# Cory D. Hauck's research while affiliated with Oak Ridge National Laboratory and other places

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## Publications (91)

A collision-based hybrid algorithm for the discrete ordinates approximation of the neutron transport equation is extended to the isotropic multigroup setting. The algorithm uses discrete energy and angle grids at two different resolutions and approximates the fission and scattering sources on the coarser grids. The coupling of a collided transport...

We consider a kinetic description of multi-species gas mixture modeled with Bhatnagar-Gross-Krook (BGK) collision operators, in which the collision frequency varies not only in time and space but also with the microscopic velocity. In this model, the Maxwellians typically used in standard BGK operators are replaced by a generalization of such targe...

In this paper, we present a predictor-corrector strategy for constructing rank-adaptive, dynamical low-rank approximations (DLRAs) of matrix-valued ODE systems. The strategy is a compromise between (i) low-rank step-truncation approaches that alternately evolve and compress solutions and (ii) strict DLRA approaches that augment the low-rank manifol...

A collision-based hybrid algorithm for the discrete ordinates approximation of the neutron transport equation is extended to the multigroup setting. The algorithm uses discrete energy and angle grids at two different resolutions and approximates the fission and scattering sources on the coarser grids. The coupling of a collided transport equation,...

In this paper, we explore applications of deep learning in statistical physics. We choose the Boltzmann equation as a typical example, where neural networks serve as a closure to its moment system. We present two types of neural networks to embed the convexity of entropy and to preserve the minimum entropy principle and intrinsic mathematical struc...

A key property of the linear Boltzmann semiconductor model is that as the collision frequency tends to infinity, the phase space density $f = f(x,v,t)$ converges to an isotropic function $M(v)\rho(x,t)$, called the drift-diffusion limit, where $M$ is a Maxwellian and the physical density $\rho$ satisfies a second-order parabolic PDE known as the dr...

We present a data-driven strategy for effective construction of a surrogate model in high-dimensional parameter space for the ion energy-angle distribution (IEAD) output of hPIC simulations of plasma-surface interactions. The methodology is based on a bin-by-bin least-squares fitting of the IEAD in the parameter space. The fitting is performed in a...

The micro-macro (mM) decomposition approach is considered for the numerical solution of the Vlasov–Poisson–Lenard–Bernstein (VPLB) system, which is relevant for plasma physics applications. In the mM approach, the kinetic distribution function is decomposed as f=E[ρf]+g, where E is a local equilibrium distribution, depending on the macroscopic mome...

We consider a kinetic description of a multi-species gas mixture modeled with Bhatnagar-Gross-Krook (BGK) collision operators, in which the collision frequency varies not only in time and space but also with the microscopic velocity. In this model, the Maxwellians typically used in standard BGK operators are replaced by a generalization of such tar...

This work presents neural network based minimal entropy closures for the moment system of the Boltzmann equation, that preserve the inherent structure of the system of partial differential equations, such as entropy dissipation and hyperbolicity. The described method embeds convexity of the moment to entropy map in the neural network approximation...

The authors developed an artificial intelligence (AI)-based algorithm for the design and optimization of a nuclear reactor core based on a flexible geometry and demonstrated a 3× improvement in the selected performance metric: temperature peaking factor. The rapid development of advanced, and specifically, additive manufacturing (3-D printing) and...

We derive a multi-species BGK model with velocity-dependent collision frequency for a non-reactive, multi-component gas mixture. The model is derived by minimizing a weighted entropy under the constraint that the number of particles of each species, total momentum, and total energy are conserved. We prove that this minimization problem admits a uni...

View Video Presentation: https://doi.org/10.2514/6.2021-2895.vid Direct simulation of physical processes on a kinetic level is prohibitively expensive in aerospace applications due to the extremely high dimension of the solution spaces. In this paper, we consider the moment system of the Boltzmann equation, which projects the kinetic physics onto t...

