Jingwei Hu

Title: Deterministic Solution of the Boltzmann Equation: Fast Spectral Methods for the Boltzmann Collision Operator
Abstract: The Boltzmann equation, an integro-differential equation for the molecular distribution function in the physical and velocity phase space, governs the fluid flow behavior at a wide range of physical conditions. Despite its wide applicability, deterministic numerical solution of the Boltzmann equation presents a huge computational challenge due to the high-dimensional, nonlinear, and nonlocal collision operator. We introduce a fast Fourier spectral method for the Boltzmann collision operator which leverages its convolutional and low-rank structure. We show that the framework is quite general and can be applied to arbitrary collision kernels, inelastic collisions, and multiple species. We then couple the fast spectral method in the velocity space with the discontinuous Galerkin discretization in the physical space to obtain a highly accurate deterministic solver for the full Boltzmann equation. Standard benchmark tests including rarefied Fourier heat transfer, Couette flow, and thermally driven cavity flow have been studied and the results are compared against direct simulation Monte Carlo (DSMC) solutions.

Cory Hauck

Title: Discontinuous Galerkin Methods and the Diffusion Limit
Abstract: Discontinuous Galerkin (DG) methods were first constructed for the purpose of solving kinetic transport equations. Since then, it has been realized that DG methods also perform well in scattering-dominated regimes, where the solution of the transport equation can be approximated asymptotically by the solution of a much simpler diffusion equation. For this reason, DG methods continue to be popular in applications where the diffusion limit is important. The effectiveness of DG in this limit can be traced back to the additional degrees of freedom per cell it uses (when compared to finite volume methods). However, these extra degrees of freedom come at a substantial cost, especially given the fact that memory is often the limiting factor when simulating realistic problems with a kinetic description.
In this talk, I will review some of the history of DG methods and their use in radiation transport simulations. I will then present two methods for reducing the memory of the standard DG approach while still capturing the asymptotic diffusion limit. Both methods rely on a hybrid approach to solving the transport equation.

Austin Minnich

Title: Transport phenomena in solids from first-principles using the linearized Boltzmann equation
Abstract: In the past decade, the calculation of transport properties such as electrical and thermal conductivities from first-principles, meaning with only atomic positions and identities as input, has become routine. The general approach is to compute the coupling strengths between electron or vibrational modes in solids using electronic structure codes and subsequently solve the Boltzmann equation to obtain the system response to an applied field. Most works to date solve the Boltzmann equation under highly restrictive assumptions to obtain transport properties. In this talk, I will present our efforts to solve the Boltzmann equation fully ab-initio for more complex transport problems in solids. In particular, I will describe our studies of the second sound phenomena of phonons, involving a collective excitation of the phonon system, as well as electronic noise in semiconductors which is obtained from the Green’s function of the Boltzmann equation.

Chanwoo Kim

Title: On passage from Boltzmann equation to Fluid models
Abstract: In this lecture we discuss some mathematical results deriving fluid models from scaled Boltzmann equation.
Relevant references are:
Bardos, Golse, Levermore: Fluid dynamical limits of kinetic equations I. Formal derivations. J. Stat. Phys. 63, 323–344 (1991)
Esposito, Guo, Kim, Marra: Stationary Solutions to the Boltzmann Equation in the Hydrodynamic Limit, Ann.PDE 4:1 (2018)

Maja Taskovic

Title: On the relativistic Landau equation
Abstract: In kinetic theory, a large system of particles is described by the particle density function. The Landau equation, derived by Landau in 1936, is one such example. It models a dilute hot plasma with fast moving particles that interact via Coulomb interactions. This model does not include the effects of Einstein’s theory of special relativity. However, when particle velocities are close to the speed of light, which happens frequently in a hot plasma, then relativistic effects become important. These effects are captured by the relativistic Landau equation, which was derived by Budker and Beliaev in 1956.
We study the Cauchy problem for the spatially homogeneous relativistic Landau equation with Coulomb interactions. The difficulty of the problem lies in the extreme complexity of the kernel in the relativistic collision operator. We present a new decomposition of such kernel. This is then used to prove the global Entropy dissipation estimate, the propagation of any polynomial moment for a weak solution, and the existence of a true weak solution for a large class of initial data. This is joint work with Robert M. Strain.

