It is well-known that the Fourier-Galerkin spectral method has been a popular approach for the numerical approximation of the deterministic Boltzmann equation with spectral accuracy rigorously proved. In this paper, we will show that such a spectral convergence of the Fourier-Galerkin spectral method also holds for the Boltzmann equation with uncertainties arising from both collision kernel and initial condition. Our proof is based on newly-established spaces and norms that are carefully designed and take the velocity variable and random variables with their high regularities into account altogether. For future studies, this theoretical result will provide a solid foundation for further showing the convergence of the full-discretized system where both the velocity and random variables are discretized simultaneously.
Numerical approximation of the Boltzmann equation presents a challenging problem due to its high-dimensional, nonlinear, and nonlocal collision operator. Among the deterministic methods, the Fourier-Galerkin spectral method stands out for its relative high accuracy and possibility of being accelerated by the fast Fourier transform. However, this method requires a domain truncation which is unphysical since the collision operator is defined in $\mathbb{R}^d$. In this paper, we introduce a Petrov-Galerkin spectral method for the Boltzmann equation in the unbounded domain. The basis functions (bo…
Numerical approximation of the Boltzmann equation is a challenging problem due to its high-dimensional, nonlocal, and nonlinear collision integral. Over the past decade, the Fourier-Galerkin spectral method has become a popular deterministic method for solving the Boltzmann equation, manifested by its high accuracy and potential of being further accelerated by the fast Fourier transform. Albeit its practical success, the stability of the method is only recently proved by Filbet, F. & Mouhot, C. in [$ Trans. Amer. Math. Soc.$ 363, no. 4 (2011): 1947-1980.] by utilizing the "spreading" property …
Spectral methods, thanks to the high accuracy and the possibility of using fast algorithms, represent an effective way to approximate collisional kinetic equations in kinetic theory. On the other hand, the loss of some local invariants can lead to the wrong long time behavior of the numerical solution. We introduce in this paper a novel Fourier-Galerkin spectral method that improves the classical spectral method by making it conservative on the moments of the approximated distribution, without sacrificing its spectral accuracy or the possibility of using fast algorithms. The method is derived …
Evaluating High Order Discontinuous Galerkin Discretization of the Boltzmann Collision Integral in $O(N^2)$ Operations Using the Discrete Fourier Transform
Department of Mathematics, California State University Nortrhidge, Nortrhdige, CA 91330, USA
In this paper we propose a novel numerical approach for the Boltzmann equation with uncertainties. The method combines the efficiency of classical direct simulation Monte Carlo (DSMC) schemes in the phase space together with the accuracy of stochastic Galerkin (sG) methods in the random space. This hybrid formulation makes it possible to construct methods that preserve the main physical properties of the solution along with spectral accuracy in the random space. The schemes are developed and analyzed in the case of space homogeneous problems as these contain the main numerical difficulties. Se…
We study uncertainty quantification for a Boltzmann-Poisson system that models electron transport in semiconductors and the physical collision mechanisms over the charges. We use the stochastic Galerkin method in order to handle the randomness associated with the problem. The main uncertainty in the Boltzmann equation concerns the initial conditions for a large number of particles, which is why the problem is formulated in terms of a probability density in phase space. The second source of uncertainty, directly related to the quantum nature of the problem, is the collision operator, as its str…
Deterministic solutions of the Boltzmann equation represent a real challenge due to the enormous computational effort which is required to produce such simulations and often stochastic methods such as Direct Simulation Monte Carlo (DSMC) are used instead due to their lower computational cost. In this work, we show that combining different technologies for the discretization of the velocity space and of the physical space coupled with suitable time integration techniques, it is possible to compute very precise deterministic approximate solutions of the Boltzmann model in different regimes, from…
We propose a deep learning based discontinuous Galerkin method (D2GM) to solve hyperbolic equations with discontinuous solutions and random uncertainties. The main computational challenges for such problems include discontinuities of the solutions and the curse of dimensionality due to uncertainties. Deep learning techniques have been favored for high-dimensional problems but face difficulties when the solution is not smooth, thus have so far been mainly used for viscous hyperbolic system that admits only smooth solutions. We alleviate this difficulty by setting up the loss function using disc…
In this paper, we focus on the construction of a hybrid scheme for the approximation of non-Maxwellian kinetic models with uncertainties. In the context of multiagent systems, the introduction of a kernel at the kinetic level is useful to avoid unphysical interactions. The methods here proposed, combine a direct simulation Monte Carlo (DSMC) in the phase space together with stochastic Galerkin (sG) methods in the random space. The developed schemes preserve the main physical properties of the solution together with accuracy in the random space. The consistency of the methods is tested with res…
Galerkin Methods for Boltzmann-Poisson transport with reflection conditions on rough boundaries
TU Wien - Institute for Analysis and Scientific Computing（维也纳工业大学分析与科学计算研究所）；The University of Texas at Austin - Institute for Computational Engineering and Sciences & Department of Mathematics（德克萨斯大学奥斯汀分校计算工程与科学学院及数学系）