The micro-macro (mM) decomposition approach is considered for the numerical solution of the Vlasov--Poisson--Lenard--Bernstein (VPLB) system, which is relevant for plasma physics applications. In the mM approach, the kinetic distribution function is decomposed as $f=\mathcal{E}[\boldsymbol{\rho}_{f}]+g$, where $\mathcal{E}$ is a local equilibrium d...

Direct simulation of physical processes on a kinetic level is prohibitively expensive in aerospace applications due to the extremely high dimension of the solution spaces. In this paper, we consider the moment system of the Boltzmann equation, which projects the kinetic physics onto the hydrodynamic scale. The unclosed moment system can be solved i...

We present a data-driven approach to construct entropy-based closures for the moment system from kinetic equations. The proposed closure learns the entropy function by fitting the map between the moments and the entropy of the moment system, and thus does not depend on the space-time discretization of the moment system and specific problem configur...

We derive a multi-species BGK model with velocity-dependent collision frequency for a non-reactive, multi-component gas mixture. The model is derived by minimizing a weighted entropy under the constraint that the number of particles of each species, total momentum, and total energy are conserved. We prove that this minimization problem admits a uni...

In this paper, properties of a recently proposed mathematical model for data flow in large-scale asynchronous computer systems are analyzed. In particular, the existence of special weak solutions based on propagating fronts is established. Qualitative properties of these solutions are investigated, both theoretically and numerically.

Implicit in any engineering design is an underlying optimization problem, although the exact objective function to be optimized is rarely stated explicitly. Nuclear systems optimization is as old as the discipline of nuclear engineering. Advanced manufacturing in the nuclear industry has opened the door for the re-examination of optimization in a w...

We present a simplified model of data flow on processors in a high-performance computing framework involving computations necessitating inter-processor communications. From this ordinary differential model, we take its asymptotic limit, resulting in a model which treats the computer as a continuum of processors and data flow as an Eulerian fluid go...

We present an error analysis for the discontinuous Galerkin method applied to the discrete-ordinate discretization of the steady-state radiative transfer equation. Under some mild assumptions, we show that the DG method converges uniformly with respect to a scaling parameter $\varepsilon$ which characterizes the strength of scattering in the system...

We present and analyze a new iterative solver for implicit discretizations of a simplified Boltzmann-Poisson system. The algorithm builds on recent work that incorporated a sweeping algorithm for the Vlasov-Poisson equations as part of nested inner-outer iterative solvers for the Boltzmann-Poisson equations. The new method eliminates the need for n...

In this paper, two modifications are introduced for improving the accuracy, versatility, and robustness of a class of hybrid methods for radiation transport. In general, such methods are constructed by splitting the radiative flux into collided and uncollided components to which low- and high-resolution angular approximations are applied, respectiv...

Several different approaches are proposed for solving fully implicit discretizations of a simplified Boltzmann-Poisson system with a linear relaxation-type collision kernel. This system models the evolution of free electrons in semiconductor devices under a low-density assumption. At each implicit time step, the discretized system is formulated as...

Solving the radiative transfer equation with the discrete ordinates (S N) method leads to a nonphysical imprint of the chosen quadrature set on the solution. To mitigate these so-called ray effects, we propose a modification of the S N method that we call artificial scattering S N (as-S N). The method adds an artificial forward-peaked scattering op...

We propose a hybrid spatial discretization for the radiative transport equation that combines a second-order discontinuous Galerkin (DG) method and a second-order finite volume (FV) method. The strategy relies on a simple operator splitting that has been used previously to combine different angular discretizations. Unlike standard FV methods with u...

Solving the radiative transfer equation with the discrete ordinates (S$_N$) method leads to a non-physical imprint of the chosen quadrature set on the solution. To mitigate these so-called ray effects, we propose a modification of the S$_N$ method, which we call artificial scattering S$_N$ (as-S$_N$). The method adds an artificial forward-peaked sc...

