Numerical analysis of a spherical harmonic discontinuous Galerkin method for scaled radiative transfer equations with isotropic scattering

In highly diffusion regimes when the mean free path $\varepsilon$ tends to zero, the radiative transfer equation has an asymptotic behavior which is governed by a diffusion equation and the corresponding boundary condition. Generally, a numerical scheme for solving this problem has the truncation error containing an $\varepsilon^{-1}$ contribution, that leads to a nonuniform convergence for small $\varepsilon$. Such phenomenons require high resolutions of discretizations, which degrades the performance of the numerical scheme in the diffusion limit. In this paper, we first provide a--priori estimates for the scaled spherical harmonic ($P_N$) radiative transfer equation. Then we present an error analysis for the spherical harmonic discontinuous Galerkin (DG) method of the scaled radiative transfer equation showing that, under some mild assumptions, its solutions converge uniformly in $\varepsilon$ to the solution of the scaled radiative transfer equation. We further present an optimal convergence result for the DG method with the upwind flux on Cartesian grids. Error estimates of $\left(1+\mathcal{O}(\varepsilon)\right)h^{k+1}$ (where $h$ is the maximum element length) are obtained when tensor product polynomials of degree at most $k$ are used

Least-Squares and Other Residual Based Techniques for Radiation Transport Calculations

In this dissertation, we develop several novel methods based on or related to least-squares transport residual for solving deterministic radiation transport problems. For the first part of this dissertation a nonlinear spherical harmonics (PN) closure (TPN) was developed based on analysis of the least-squares residual for time-dependent PN equations in 1D slab geometry. The TPN closure suppresses the oscillations induced by Gibbs phenomenon in time-dependent transport calculations effectively. Simultaneously, a nonlinear viscosity term based on the spatial and temporal variations is realized and used in the extension to filtered PN method (NFPN). NFPN determines the angular viscosity on the fly and potentially fixed the issue existed in linear FPN that filtering strength needs to be predefined by iteratively solving the problem. We further developed another type of NFPN and demonstrate both of the two NFPN preserve the thick diffusion limit for thermal radiative transfer problems theoretically and numerically. We also developed several novel methods along with error analyses for steady-state neutron transport calculations based on least-squares methods. Firstly, a relaxed L1 finite element method was developed based on nonlinearly weighting the least-squares formulation by the pointwise transport residual. In problems such as void and near-void situations where least-squares accuracy is poor, the L1 method improves the solution. Further, a non-converged RL1 still can present comparable accuracy. We then developed a least-squares method based on a novel contiguous-discontinuous functional. A proof is provided for the conservation preservation for such a method, which is significant for problems such as k-eigenvalue calculations. Also, a second order accuracy is observed with much lower error magnitudes in several quantities of interest for heterogeneous problems compared with self-adjoint angular flux (SAAF) solution. Lastly, we extended the CD methodology with 1/σt-weighted least-squares functional to derive a CD-SAAF method and developed a SN-PN angular hybrid scheme. The hybrid scheme can employ high order SN in regions with strong transport feature to couple with low order PN in regions with diffusive flux. In k-eigenvalue calculations, it shows superb accuracy with low degrees of freedom

The Inertial Range of Turbulence in the Inner Heliosheath and in the Local Interstellar Medium

The governing mechanisms of magnetic field annihilation in the outer heliosphere is an intriguing topic. It is currently believed that the turbulent fluctuations pervade the inner heliosheath (IHS) and the Local Interstellar Medium (LISM). Turbulence, magnetic reconnection, or their reciprocal link may be responsible for magnetic energy conversion in the IHS.   As 1-day averaged data are typically used, the present literature mainly concerns large-scale analysis and does not describe inertial-cascade dynamics of turbulence in the IHS. Moreover, lack of spectral analysis make IHS dynamics remain critically understudied. Our group showed that 48-s MAG data from the Voyager mission are appropriate for a power spectral analysis over a frequency range of five decades, from 5e-8 Hz to 1e-2 Hz [Gallana et al., JGR 121 (2016)]. Special spectral estimation techniques are used to deal with the large amount of missing data (70%). We provide the first clear evidence of an inertial-cascade range of turbulence (spectral index is between -2 and -1.5). A spectral break at about 1e-5 Hz is found to separate the inertial range from the enegy-injection range (1/f energy decay). Instrumental noise bounds our investigation to frequencies lower than 5e-4 Hz. By considering several consecutive periods after 2009 at both V1 and V2, we show that the extension and the spectral energy decay of these two regimes may be indicators of IHS regions governed by different physical processes. We describe fluctuations’ regimes in terms of spectral energy density, anisotropy, compressibility, and statistical analysis of intermittency.   In the LISM, it was theorized that pristine interstellar turbulence may coexist with waves from the IHS, however this is still a debated topic. We observe that the fluctuating magnetic energy cascades as a power law with spectral index in the range [-1.35, -1.65] in the whole range of frequencies unaffected by noise. No spectral break is observed, nor decaying turbulence