Discontinuous Galerkin Deterministic Solvers for a Boltzmann-Poisson Model of Hot Electron Transport by Averaged Empirical Pseudopotential Band Structures
University of Texas at Austin（德克萨斯大学奥斯汀分校）；Michigan State University（密歇根州立大学）；University of Catania（卡塔尼亚大学）；Brown University（布朗大学）
A Reduced-Order Model for Nonlinear Radiative Transfer Problems Based on Moment Equations and POD-Petrov-Galerkin Projection of the Normalized Boltzmann Transport Equation
A Nodal Discontinuous Galerkin Method with Rank-Adaptive Velocity Space Representation for the Multiscale BGK Model
The University of Texas at Austin（德克萨斯大学奥斯汀分校）；Swarthmore College（斯沃斯莫尔学院）；Virginia Polytechnic and State University（弗吉尼亚理工大学）；University of Delaware（特拉华大学）；Los Alamos National Laboratory（洛斯阿拉莫斯国家实验室）
The study of uncertainty propagation is of fundamental importance in plasma physics simulations. To this end, in the present work we propose a novel stochastic Galerkin (sG) particle {method} for collisional kinetic models of plasmas under the effect of uncertainties. This class of methods is based on a generalized polynomial chaos (gPC) expansion of the particles' position and velocity. In details, we introduce a stochastic particle approximation for the Vlasov-Poisson system with a BGK term describing plasma collisions. A careful reformulation of such dynamics is needed to perform the sG pro…
Spectral methods, thanks to their high accuracy and the possibility to use fast algorithms, represent an effective way to approximate the collisional kinetic equations of Boltzmann type, such as the Boltzmann-Nordheim equation. This equation, modeled on the seminal Boltzmann equation, describes using a statistical physics formalism the time evolution of a gas composed of bosons or fermions. Using the spectral-Galerkin algorithm introduced in [F. Filbet, J. Hu, and S. Jin, ESAIM: Math. Model. Numer. Anal., 2011], together with some novel parallelization techniques, we investigate some of the co…
The quantification of multivariate uncertainties in partial differential equations can easily exceed any computing capacity unless proper measures are taken to reduce the complexity of the model. In this work, we propose a multidimensional Galerkin Proper Orthogonal Decomposition that optimally reduces each dimension of a tensorized product space. We provide the analytical framework and results that define and quantify the low-dimensional approximation. We illustrate its application for uncertainty modeling with Polynomial Chaos Expansions and show its efficiency in a numerical example.
We consider the method-of-moments approach to solve the Boltzmann equation of rarefied gas dynamics, which results in the following moment-closure problem. Given a set of moments, find the underlying probability density function. The moment-closure problem has infinitely many solutions and requires an additional optimality criterion to single-out a unique solution. Motivated from a discontinuous Galerkin velocity discretization, we consider an optimality criterion based upon L2-minimization. To ensure a positive solution to the moment-closure problem, we enforce positivity constraints on L2-mi…
We investigate a numerical behaviour of robust deterministic optimal control problem subject to a convection diffusion equation containing uncertain inputs. Stochastic Galerkin approach, turning the original optimization problem containing uncertainties into a large system of deterministic problems, is applied to discretize the stochastic domain, while a discontinuous Galerkin method is preferred for the spatial discretization due to its better convergence behaviour for optimization problems governed by convection dominated PDEs. Error analysis is done for the state and adjoint variables in th…
Hypocoercivity based Sensitivity Analysis and Spectral Convergence of the Stochastic Galerkin Approximation to Collisional Kinetic Equations with Multiple Scales and Random Inputs
The Institute for Computational Engineering and Sciences (ICES), University of Texas at Austin（德克萨斯大学奥斯汀分校计算工程与科学研究所）；Department of Mathematics, University of Wisconsin-Madison（威斯康星大学麦迪逊分校数学系）
Understanding fundamental kinetic processes is important for many problems, from plasma physics to gas dynamics. A first-principles approach to these problems requires a statistical description via the Boltzmann equation, coupled to appropriate field equations. In this paper we present a novel version of the discontinuous Galerkin (DG) algorithm to solve such kinetic equations. Unlike Monte-Carlo methods we use a continuum scheme in which we directly discretize the 6D phase-space using discontinuous basis functions. Our DG scheme eliminates counting noise and aliasing errors that would otherwi…
Wave propagation in real media is affected by various non-trivial physical phenomena, e.g., anisotropy, an-elasticity and dissipation. Assumptions on the stress-strain relationship are an integral part of seismic modeling and determine the deformation and relaxation of the medium. Stress-strain relationships based on simplified rheologies will incorrectly predict seismic amplitudes, which are used for quantitative reservoir characterization. Constitutive equations for the rheological model include the generalized Hooke's law and Boltzmann's superposition principal with dissipation models based…
We introduce Spline Moment Equations (SME) for kinetic equations using a new weighted spline ansatz of the distribution function and investigate the ansatz, the model, and its performance by simulating the one-dimensional Boltzmann-BGK equation. The new basis is composed of weighted constrained splines for the approximation of distribution functions that preserves mass, momentum, and energy. This basis is then used to derive moment equations using a Galerkin approach for a shifted and scaled Boltzmann-BGK equation, to allow for an accurate and efficient discretization in velocity space with an…
Spectral Gap Computations for Linearized Boltzmann Operators
The Institute for Computational Engineering and Sciences（计算工程与科学研究所）；Department of Mathematics（数学系）；ICES（计算工程与科学研究所）；The University of Texas at Austin（德克萨斯大学奥斯汀分校）