In this paper, properties of a recently proposed mathematical model for data flow in large-scale asynchronous computer systems are analyzed. In particular, the existence of special weak solutions based on propagating fronts is established. Qualitative properties of these solution are investigated, both theoretically and numerically.

We present a simplified model of data flow on processors in a high performance computing framework involving computations necessitating inter-processor communications. From this ordinary differential model, we take its asymptotic limit, resulting in a model which treats the computer as a continuum of processors and data flow as an Eulerian fluid go...

We investigate overdetermined linear inverse problems for which the forward operator may not be given accurately. We introduce a new tool called the structure, based on the Wasserstein distance, and propose the use of this to diagnose and remedy forward operator error. Computing the structure turns out to use an easy calculation for a Euclidean hom...

We present and analyze a discrete ordinates (\(\text {S}_N\)) discretization of a filtered radiative transport equation (RTE). Under certain conditions, \(\text {S}_N\) discretizations of the standard RTE create numeric artifacts, known as “ray-effects”; the goal of the filter is to remove such artifacts. We analyze convergence of the filtered disc...

The discrete ordinates discontinuous Galerkin ($S_N$-DG) method is a well-established and practical approach for solving the radiative transport equation. In this paper, we study a low-memory variation of the upwind $S_N$-DG method. The proposed method uses a smaller finite element space that is constructed by coupling spatial unknowns across collo...

Several different approaches are proposed for solving fully implicit discretizations of a simplified Boltzmann-Poisson system with a linear relaxation-type collision kernel. This system models the evolution of free electrons in semiconductor devices under a low-density assumption. At each implicit time step, the discretized system is formulated as...

We develop implicit-explicit (IMEX) schemes for neutrino transport in a background material in the context of a two-moment model that evolves the angular moments of a neutrino phase-space distribution function. Considering the upper and lower bounds that are introduced by Pauli’s exclusion principle on the moments, an algebraic moment closure based...

Building on the framework of Zhang & Shu [1,2], we develop a realizability-preserving method to simulate the transport of particles (fermions) through a background material using a two-moment model that evolves the angular moments of a phase space distribution function f. The two-moment model is closed using algebraic moment closures; e.g., as prop...

We analyze the properties and compare the performance of several positivity limiters for spectral approximations with respect to the angular variable of linear transport equations. It is well-known that spectral methods suffer from the occurrence of (unphysical) negative spatial particle concentrations due to the fact that the underlying polynomial...

We introduce an extension of the fast semi-Lagrangian scheme developed in J Comput Phys 255:680–698 (2013) in an effort to increase the spatial accuracy of the method. The basic idea of this extension is to modify the first-order accurate transport step of the original semi-Lagrangian scheme to allow for a general piecewise polynomial reconstructio...

In this work, we describe extensions of a hybrid method for time-dependent linear, kinetic radiation transport problems to high-order time integration schemes of the diagonally-implicit Runge–Kutta (DIRK) and space–time discontinuous Galerkin (STDG) types. The hybrid methods are constructed by splitting the radiation flux into collided and uncollid...

We show the convergence of the zero relaxation limit in systems of $2 \times 2$ hyperbolic conservation laws with stochastic initial data. Precisely, solutions converge to a solution of the local equilibrium approximation as the relaxation time tends to zero. The initial data are assumed to depend on finitely many random variables, and the converge...

We investigate overdetermined linear inverse problems for which the forward operator may not be given accurately. We introduce a new tool called the structure, based on the Wasserstein distance, and propose the use of this to diagnose and remedy forward operator error. Computing the structure turns out to use an easy calculation for a Euclidean hom...

Building on the framework of Zhang \& Shu \cite{zhangShu_2010a,zhangShu_2010b}, we develop a realizability-preserving method to simulate the transport of particles (fermions) through a background material using a two-moment model that evolves the angular moments of a phase space distribution function $f$. The two-moment model is closed using algebr...

