A deterministic-stochastic method for computing the Boltzmann collision integral in $\mathcal{O}(MN)$ operations

We developed and implemented a numerical algorithm for evaluating the Boltzmann collision integral with \begin{document}$O(MN)$\end{document} operations, where \begin{document}$N$\end{document} is the number of the discrete velocity points and \begin{document}$M . At the base of the algorithm are nodal-discontinuous Galerkin discretizations of the collision operator on uniform grids and a bilinear convolution form of the Galerkin projection of the collision operator. Efficiency of the algorithm is achieved by applying singular value decomposition compression of the discrete collision kernel and by approximating the kinetic solution by a sum of Maxwellian streams using a stochastic likelihood maximization algorithm. Accuracy of the method is established on solutions to the problem of spatially homogeneous relaxation.

[1]  Yonghao Zhang,et al.  A fast spectral method for the Boltzmann equation for monatomic gas mixtures , 2015, J. Comput. Phys..

[2]  Lorenzo Pareschi,et al.  Solving the Boltzmann Equation in N log2N , 2006, SIAM J. Sci. Comput..

[3]  C. D. Levermore,et al.  Moment closure hierarchies for kinetic theories , 1996 .

[4]  Hans Babovsky,et al.  A Discrete Boltzmann Equation Based on a Cub-Octahedron in R3 , 2008, SIAM J. Sci. Comput..

[5]  Chenglong Zhang,et al.  A conservative discontinuous Galerkin scheme with O(N2) operations in computing Boltzmann collision weight matrix , 2014 .

[6]  Alexei Heintz,et al.  A new consistent discrete‐velocity model for the Boltzmann equation , 2002 .

[7]  A. Alekseenko,et al.  Deterministic solution of the spatially homogeneous Boltzmann equation using discontinuous Galerkin discretizations in the velocity space , 2013, J. Comput. Phys..

[8]  P. Vedula,et al.  Validation of a collisional Lattice Boltzmann Method , 2011 .

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

[10]  N. Gimelshein,et al.  Origins of radiometric forces on a circular vane with a temperature gradient , 2009, Journal of Fluid Mechanics.

[11]  Lorenzo Pareschi,et al.  A Fourier spectral method for homogeneous boltzmann equations , 1996 .

[12]  E. Josyula,et al.  Deterministic solution of the Boltzmann equation using a discontinuous Galerkin velocity discretization , 2012 .

[13]  S. Rjasanow,et al.  A Weak Formulation of the Boltzmann Equation Based on the Fourier Transform , 2007 .

[14]  D. Goldstein,et al.  Generation of a hybrid DSMC/CFD solution for gas mixtures with internal degrees of freedom , 2012 .

[15]  Yonghao Zhang,et al.  Deterministic numerical solutions of the Boltzmann equation using the fast spectral method , 2013, J. Comput. Phys..

[16]  Irene M. Gamba,et al.  SHOCK AND BOUNDARY STRUCTURE FORMATION BY SPECTRAL-LAGRANGIAN METHODS FOR THE INHOMOGENEOUS BOLTZMANN TRANSPORT EQUATION * , 2010 .

[17]  Jonathan M. Burt,et al.  Novel Cartesian Implementation of the Direct Simulation Monte Carlo Method , 2012 .

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

[19]  Ivo D. Dinov,et al.  Expectation Maximization and Mixture Modeling Tutorial , 2008 .

[20]  Armando Majorana.  A numerical model of the Boltzmann equation related to the discontinuous Galerkinmethod , 2011 .

[21]  Prakash Vedula,et al.  Quadrature-Based Moment Model for Moderately Dense Polydisperse Gas-Particle Flows , 2010 .

[22]  Lorenzo Pareschi,et al.  Fast algorithms for computing the Boltzmann collision operator , 2006, Math. Comput..

[23]  Luc Devroye,et al.  General Principles in Random Variate Generation , 1986 .

[24]  D. Kalman.  A Singularly Valuable Decomposition: The SVD of a Matrix , 1996 .

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

[26]  F. G. Tcheremissine,et al.  Method for solving the Boltzmann kinetic equation for polyatomic gases , 2012 .

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

[28]  Christopher R. Schrock,et al.  Convergence of A Distributional Monte Carlo Method for the Boltzmann Equation , 2012 .

[29]  Iain D. Boyd.  Vectorization of a Monte Carlo simulation scheme for nonequilibrium gas dynamics , 1991 .

[30]  Irene M. Gamba,et al.  A spectral-Lagrangian Boltzmann solver for a multi-energy level gas , 2013, J. Comput. Phys..