Nonlinear Boltzmann Equation Research Articles

Green's function and pointwise space-time behaviors of the three-dimensional relativistic Boltzmann equation

The pointwise space-time behavior of the Green's function of the three-dimensional relativistic Boltzmann equation is studied in this paper. It is shown that the Green's function has a decomposition of the macroscopic diffusive waves and Huygens waves with the speed a2+b2 at low-frequency, the singular kinetic wave and the remainder term decaying exponentially in space and time. In addition, we establish the pointwise space-time estimate of the global solution to the nonlinear relativistic Boltzmann equation with non-smooth initial data based on the Green's function.

The Boltzmann equation for plane Couette flow

In the paper, we study the plane Couette flow of a rarefied gas between two parallel infinite plates at y=\pm L moving relative to each other with opposite velocities (\pm \alpha L,0,0) along the x -direction. Assuming that the stationary state takes the specific form of F(y,v_{x}-\alpha y,v_{y},v_{z}) with the x -component of the molecular velocity sheared linearly along the y -direction, such steady flow is governed by a boundary value problem for a steady nonlinear Boltzmann equation driven by an external shear force under the homogeneous nonmoving diffuse reflection boundary condition. In the case of the Maxwell molecule collisions, we establish the existence of spatially inhomogeneous nonequilibrium stationary solutions to the steady problem for any small enough shear rate \alpha>0 via an elaborate perturbation approach using Caflisch’s decomposition together with Guo’s L^{\infty}\cap L^{2} theory. The result indicates a polynomial tail at large velocities for the stationary distribution. Moreover, the large time asymptotic stability of the stationary solution with exponential convergence is also obtained and as a consequence the nonnegativity of the steady profile is justified.

Determining the collision kernel in the Boltzmann equation near the equilibrium

We consider an inverse problem for the nonlinear Boltzmann equation near the equilibrium. Our goal is to determine the collision kernel in the Boltzmann equation from the knowledge of the Albedo operator. Our approach relies on a linearization technique as well as the injectivity of the Gauss-Weierstrass transform.

Multidisciplinary benchmarks of a conservative spectral solver for the nonlinear Boltzmann equation

The Boltzmann equation describes the evolution of the phase-space probability distribution of classical particles under binary collisions. Approximations to it underlie the basis for several scholarly fields, including aerodynamics and plasma physics. While these approximations are appropriate in their respective domains, they can be violated in niche but diverse applications which require direct numerical solution of the original nonlinear Boltzmann equation. An expanded implementation of the Galerkin–Petrov conservative spectral algorithm is employed to study a wide variety of physical problems. Enabled by distributed precomputation, solutions of the spatially homogeneous Boltzmann equation can be achieved in seconds on modern personal hardware, while spatially-inhomogeneous problems are solvable in minutes. Several benchmarks are presented focusing on accuracy compared to both analytic theoretical predictions and other Boltzmann solvers. These benchmarks span several physical domains including weakly ionized plasma, gaseous fluids, and atomic-plasma interaction.

Adjoint DSMC for nonlinear spatially-homogeneous Boltzmann equation with a general collision model

We derive an adjoint method for the Direct Simulation Monte Carlo (DSMC) method for the spatially homogeneous Boltzmann equation with a general collision law. This generalizes our previous results in Caflisch et al. (2021) [9], which was restricted to the case of Maxwell molecules, for which the collision rate is constant. The main difficulty in generalizing the previous results is that a rejection sampling step is required in the DSMC algorithm in order to handle the variable collision rate. We find a new term corresponding to the so-called score function in the adjoint equation and a new adjoint Jacobian matrix capturing the dependence of the collision parameter on the velocities. The new formula works for a much more general class of collision models.

Global solvability of a nonlinear Boltzmann equation

In this paper, based on the splitting method scheme, the existence and uniqueness theorem on the whole time interval t ∈ [0, T), T ≤ ∞ for the full nonlinear Boltzmann equation in the nonequilibrium case is proved where the intermolecular interactions are hard-sphere molecule and central forces. Considering the existence of a bounded solution in the space C, the strict positivity of the solution to the full nonlinear Boltzmann equation is proved when the initial function is positive. On the basis of this some mathematical justification of the H−theorem of Boltzmann is shown.