Chengyun Hua

Title: Direct solution to the space-time dependent Peierls-Boltzmann transport equation using an eigendecomposition method
Co-authors: Austin Minnich, Lucas Lindsay
Abstract: Nonlocal thermal transport is generally described by the Peierls-Boltzmann transport equation (PBE). However, solving the PBE for a general space-time dependent problem remains a challenging task due to the high dimensionality of the integro-differential equation. In this work, we present a direct solution to the space-time dependent PBE with a linearized collision matrix using an eigendecomposition method. Furthermore, we show that there exists a generalized Fourier type relation that links heat flux to the local temperature, and this constitutive relation is valid from ballistic to diffusive regimes. Combining this approach with ab initio calculations of phonon properties, we demonstrate that the derived solution gives a more accurate description of thermal transport in crystals that exhibit weak anharmonicity than the commonly-used single-mode relaxation time approximation and thus will lead to an improved understanding of phonon transport in solids.

Navaneeth Ravichandran

Title: Higher-order scattering among phonons in the Peierls-Boltzmann framework
Abstract: Conventionally, thermal conductivity of solids is obtained from the solution of the Peierls-Boltzmann equation (PBE), where the collision term only contains scattering events among three heat carriers (phonons) at a time. While this approach quantitatively predicts the thermal conductivity of most solids, it becomes insufficient for the ultrahigh thermal conductivity material - Boron Arsenide (BAs), resulting in the need to consider higher-order scattering among four phonons. In this talk, I will describe the formulation and solution of the PBE with the collision term including both three-phonon and four-phonon scattering terms, to obtain the thermal conductivity of BAs in good agreement with experiments. I will show the importance of distinguishing between momentum conserving and momentum dissipating phonon scattering processes, both at the three-phonon and the four-phonon level, to correctly capture the ultrahigh thermal conductivity of BAs. I will also discuss the computational complexity in accounting for four-phonon scattering within the PBE framework and our approaches to overcome them.

Lee Ricketson

Title: An implicit, energy-conserving and asymptotic-preserving time-integration scheme particle-in-cell simulation of magnetized kinetic plasmas

Authors: L.F. Ricketson (LLNL), L. Chacon (LANL)

Abstract: Charged particle motion in a strong magnetic field – like those present in magnetic confinement fusion devices – is characterized by rapid gyration about magnetic field lines superimposed on much slower drift motions. Any kinetic simulation of plasma dynamics in such a strong field must reckon with this time-scale separation. Traditionally, this is done by asymptotic expansion on the Vlasov equation to analytically eliminate the fast gyromotion. However, it is increasingly evident that this asymptotic expansion is only valid in a portion of modern fusion devices. We thus present an asymptotic-preserving time integrator for charged particle motion that recovers the so-called “guiding center” limit when stepping over the gyration time-scales while still converging to the full solution. The resulting method, when implemented in an implicit particle-in-cell scheme, enables one to step over the gyration scale only when it is rigorously justified. Our scheme differs from previous efforts in that it preserves energy conservation, which is shown in numerical examples to lead to tremendous improvement in the accuracy of the scheme.

*Work performed under the auspices of the U.S. Department of Energy by LLNL and LANL under contracts DE-AC52-07NA27344, DE-AC52-06NA25396, and supported by the Exascale Computing Project (17-SC-20-SC), a collaborative effort of the U.S. Department of Energy Office of Science and the National Nuclear Security Administration.

Ming Tse Paul Laiu

Title: A fast implicit solver for semiconductor models in one space dimension

Abstract: We propose several different approaches 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 a fixed-point problem, which can then be solved with a variety of methods. A key algorithmic component in all the approaches considered here is a recently developed sweeping algorithm for Vlasov-Poisson systems. A synthetic acceleration scheme has been implemented to accelerate the convergence of iterative solvers by using the solution to a drift-diffusion equation as a preconditioner. The performance of four iterative solvers and their accelerated variants has been compared on problems modeling semiconductor devices with various electron mean-free-path.