We present a positive and asymptotic preserving numerical scheme for solving linear kinetic, transport equations that relax to a diffusive equation in the limit of infinite scattering. The proposed scheme is developed using a standard spectral angular discretization and a classical micro-macro decomposition. The three main ingredients are a semi-im...

We present a new entropy-based moment method for the velocity discretization of kinetic equations. This method is based on a regularization of the optimization problem defining the original entropy-based moment method, and this gives the new method the advantage that the moment vectors of the solution do not have to take on realizable values. We sh...

We present a domain decomposition algorithm to accelerate the solution of Eulerian-type discretizations of the linear, steady-state Vlasov equation. The steady-state solver then forms a key component in the implementation of fully implicit or nearly fully implicit temporal integrators for the nonlinear Vlasov–Poisson system. The solver relies on a...

We have extended a recently developed multispecies, multitemperature Bhatnagar-Gross-Krook model [Haack et al., J. Stat. Phys. 168, 822 (2017)], to include multiphysics capabilities that enable modeling of a wider range of physical conditions. In terms of geometry, we have extended from the spatially homogeneous setting to one spatial dimension. In...

The work presented in this paper is related to the development of positivity preserving Discontinuous Galerkin (DG) methods for Boltzmann - Poisson (BP) computational models of electronic transport in semiconductors. We pose the Boltzmann Equation for electron transport in curvilinear coordinates for the momentum. We consider the 1D diode problem w...

In this work, we prove rigorous convergence properties for a semi-discrete, moment-based approximation of a model kinetic equation in one dimension. This approximation is equivalent to a standard spectral method in the velocity variable of the kinetic distribution and, as such, is accompanied by standard algebraic estimates of the form $N^{-q}$, wh...

We derive a conservative multispecies BGK model that follows the spirit of the original, single species BGK model by making the specific choice to conserve species masses, total momentum, and total kinetic energy and to satisfy Boltzmann’s \(\mathcal {H}\)-Theorem. The derivation emphasizes the connection to the Boltzmann operator which allows for...

We propose to apply a low dimensional manifold model to scientific data interpolation from regular and irregular samplings with a significant amount of missing information. The low dimensionality of the patch manifold for general scientific data sets has been used as a regularizer in a variational formulation. The problem is solved via alternating...

In this work, we describe the implementation of an arbitrarily high-order hybrid solver for linear, kinetic, radiative transport equations. The hybrid method is derived from a splitting of the radiative flux into free-streaming and collisional components to which high- and low-resolution discrete ordinates methods are applied, respectively. Arbitra...

We propose a simple fast spectral method for the Boltzmann collision operator with general collision kernels. In contrast to the direct spectral method \cite{PR00, GT09} which requires $O(N^6)$ memory to store precomputed weights and has $O(N^6)$ numerical complexity, the new method has complexity $O(MN^4\log N)$, where $N$ is the number of discret...

In this paper we genealize the fast semi-Lagrangian scheme developed in [J. Comput. Phys., Vol. 255, 2013, pp 680-698] to the case of high order reconstructions of the distribution function. The original first order accurate semi-Lagrangian scheme is supplemented with polynomial reconstructions of the distribution function and of the collisional op...

In this work, we provide a fully-implicit implementation of the time-dependent, filtered spherical harmonics (FPN) equations for non-linear, thermal radiative transfer. We investigate local filtering strategies and analyze the effect of the filter on the conditioning of the system in the streaming limit, showing in particular that the filter improv...

We propose a positive-preserving moment closure for linear kinetic transport equations based on a filtered spherical harmonic (FPN) expansion in the angular variable. The recently proposed FPN moment equations are known to suffer from the occurrence of (unphysical) negative particle concentrations. The origin of this problem is that the FPN approxi...

We analyze the global convergence properties of the filtered spherical harmonic (FPN) equations for radiation transport. The well-known spherical harmonic (PN) equations are a spectral method (in angle) for the radiation transport equation and are known to suffer from Gibbs phenomena around discontinuities. The filtered equations include additional...

