, we define a directed, weighted graph whose vertices correspond to mesh elements, and whose edges correspond to mesh element upwind dependencies. This graph is made acyclic by removing select edges in a way that approximately minimizes the sum of removed edge weights. Once the set of removed edges is determined, transport sweeps are performed by lagging the upwind dependency associated with the removed edges. The proposed algorithm is tested on several 2D and 3D meshes composed of high-order curved mesh elements.

We present a numerical method for solving the time-independent thermal radiative transfer (TRT) equation or the neutron transport (NT) equation when the opacity or cross-section varies rapidly in energy (frequency). The approach is based on a rigorous homogenization of the TRT/NT equation in the energy (frequency) variable. Discretization of the homogenized TRT/NT equation results in a multigroup-type system, and can therefore be solved by standard methods. We demonstrate the accuracy and efficiency of the approach on three model problems. First we consider the Elsasser band model with constant temperature and a small line spacing. Second, we consider a neutron transport application for fast neutrons incident on iron, where the characteristic resonance spacing necessitates about 16,000 energy discretization parameters if Planck-weighted cross sections are used. Third, we consider an atmospheric TRT problem with an opacity corresponding to water vapor. For all three problems, we demonstrate that we can achieve between 0.1 and 1 percent relative error in the solution, and with several orders of magnitude fewer parameters than a standard multigroup formulation with a comparable accuracy.

The manuscript presents a new technique for computing the exponential of skew-Hermitian operators. Principal advantages of the proposed method include: stability even for large time-steps, the possibility to parallelize in time over many characteristic wavelengths, and large speed-ups over existing methods in situations where simulation over long times are required. Numerical examples involving the 2D rotating shallow water equations and the 2D wave equation in an inhomogenous medium are presented, and the method is compared to the 4th order Runge-Kutta (RK4) method and to the use of Chebyshev polynomials. Is is demonstrated that the new method achieves high accuracy over long time intervals, and with speeds that are orders of magnitude faster than both RK4 and the use of Chebyshev polynomials.

The need to compute small con-eigenvalues and the associated con-eigenvectors of positive-definite Cauchy matrices naturally arises when constructing rational approximations with a (near) optimally small $L^{\infty}$ error. Specifically, given a rational function with $n$ poles in the unit disk, a rational approximation with $m\ll n$ poles in the unit disk may be obtained from the $m$th con-eigenvector of an $n\times n$ Cauchy matrix, where the associated con-eigenvalue $\lambda_{m}>0$ gives the approximation error in the $L^{\infty}$ norm. Unfortunately, standard algorithms do not accurately compute small con-eigenvalues (and the associated con-eigenvectors) and, in particular, yield few or no correct digits for con-eigenvalues smaller than the machine roundoff. We develop a fast and accurate algorithm for computing con-eigenvalues and con-eigenvectors of positive-definite Cauchy matrices, yielding even the tiniest con-eigenvalues with high relative accuracy. The algorithm computes the $m$th con-eigenvalue in $\mathcal{O}(m^{2}n)$ operations and, since the con-eigenvalues of positive-definite Cauchy matrices decay exponentially fast, we obtain (near) optimal rational approximations in $\mathcal{O}(n(\log\delta^{-1})^{2})$ operations, where $\delta$ is the approximation error in the $L^{\infty}$ norm. We derive error bounds demonstrating high relative accuracy of the computed con-eigenvalues and the high accuracy of the unit con-eigenvectors. We also provide examples of using the algorithm to compute (near) optimal rational approximations of functions with singularities and sharp transitions, where approximation errors close to machine precision are obtained. Finally, we present numerical tests on random (complex-valued) Cauchy matrices to show that the algorithm computes all the con-eigenvalues and con-eigenvectors with nearly full precision.