Numerical Regularized Moment Method of Arbitrary Order for Boltzmann-BGK Equation

We introduce a numerical method for solving Grad's moment equations or regularized moment equations for an arbitrary order of moments. In our algorithm, we do not explicitly need the moment equations. Instead, we directly start from the Boltzmann equation and perform Grad's moment method [H. Grad, Commun. Pure Appl. Math., 2 (1949), pp. 331-407] and the regularization technique [H. Struchtrup and M. Torrilhon, Phys. Fluids, 15 (2003), pp. 2668-2680] numerically. We define a conservative projection operator and propose a fast implementation, which makes it convenient to add up two distributions and provides more efficient flux calculations compared with the classic method using explicit expressions of flux functions. For the collision term, the BGK model is adopted so that the production step can be done trivially based on the Hermite expansion. Extensive numerical examples for one- and two-dimensional problems are presented. Convergence in moments can be validated by the numerical results for different numbers of moments.

[1]  Guangjun Liu,et al.  A method for constructing a model form for the Boltzmann equation , 1990 .

[2]  D. R. Emerson,et al.  A computational strategy for the regularized 13 moment equations with enhanced wall-boundary conditions , 2007, J. Comput. Phys..

[3]  P. Lax,et al.  On Upstream Differencing and Godunov-Type Schemes for Hyperbolic Conservation Laws , 1983 .

[4]  Manuel Torrilhon,et al.  Characteristic waves and dissipation in the 13-moment-case , 2000 .

[5]  Lowell H. Holway,et al.  New Statistical Models for Kinetic Theory: Methods of Construction , 1966 .

[6]  S. Chapman,et al.  On the Law of Distribution of Molecular Velocities, and on the Theory of Viscosity and Thermal Conduction, in a Non-Uniform Simple Monatomic Gas , 1916 .

[7]  E. Toro.  Riemann Solvers and Numerical Methods for Fluid Dynamics , 1997 .

[8]  R. Illner,et al.  The mathematical theory of dilute gases , 1994 .

[9]  A. Bobylev,et al.  The Chapman-Enskog and Grad methods for solving the Boltzmann equation , 1982 .

[10]  Manuel Torrilhon,et al.  Modeling Micro Mass and Heat Transfer for Gases Using Extended Continuum Equations , 2009 .

[11]  H. Struchtrup,et al.  Regularization of Grad’s 13 moment equations: Derivation and linear analysis , 2003 .

[12]  Manuel Torrilhon,et al.  The Riemann-problem in extended thermodynamics , 2001 .

[13]  Jie Shen,et al.  Spectral and High-Order Methods with Applications , 2006 .

[14]  H. Struchtrup.  Macroscopic transport equations for rarefied gas flows , 2005 .

[15]  H. Grad.  On the kinetic theory of rarefied gases , 1949 .

[16]  J. D. Aua.  The shock tube study in extended thermodynamics , 2001 .

[17]  Ingo Müller,et al.  Waves in Extended Thermodynamics , 1998 .

[18]  H. Grad.  Principles of the Kinetic Theory of Gases , 1958 .

[19]  Harold Grad,et al.  The profile of a steady plane shock wave , 1952 .

[20]  Manuel Torrilhon,et al.  Stability and consistency of kinetic upwinding for advection–diffusion equations , 2006 .

[21]  Jason M. Reese,et al.  Computational framework for the regularized 20‐moment equations for non‐equilibrium gas flows , 2008 .

[22]  J. Maxwell,et al.  On Stresses in Rarified Gases Arising from Inequalities of Temperature , 2022 .

[23]  Ernst Rank,et al.  Discretization of the Boltzmann equation in velocity space using a Galerkin approach , 2000 .

[24]  Lorenzo Pareschi,et al.  Modeling and Computational Methods for Kinetic Equations , 2012 .

[25]  J. Boon.  The Lattice Boltzmann Equation for Fluid Dynamics and Beyond , 2003 .

[26]  X. He,et al.  Discretization of the Velocity Space in the Solution of the Boltzmann Equation , 1997, comp-gas/9712001.

[27]  M. Abramowitz,et al.  Handbook of Mathematical Functions With Formulas, Graphs and Mathematical Tables (National Bureau of Standards Applied Mathematics Series No. 55) , 1965 .

[28]  Luc Mieussens,et al.  DISCRETE VELOCITY MODEL AND IMPLICIT SCHEME FOR THE BGK EQUATION OF RAREFIED GAS DYNAMICS , 2000 .

[29]  Manuel Torrilhon,et al.  Two-Dimensional Bulk Microflow Simulations Based on Regularized Grad's 13-Moment Equations , 2006, Multiscale Model. Simul..

[30]  M. H. Ernst,et al.  Nonlinear model-Boltzmann equations and exact solutions , 1981 .

[31]  Manuel Torrilhon,et al.  Regularized 13-moment equations: shock structure calculations and comparison to Burnett models , 2004, Journal of Fluid Mechanics.

[32]  E. M. Shakhov.  Generalization of the Krook kinetic relaxation equation , 1968 .

[33]  P. Bhatnagar,et al.  A Model for Collision Processes in Gases. I. Small Amplitude Processes in Charged and Neutral One-Component Systems , 1954 .

[34]  Manuel Torrilhon,et al.  Boundary conditions for regularized 13-moment-equations for micro-channel-flows , 2008, J. Comput. Phys..

[35]  Manuel Torrilhon,et al.  Explicit fluxes and productions for large systems of the moment method based on extended thermodynamics , 2003 .

[36]  Manuel Torrilhon,et al.  Essentially optimal explicit Runge–Kutta methods with application to hyperbolic–parabolic equations , 2007, Numerische Mathematik.