Time-dependent projection operator and nonlinear generalized master equations.

By introducing a special time-dependent projection operator P(t), the nonlinear version of the Nakajima-Zwanzig inhomogeneous generalized master equation(GME) for the relevant part of the N-particle distribution function containing the irrelevant initial correlation term is derived. For this projection operator the relevant part of the N-particle distribution function is a product of the distribution function of a group of s (s≤N) selected particles F_{s}(t) and the distribution function F_{N-s}(t) of N-s particles constituting the "environment" for a group of s selected particles. Thus, the linear projection operator approach results in the exact nonlinear GME for the relevant part of the distribution function, which is equivalent to the exact nonlinear GME for the distribution function of the complex of s particles of interest. This equationis further specified in the first approximation in the particle density, and then considered for one-particle and two-particle distribution functions. Given that there is no satisfactory way of dealing with the initial correlation term, the approach is suggested to rigorously include this term into consideration by converting the obtained inhomogeneous GME into homogeneous form containing initial correlations in the kernel governing the evolution of the relevant part of the distribution function. The homogeneous nonlinear GME for a one-particle distribution function is considered and the conditions for its equivalence to the nonlinear Boltzmann equationare discussed.

An Entropic Method for Discrete Systems with Gibbs Entropy

We consider general systems of ordinary differential equations with monotonic Gibbs entropy and introduce an entropic scheme that simply imposes an entropy fix after every time step of any existing time integrator. It is proved that in the general case, our entropy fix has only infinitesimal influence on the numerical order of the original scheme, and in many circumstances, it can be shown that the scheme does not affect the numerical order. Numerical experiments on the linear Fokker--Planck equation and nonlinear Boltzmann equation are carried out to support our numerical analysis.

Boundary Layer Solution of the Boltzmann Equation for Diffusive Reflection Boundary Conditions in Half-Space

We study steady Boltzmann equation in half-space, which arises in the Knudsen boundary layer problem, with diffusive reflection boundary conditions. Under certain admissible conditions and the source term decaying exponentially, we establish the existence of boundary layer solution for both linear and nonlinear Boltzmann equation in half-space with diffusive reflection boundary condition in $L^\infty_{x,v}$ when the far-field Mach number of the Maxwellian is zero. The continuity and the spacial decay of the solution are obtained. The uniqueness is established under some constraint conditions.

An entropy stable scheme for the non-linear Boltzmann equation

The kinetic equations govern the behavior of gaseous flows, compressibility, turbulence, reactions with internal energy exchange, that form the essence of numerous physical processes. Despite their wide applicability, their six-dimensional (seven including time) nature presents a huge computational challenge. With the advent of the modern high performance computing systems and few recent advances in numerical methods, it is now possible to numerically study the behavior of these systems. However, to understand the instability of rich molecular processes and find a structure in the apparent chaos, a scheme that is efficient in multi-dimensions, exhibits high parallel-efficiency, and foremost produces an entropy solution as per the theory of solutions of hyperbolic systems may prove useful. In this work, first, we construct an entropy stable flux for non-linear inhomogeneous (full) Boltzmann equation. Second, to ensure geometrical flexibility, we couple the scheme with a class of high order discontinuous Galerkin discretization (Jaiswal 2019 [41]) which satisfy summation-by-part (SBP) discretely. Third, utilizing SBP, we prove that the resulting semi-discrete scheme is locally and globally conservative in flat spaces; preserves the entropy-decay (H-theorem) property; efficient and simple; and therefore suitable for dealing with highly complex non-smooth flow problems. Fourth, we show that the fully-discrete kinetic scheme, utilizing an implicit-explicit time-discretization, in the limit of vanishing Knudsen number, becomes an entropy-stable explicit scheme applied to the Euler system. Fifth, we carry out a series of verification tests to illustrate the stability, accuracy, and conservation properties of the proposed method. These tests involve over 85 million degrees of freedom and over 50 billion operations per time-step.