Brian Cornille

Title: A Comparison of Galerkin and First Order System Least Squares Finite Element Formulations of Electron Magnetohydrodynamics
Abstract: In multi-timescale extended magnetohydrodynamics (MHD) simulation, representing Hall term effects is particularly challenging. When substituted into Faraday's Law, the Hall term introduces an advection-like term, which hampers mathematical coercivity in Galerkin finite-element methods. The first order system least squares (FOSLS) approach can be used to make the system coercive and stabilize the advection-like term in implicit computations [1]. To better understand this numerical challenge, we consider a limit of the extended MHD system known as electron MHD (EMHD), which largely isolates the Hall term. In EMHD the FOSLS approach presents many choices for independent variable pairs. Here we study two options, vector fields {B, J} in the function space {H1, H1} and {B, E} in {H1, H(curl)}. We also consider two Galerkin formulations with B in H1 and H(curl). These formulations are being implemented using the MFEM library. Numerical properties of all four formulations are compared and contrasted. The multiscale nature of the EMHD tearing mode in a slab geometry provides a challenging test case that exposes strengths and weaknesses of the different formulations and that permits validation with analytical theory.
[1] C. A. Leibs, T. A. Manteuffel, Nested Iteration and First-Order Systems Least Squares for a Two-Fluid Electromagnetic Darwin Model. SIAM Journal on Scientific Computing. 37, S314–S333 (2015).
Acknowledgements: Supported by DOE CSGF under grant DE-FG02-97ER25308 and by DOE grant DE-SC0018001.

Gunjan Pahlani

Title: Objective Molecular Dynamids
Abstract: One of the pervasive bottlenecks in science and engineering is the time-scale limitation of molecular dynamics (MD). Using accurate atomic forces, how do we perform an MD simulation on a large number of atoms for experimentally accessible time scales? In this work, we are developing the method of Objective Molecular Dynamics for this purpose. This is a method of simulation in which only a few (say 50-1000) atoms are actually simulated, but the full infinite set of atoms satisfy exactly the MD equations. We present a method, capable of simulating three parameter family of incompressible flows as well as compressible flow and unsteady flows. It allows us to calculate viscometric properties from a molecular-level simulation in the absence of constitutive equations, fluids in regime currently inaccessible to theory or experiment undergoing chemical reactions, high rates of shear, expansion or phase change which are far from equilibrium. We illustrate this method using Couette and Extensional flow. From a dynamical systems viewpoint, this is an (unstudied) invariant manifold of molecular dynamics. This invariant manifold provided by OMD is inherited by Boltzmann equation. We present fascinating connections with the Boltzmann equation and continuum mechanics.

William Taitano

Title: High-Order Low-Order Nonlinear Convergence Accelerator for the Rosenbluth-Fokker-Planck Collision Operator and Applications to Multiscale Problems in Inertial Confinement Fusion (ICF)
Abstract: In weakly coupled plasmas, the integro-differential Fokker-Planck collision operator describes the dynamical collisional relaxation of the plasma distribution function in the velocity space. The operator is quadratically nonlinear, features exact conservation of mass, momentum, and energy, satisfies the Boltzmann H-theorem, preserves positivity of the distribution function, and features a unique equilibrium solution (i.e., a drifting Maxwellian). Additionally, the collisional velocity space transport coefficients are obtained from integrals of the distribution function (the so-called Rosenbluth potentials). The integral formulation, together with the above-mentioned properties make the numerical solution aspects of this system exceedingly challenging to deal with.
Often, collision times, τcol, are much shorter than dynamical timescales of interest (e.g., in ICF problems, often >104), and the use of implicit methods (i.e., Newton, accelerated Picard [1], etc.) is warranted. Fully implicit methods are advocated here, owing for the need to exactly preserve conservation laws, and to ensure asymptotic convergence to the Maxwellian when appropriate [2]. However, fully implicit methods demand nonlinear iterative solvers, which developing an effective one is challenging in this context owing to the non-locality of the formulation (via the Rosenbluth potentials), leading to dense linear systems.
To effectively deal with the numerical challenges arising from non-locality, we explore a multiscale iterative strategy based on high-order/low-order (HOLO) convergence accelerator scheme [3,4] for the full nonlinear Rosenbluth-Fokker-Planck collision operator. HOLO employ an LO (fluid) moment system to accelerate the convergence of the HO (kinetic) system. In turn, the LO system is closed self-consistently with the HO system. The LO system is comprised of equations for plasma number density, drift velocity, and temperature. These quantities, in turn, inform an LTE approximation for the Rosenbluth potentials plus a perturbation term [of O(Δt/τcol)<< 1] computed from the HO solution. This reformulation shifts the non-local contributions through the Rosenbluth potentials from the HO system to the LO one, which lives in a low-dimensional space and where they can be dealt with efficiently. Numerical experiments in challenging applications in ICF (i.e., shocks and implosions) will demonstrate the enabling capabilities of the HOLO scheme when Δt >> τcol.
[1] Anderson J. Assoc. Comput. Mach., 12, 547-560 (1965).
[2] Taitano et al., J. Comp. Phys., 297, 357-380 (2015).
[3] Taitano et al., J. Comp. Phys., 284, 737-757 (2015).
[4] Chacón et al., J. Comp. Phys., 330, 21-45 (2017).

