Symmetry of the nonlinear collision operator matrix and new prospects in the moment method for solving the Boltzmann equation

1999 ◽  
Vol 44 (9) ◽  
pp. 1005-1008
Author(s):  
A. Ya. Énder ◽  
I. A. Énder
Keyword(s):  
Boltzmann Equation ◽  
Moment Method ◽  
Collision Operator ◽  
Operator Matrix ◽  
The Boltzmann Equation ◽  
The Moment

Convergence of A Distributional Monte Carlo Method for the Boltzmann Equation

2012 ◽  
Vol 4 (1) ◽  
pp. 102-121 ◽  
Author(s):  
Christopher R. Schrock ◽  
Aihua W. Wood
Keyword(s):  
Monte Carlo ◽  
Boltzmann Equation ◽  
Substantial Reduction ◽  
Direct Simulation Monte Carlo ◽  
Collision Operator ◽  
Direct Simulation ◽  
The Boltzmann Equation ◽  
Distributional Method ◽  
Point Measure ◽  
Simulation Monte Carlo

AbstractDirect Simulation Monte Carlo (DSMC) methods for the Boltzmann equation employ a point measure approximation to the distribution function, as simulated particles may possess only a single velocity. This representation limits the method to converge only weakly to the solution of the Boltzmann equation. Utilizing kernel density estimation we have developed a stochastic Boltzmann solver which possesses strong convergence for bounded and L∞ solutions of the Boltzmann equation. This is facilitated by distributing the velocity of each simulated particle instead of using the point measure approximation inherent to DSMC. We propose that the development of a distributional method which incorporates distributed velocities in collision selection and modeling should improve convergence and potentially result in a substantial reduction of the variance in comparison to DSMC methods. Toward this end, we also report initial findings of modeling collisions distributionally using the Bhatnagar-Gross-Krook collision operator.

On the Kinetic Theory of the Enskog Fluid. Viscosity and Viscoelasticity, Heat Conduction and Thermal Pressure

1981 ◽  
Vol 36 (6) ◽  
pp. 545-553 ◽  
Author(s):  
G. Schmidt ◽  
W. E. Köhler ◽  
S. Hess
Keyword(s):  
Heat Conduction ◽  
Boltzmann Equation ◽  
Kinetic Theory ◽  
Moment Method ◽  
Solution Procedure ◽  
Fluid Viscosity ◽  
Thermal Pressure ◽  
Dense Fluid ◽  
The Moment ◽  
Dependent Viscosity

Abstract The moment method is applied to the linearized Enskog-Boltzmann equation for a dense gas. Thus an enlarged set of equations of thermo-hydrodynamics is obtained which allows to go beyond ordinary hydrodynamics. The resulting expressions for the density dependent viscosity and heat conductivity coincide with those previously obtained with the help of the Chapman- Enskog solution procedure. In addition, however, the frequency dependence of the viscosity is treated and it is demonstrated that the thermal pressure does not vanish in a dense fluid

DISSIPATIVE DISCRETIZATION METHODS FOR APPROXIMATIONS TO THE BOLTZMANN EQUATION

2001 ◽  
Vol 11 (01) ◽  
pp. 133-148 ◽  
Author(s):  
CHRISTIAN RINGHOFER
Keyword(s):  
Boltzmann Equation ◽  
Discrete System ◽  
Finite Difference Methods ◽  
Collision Operator ◽  
Entropy Condition ◽  
Discretization Methods ◽  
The Boltzmann Equation ◽  
Galerkin Approximations ◽  
Difference Methods ◽  
Model Equations

This paper deals with the spatial discretization of partial differential equations arising from Galerkin approximations to the Boltzmann equation, which preserves the entropy properties of the original collision operator. A general condition on finite difference methods is derived, which guarantees that the discrete system satisfies the appropriate equivalent of the entropy condition. As an application of this concept, entropy producing difference methods for the hydrodynamic model equations and for spherical harmonics expansions are presented.

A Stochastic Galerkin Method for the Boltzmann Equation with Multi-Dimensional Random Inputs Using Sparse Wavelet Bases

2017 ◽  
Vol 10 (2) ◽  
pp. 465-488 ◽  
Author(s):  
Ruiwen Shu ◽  
Jingwei Hu ◽  
Shi Jin
Keyword(s):  
Boltzmann Equation ◽  
Galerkin Method ◽  
Collision Operator ◽  
Collision Kernel ◽  
Wavelet Bases ◽  
Random Inputs ◽  
Stochastic Galerkin Method ◽  
The Boltzmann Equation ◽  
Accuracy Result ◽  
Stochastic Galerkin

AbstractWe propose a stochastic Galerkin method using sparse wavelet bases for the Boltzmann equation with multi-dimensional random inputs. Themethod uses locally supported piecewise polynomials as an orthonormal basis of the random space. By a sparse approach, only a moderate number of basis functions is required to achieve good accuracy in multi-dimensional random spaces. We discover a sparse structure of a set of basis-related coefficients, which allows us to accelerate the computation of the collision operator. Regularity of the solution of the Boltzmann equation in the random space and an accuracy result of the stochastic Galerkin method are proved in multi-dimensional cases. The efficiency of the method is illustrated by numerical examples with uncertainties from the initial data, boundary data and collision kernel.

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