Adjoint Dsmc for Nonlinear Spatially-Homogeneous Boltzmann Equation with a General Collision Model

Adjoint Dsmc for Nonlinear Spatially-Homogeneous Boltzmann Equation with a General Collision Model

Non-linear Boltzmann equation on hybrid-unstructured non-conforming multi-domains

Adaptivity is crucial for addressing practical challenges of the next decades. In this regard, recent surveys [117] highlight that it continues to be a major bottleneck in computational fluid dynamics workflow. We introduce mixed non-conforming discontinuous Galerkin discretization for the full Boltzmann equation in 2D/3D. These schemes have been designed for efficiency — motivated in part by spectral and isogeometric weighted collocation methods — and retain an optimal O(p+1) convergence for a p order approximation for non-linear kinetic systems on non-orthogonal grids. In this setting, it is possible to analyze highly complex problems of industrial strength i.e., structured, unstructured, mixed, irregular, multi-domain (multi-block) adaptive geometries at massively parallel scale (ten thousand cores or beyond). Mixed domains permit flexible mesh generation, whereas local nature of discontinuous Galerkin permits construction of adaptive numerical schemes that scale well. To address flows in mixing regime (low/high rarefaction), we couple the scheme with an asymptotic preserving implicit-explicit time discretization. These schemes are iteration free and applicable for a wide range of rarefied flows from free-molecular to continuum. To ensure stability in presence of shocks, we describe a method of constructing limiters on non-conforming grids. Finally, we show that the computational overhead for solving kinetic equations on non-conforming structured/unstructured domains is negligible relative to conforming domains. So, there is no reason to not prefer non-conforming unstructured domains.

Macroscopic approximation of a Fermi-Dirac statistics: Unbounded velocity space setting

An approximation by diffusion of a nonlinear Boltzmann equation modeling a Fermi-Dirac statistics is analyzed for an unbounded velocity space and Poisson coupling. A careful analysis of the entropy and entropy-dissipation allows to control the distribution function and to pass to the limit using duality method.

On the multi-species Boltzmann equation with uncertainty and its stochastic Galerkin approximation

In this paper the nonlinear multi-species Boltzmann equation with random uncertainty coming from the initial data and collision kernel is studied. Well-posedness and long-time behavior – exponential decay to the global equilibrium – of the analytical solution, and spectral gap estimate for the corresponding linearized gPC-based stochastic Galerkin system are obtained, by using and extending the analytical tools provided in [M. Briant and E.S. Daus,Arch. Ration. Mech. Anal.3(2016) 1367–1443] for the deterministic problem in the perturbative regime, and in [E.S. Daus, S. Jin and L. Liu,Kinet. Relat. Models12(2019) 909–922] for the single-species problem with uncertainty. The well-posedness result of the sensitivity system presented here has not been obtained so far neither in the single species case nor in the multi-species case.

Adjoint DSMC for nonlinear Boltzmann equation constrained optimization

Applications for kinetic equations such as optimal design and inverse problems often involve finding unknown parameters through gradient-based optimization algorithms. Based on the adjoint-state method, we derive two different frameworks for approximating the gradient of an objective functional constrained by the nonlinear Boltzmann equation. While the forward problem can be solved by the DSMC method, it is difficult to efficiently solve the high-dimensional continuous adjoint equation obtained by the “optimize-then-discretize” approach. This challenge motivates us to propose an adjoint DSMC method following the “discretize-then-optimize” approach for Boltzmann-constrained optimization. We also analyze the properties of the two frameworks and their connections. Several numerical examples are presented to demonstrate their accuracy and efficiency.