Vincent Heningburg

Title: Filtered Discrete Ordinates Equations for Radiative Transport
Abstract: This talk covers the discrete ordinates discretization of a filtered radiative transport equation (RTE). Under certain conditions, discrete ordinates 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 discrete ordinates solution to the solution of the non-filtered RTE, taking into account the effect of the filter as well as the usual quadrature and truncation errors that arise in discretize ordinate methods. We solve the filtered discrete ordinates equations numerically with a discontinuous Galerkin spatial discretization and implicit time stepping. The form of the filter enables the resulting linear systems to be solved in an established Krylov framework. We demonstrate, via the simulation of two benchmark problems, the effectiveness of the filtering approach in reducing ray effects. In addition, we also examine efficiency of the method, in particular the balance between improved accuracy and additional cost of including the filter.

Hongxu Chen

Title: Boltzmann equation with Cercignani-Lampis boundary condition in bounded domain.
Abstract: When dilute gas is confined in a bounded domain, boundary effects play an important role. [1] established the classical solution for the Boltzmann equation with various boundary condition: diffuse, specular reflection and bounce back. The Cercignani-Lampis(C-L) boundary condition, derived in [2], is a model for gas-surface interaction that encompass the pure diffusion and pure reflection via two accommodation coefficients. The complexity of the scattering kernel in the boundary condition brings the difficulty in the analysis. In this talk I will present the local well-posedness of the Boltzmann equation with the C-L boundary condition. In particular, I will focus on the L^\infty estimate. The main idea is that we trace back along the characteristic and use a new decomposition for the integral on the boundary.
[1] Guo, Yan, et al. "Regularity of the Boltzmann equation in convex domains." Inventiones mathematicae 207.1 (2017): 115-290.
[2] Cercignani, Carlo, and Maria Lampis. "Kinetic models for gas-surface interactions." transport theory and statistical physics1.2 (1971): 101-114.

David Reynolds

Title: A Topological Model for Collective Motion and its Kinetic Formulation
Abstract: We begin with some of the basics of collective motion, including various applications. Introduce a new topological model that describes collective motion, and derive the kinetic formulation of the model through the mean-field limit. We then finish by discussing various results concerning well-posedness of the model and alignment of velocities.

Trevor Leslie

Title: The Structure of Limiting Flocks in Hydrodynamic Euler Alignment Models
Abstract: In this talk, we discuss flocking in the context of the hydrodynamic Euler Alignment model on the 1D torus, focusing on the case where the interaction protocol has small support. We show that for a large class of interaction kernels, the deviation of the density from a uniform distribution is controlled for large times by a certain conserved quantity of the system. This is joint work with Roman Shvydkoy.

Ke Chen

Title: A fast random sampling algorithm for radiative transfer equation

Abstract: Random sampling has been used successfully to find low rank structure in the data science. We describe a fast algorithm for solving radiative transfer equation (RTE) that uses random sampling to capture low-rank feature of solution spaces arising in a domain decomposition framework. This algorithm does not rely on detailed analytical understanding of various limiting regimes of RTE. Numerical examples demonstrate accuracy and efficacy of our approach.

Kit Newton

Title: A Bayesian perspective on diffuse optical tomography
Abstract: The process of reconstructing properties of biological tissue using measurements of incoming and outgoing light intensity is known as optical tomography. It may be described mathematically by the inverse radiative transfer equation: optical tomography amounts to reconstructing the scattering coefficient in the RTE using the boundary measurements. In the strong scattering regime, the RTE is asymptotically equivalent to the diffusion equation (DE). In the Bayesian framework, we examine the posterior distribution of the scattering coefficient after the measurements have been taken. Sampling from this distribution is computationally expensive, so we employ a two-level MCMC technique, using the DE posterior distribution to make sampling from the RTE posterior distribution computationally feasible.

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