Aeronautical engineering: A continuing bibliography with indexes (supplement 286)

This bibliography lists 845 reports, articles, and other documents introduced into the NASA scientific and technical information system in Dec. 1992. Subject coverage includes: design, construction and testing of aircraft and aircraft engines; aircraft components, equipment, and systems; ground support systems; and theoretical and applied aspects of aerodynamics and general fluid dynamics

Aeronautical engineering: A continuing bibliography with indexes (supplement 272)

This bibliography lists 719 reports, articles, and other documents introduced into the NASA scientific and technical information system in November, 1991. Subject coverage includes: design, construction and testing of aircraft and aircraft engines; aircraft components, equipment, and systems; ground support systems; and theoretical and applied aspects of aerodynamics and general fluid dynamics

Improved Fully-Implicit Spherical Harmonics Methods for First and Second Order Forms of the Transport Equation Using Galerkin Finite Element

In this dissertation, we focus on solving the linear Boltzmann equation -- or transport equation -- using spherical harmonics (PN) expansions with fully-implicit time-integration schemes and Galerkin Finite Element spatial discretizations within the Multiphysics Object Oriented Simulation Environment (MOOSE) framework. The presentation is composed of two main ensembles. On one hand, we study the first-order form of the transport equation in the context of Thermal Radiation Transport (TRT). This nonlinear application physically necessitates to maintain a positive material temperature while the PN approximation tends to create oscillations and negativity in the solution. To mitigate these flaws, we provide a fully-implicit implementation of the Filtered PN (FPN) method and investigate local filtering strategies. After analyzing its effect on the conditioning of the system and showing that it improves the convergence properties of the iterative solver, we numerically investigate the error estimates derived in the linear setting and observe that they hold in the non-linear case. Then, we illustrate the benefits of the method on a standard test problem and compare it with implicit Monte Carlo (IMC) simulations. On the other hand, we focus on second-order forms of the transport equation for neutronics applications. We mostly consider the Self-Adjoint Angular Flux (SAAF) and Least-Squares (LS) formulations, the former being globally conservative but void incompatible and the latter having -- in all generality -- the opposite properties. We study the relationship between these two methods based on the weakly-imposed LS boundary conditions. Equivalences between various parity-based PN methods are also established, in particular showing that second-order filters are not an appropriate fix to retrieve void compatibility. The importance of global conservation is highlighted on a heterogeneous multigroup k-eigenvalue test problem. Based on these considerations, we propose a new method that is both globally conservative and compatible with voids. The main idea is to solve the LS form in the void regions and the SAAF form elsewhere. For the LS form to be conservative in void, a non-symmetric fix is required, yielding the Conservative LS (CLS) formulation. From there, a hybrid SAAF-- CLS method can be derived, having the desired properties. We also show how to extend it to near-void regions and time-dependent problems. While such a second-order form already existed for discrete-ordinates (SN) discretizations (Wang et al. 2014), we believe that this method is the first of its kind, being well-suited to both SN and PN discretizations

Aeronautical engineering: A continuing bibliography with indexes (supplement 289)

This bibliography lists 792 reports, articles, and other documents introduced into the NASA scientific and technical information system in Mar. 1993. Subject coverage includes: design, construction and testing of aircraft and aircraft engines; aircraft components, equipment, and systems; ground support systems; and theoretical and applied aspects of aerodynamics and general fluid dynamics