Reconstruction of the Collision Kernel in the Nonlinear Boltzmann Equation

Related DatabasesWeb of Science You must be logged in with an active subscription to view this.Article DataHistorySubmitted: 3 April 2020Accepted: 07 December 2020Published online: 16 February 2021Keywordsinverse problems, nonlinearity, Boltzmann equation, collision operatorAMS Subject Headings35R30, 35Q20Publication DataISSN (print): 0036-1410ISSN (online): 1095-7154Publisher: Society for Industrial and Applied MathematicsCODEN: sjmaah

Subdynamics of fluctuations in an equilibrium classical many-particle system and generalized linear Boltzmann and Landau equations.

Exact completely closed homogeneous generalized master equations (GMEs) governing the evolution in time of equilibrium two-time correlation functions for dynamic variables of a subsystem of s particles (s<N) selected from N≫1 particles of a classical many-body system are obtained. These time-convolution and time-convolutionless GMEs differ from the known GMEs (e.g., Nakajima-Zwanzig GME) by the absence of inhomogeneous terms containing correlations between all N particles at the initial moment of time and preventing the closed description of s-particle subsystem evolution. Closed homogeneous GMEs describing the subdynamics of fluctuations are obtained by applying a special projection operator to the Liouville-type equation governing the dynamics of the correlation function with the related to the Gibbs distribution initial state, which is more natural than the conventional factorized initial state. No common approximation, like the "molecular chaos," is needed. In the linear approximation in the particles' density, the linear generalized Boltzmann equation accounting for initial correlations and valid at all timescales is obtained. This equation for a weak interparticle interaction converts into the generalized linear Landau equation in which the initial correlations are also accounted for. The connection of these equations to the nonlinear Boltzmann and Landau equations is discussed. The same approach is applicable to studying the kinetics of the conventional reduced s-particle distribution functions for a classical N-particle system driven from an equilibrium state by an external force.

Investigations to the price evolutions of goods exchange with CES utility functions

The price evolutions of goods exchange in multi-agent market are investigated by employing the kinetic theory. Assume that a closed market consists of two groups (A and B) of agents, who have the same preferences for goods. The constant elasticity of substitution (CES) utility function which contains Cobb–Douglas utility function and Leontief utility function, is used to describe the agent’s preferences. A system of linear Boltzmann type equation is given to describe the probability distributions of goods for agents of the two groups, in which the agents are allowed to adopt certain trading strategies to maximize their utility and improve their wealth conditions. We assume that agents in group A put all their goods into the market for maximum utility, while agents in group B aim to acquire maximal utility from goods exchange by suitably selecting the percentage of goods which are exchanged. The effects of different trading strategies on commodity price and wealth distribution are discussed. The results show that the trading strategies of agents in group B have effectively improved their wealth status under certain assumptions. Also, a general system of nonlinear Boltzmann equation for the probability distributions of goods is given to describe the pricing issues of goods exchange for agents. The numerical experiments of nonlinear Boltzmann equations demonstrate how the trading strategies and preferences of agents modify the price of goods.

Construction of the collision integral kernels for the hard spheres model

Recurrence procedure for constructing kernels of the integral operators arising in expansion of the nonlinear Boltzmann collision integral in Legendre polynomials is realised in “Mathematica” system for the hard spheres gas. All the kernels with increasing indices corresponding to the Legendre polynomials indices are subsequently calculated up to sum of indices equal to 40. The main features of the kernels as functions of their indices are considered. The kernels can be used to solve the nonlinear Boltzmann equation in strongly anisotropic case.

On stochastic Galerkin approximation of the nonlinear Boltzmann equation with uncertainty in the fluid regime

The Boltzmann equation may contain uncertainties in initial/boundary data or collision kernel. To study the impact of these uncertainties, a stochastic Galerkin (sG) method was proposed in [18] and studied in the kinetic regime. When the system is close to the fluid regime (the Knudsen number is small), the method would become prohibitively expensive due to the stiff collision term. In this work, we develop efficient sG methods for the Boltzmann equation that work for a wide range of Knudsen numbers, and investigate, in particular, their behavior in the fluid regime.