We present computational advances and results in the implementation of an entropy-based moment closure, MN, in the context of linear kinetic equations, with an emphasis on heterogeneous and large-scale computing platforms. Entropy-based closures are known in several cases to yield more accurate results than closures based on standard spectral appro...

The spherical harmonic equations for radiative transport are a linear, hyperbolic set of balance laws that describe the state of a system of particles as they advect through and collide with a material medium. For regimes in which the collisionality of the system is light to moderate, significant qualitative differences have been observed between s...

We extend the positivity-preserving method of Zhang & Shu (2010, JCP, 229,
3091-3120) to simulate the advection of neutral particles in phase space using
curvilinear coordinates. The ability to utilize these coordinates is important
for non-equilibrium transport problems in general relativity and also in
science and engineering applications with sp...

We propose a semi-Lagrangian numerical algorithm for a time-dependent, anisotropic temperature transport equation in magnetized plasmas in regimes with negligible variation of the magnitude of the magnetic field B along field lines. The approach is based on a formal integral solution of the parallel (i.e., along the magnetic field) transport equati...

We discuss several moment closure models for linear kinetic equations that have been developed over the past few years as alternatives to classical spectral and collocation methods. We then present numerical simulation results for a challenging benchmark problem known as the line source and observe the relative strengths and weaknesses of each clos...

The radiative transfer equation describes the propagation of radiation through a material medium. While it provides a highly accurate description of the radiation field, the large phase space on which the equation is defined makes it numerically challenging. As a consequence, significant effort has gone into the development of accurate approximatio...

Entropy-based (M_N) moment closures for kinetic equations are defined by a
constrained optimization problem that must be solved at every point in a
space-time mesh, making it important to solve these optimization problems
accurately and efficiently. We present a complete and practical numerical
algorithm for solving the dual problem in one-dimensio...

We investigate the approximate dynamics of several differential equations
when the solutions are restricted to a sparse subset of a given basis. The
restriction is enforced at every time step by simply applying soft thresholding
to the coefficients of the basis approximation. By reducing or compressing the
information needed to represent the soluti...

We present a hybrid method for simulating kinetic equations with multiscale phenomena in the context of linear transport. The method consists of (i) partitioning the kinetic equation into collisional and noncollisional components, (ii) applying a different numerical method to each component, and (iii) repartitioning the kinetic distribution after e...

We study the statistical properties of a cellular automata model of traffic
flow with the look-ahead potential. The model defines stochastic rules for the
movement of cars on a lattice. We analyze the underlying statistical
assumptions needed for the derivation of the coarse-grained model and
demonstrate that it is possible to relax some of them to...

We derive a hierarchy of closures based on perturbations of well-known
entropy-based closures; we therefore refer to them as perturbed entropy-based
models. Our derivation reveals final equations containing an additional
convective and diffusive term which are added to the flux term of the standard
closure. We present numerical simulations for the...

The Mâ model for radiative transfer coupled to a material energy equation in planar geometry is studied in this paper. For this model to be well-posed, its moment variables must fulfill certain realizability conditions. Our main focus is the design and implementation of an explicit Runge-Kutta discontinuous Galerkin method which, under a more rest...

We present a numerical algorithm to implement entropy-based (M{sub N}) moment models in the context of a simple, linear kinetic equation for particles moving through a material slab. The closure for these models - as is the case for all entropy-based models - is derived through the solution of constrained, convex optimization problem. The algorithm...

We compute high-order entropy-based (M N) models for a linear transport equation on a one-dimensional, slab geometry. We simulate two test problems from the literature: the two-beam instability and the plane-source problem. In the former case, we compute solutions for systems up to order N = 5; in the latter, up to N = 15. The most notable outcome...

We present a novel application of filters to the spherical harmonics (PN) expansion for radiative transfer problems in the high-energy-density regime. The filter we use is based on non-oscillatory spherical splines and a filter strength chosen to (i) preserve the equilibrium diffusion limit and (ii) vanish as the expansion order tends to infinity....

This Letter presents a novel application of filters to the spherical harmonics (PN) expansion for radiative transfer problems in the high-energy-density regime. The filter, which is based on non-oscillatory spherical splines, preserves both the equilibrium diffusion limit and formal convergence properties of the unfiltered expansion. While the meth...

We introduce a modification to the standard spherical harmonic closure used with linear kinetic equations of particle transport. While the standard closure is known to produce negative particle concentrations, the modification corrects this defect by requiring that the ansatz used to close the equations itself be a nonnegative function. We impose t...

The occurrence of oscillations in a well-known asymptotic preserv-ing (AP) numerical scheme is investigated in the context of a linear model of diffusive relaxation, known as the P 1 equations. The scheme is derived with operator splitting methods that separate the P 1 system into slow and fast dy-namics. A careful analysis of the scheme shows that...

In this paper, we introduce a regularization of the P N equations for one-dimensional, slab geometries. These equations are used to describe particle transport through a material medium. Our regularization is based on a temporal splitting of fast and slow dynamics in the P N system. It uses ideas first introduced in [14] for 2 × 2 systems to addres...

A common method for constructing a function from a finite set of moments is to solve a constrained minimization problem. The idea is to find, among all functions with the given moments, that function which minimizes a physically motivated, strictly convex functional. In the kinetic theory of gases, this functional is the kinetic entropy; the given...

We present a new way of using spherical harmonics expansions to solve transport problems. Our ap- proach uses filtered expansions to give positive solutions a nd reduce wave effects in the solutions. We present two specific filters: one based on maintaining positi vity in a P1 expansion, and the other that is a function of the total cross-section a...

This study will investigate several numerical methods for modeling the transfer of neutral particles through a material medium, such as neutrons within a nuclear reactor. In a kinetic description, the particle evolution is governed by a transport equation, generally of the form (1.1) {sub t}F + v · F = C(F). Here x R³ is a spatial coordinate, v R³...

We present recent progress in the development of two substantially different approaches for simulating the so-called of P{sub N} equations. These are linear hyperbolic systems of PDEs that are used to model particle transport in a material medium, that in highly collisional regimes, are accurately approximated by a simple diffusion equation. This l...

We present a hybrid method for simulating kinetic equations with multi-scale phenomena in the context of linear transport. The method consists of (i) partitioning the kinetic equation into collisional and non-collisional components; (ii) applying a different numerical method to each compo-nent; and (iii) re-partitioning the kinetic distribution aft...

## Citations

... The collision frequencies per density are assumed to be dependent only on and and not on the microscopic velocity . For references taking into account also a dependency on the microscopic velocity see [64] for the one species case, [38] for the gas mixture case and [39] for the numerics of the gas mixture case. ...

... Such relaxation terms include the Bhatnagar-Gross-Krook (BGK) model [10] and its extensions, the ellipsoidal-statistical BGK model (ES-BGK) [11] and the Shakhov model (S-model) [12]. However, the correct incorporation of complex collisional physical phenomena, such as chemical reactions, transitions of internal energy, and complicated scattering laws within the framework of these model equations is still an open question and a topic of active research [13,14,15]. Despite these drawbacks of the linear models, they offer advantages in terms of being scalable to dense regimes [16,17]. ...

... Using (eqs. 19,21), we obtain the following relations Fig. 6 shows the profile of the radiation energy density at time t = 4 for different resolutions compared with the semi-analytic solution as in ). Despite the pitfalls of this test, the results of our simulations show an approximately second-order convergence for several resolutions, as it is displayed in Fig. 7. Furthermore, we have investigated the effect of an additional AMR grid level following the gaussian profile, observing still second order convergence. ...