A Conservative Discontinuous Galerkin Solver for the Space Homogeneous Boltzmann Equation for Binary Interactions

In the present work, we propose a deterministic numerical solver for the space homogeneous Boltzmann equation based on discontinuous Galerkin (DG) methods. Such an application has been rarely studi...

[1]  Yingda Cheng,et al.  Positivity-preserving discontinuous Galerkin schemes for linear Vlasov-Boltzmann transport equations , 2012, Math. Comput..

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

[3]  C. Villani,et al.  Upper Maxwellian Bounds for the Spatially Homogeneous Boltzmann Equation , 2007 .

[4]  J. Broadwell,et al.  Study of rarefied shear flow by the discrete velocity method , 1964, Journal of Fluid Mechanics.

[5]  Irene M. Gamba,et al.  Spectral Gap Computations for Linearized Boltzmann Operators , 2018, 1807.09868.

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

[7]  E. H. van Brummelen,et al.  A discontinuous Galerkin finite-element method for a 1D prototype of the Boltzmann equation , 2011, J. Comput. Phys..

[8]  V. Aristov.  Direct Methods for Solving the Boltzmann Equation and Study of Nonequilibrium Flows , 2001 .

[9]  S. Rjasanow,et al.  Difference scheme for the Boltzmann equation based on the Fast Fourier Transform , 1997 .

[10]  George Bosilca,et al.  Open MPI: Goals, Concept, and Design of a Next Generation MPI Implementation , 2004, PVM/MPI.

[11]  Yingda Cheng,et al.  Study of conservation and recurrence of Runge–Kutta discontinuous Galerkin schemes for Vlasov–Poisson systems , 2012, J. Sci. Comput..

[12]  Graeme A. Bird,et al.  Molecular Gas Dynamics , 1976 .

[13]  Chi-Wang Shu,et al.  Runge–Kutta Discontinuous Galerkin Methods for Convection-Dominated Problems , 2001, J. Sci. Comput..

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

[15]  Yingda Cheng,et al.  A discontinuous Galerkin solver for Boltzmann-Poisson systems in nano devices , 2009, 0902.3514.

[16]  Yingda Cheng,et al.  Discontinuous Galerkin methods for the Boltzmann‐Poisson systems in semiconductor device simulations , 2011 .

[17]  Lorenzo Pareschi,et al.  Numerical Solution of the Boltzmann Equation I: Spectrally Accurate Approximation of the Collision Operator , 2000, SIAM J. Numer. Anal..

[18]  J. Cooper.  SINGULAR INTEGRALS AND DIFFERENTIABILITY PROPERTIES OF FUNCTIONS , 1973 .

[19]  Chenglong Zhang,et al.  A conservative scheme for Vlasov Poisson Landau modeling collisional plasmas , 2016, J. Comput. Phys..

[20]  W. Wagner,et al.  A Stochastic Weighted Particle Method for the Boltzmann Equation , 1996 .

[21]  Henri Cabannes.  Global solution of the initial value problem for the discrete Boltzmann equation , 1978 .

[22]  S. Rjasanow,et al.  Fast deterministic method of solving the Boltzmann equation for hard spheres , 1999 .

[23]  A. Bobylev,et al.  Exact solutions of the nonlinear Boltzmann equation and the theory of relaxation of a Maxwellian gas , 1984 .

[24]  Yingda Cheng,et al.  A brief survey of the discontinuous Galerkin method for the Boltzmann-Poisson equations , 2011 .

[25]  Kenichi Nanbu,et al.  Direct simulation scheme derived from the Boltzmann equation. I - Monocomponent gases. II - Multicom , 1980 .

[26]  Yingda Cheng,et al.  Discontinuous Galerkin Methods for the Vlasov-Maxwell Equations , 2013, SIAM J. Numer. Anal..

[27]  Matthias Abend.  Stochastic Numerics For The Boltzmann Equation , 2016 .

[28]  Ricardo Alonso,et al.  A New Approach to the Creation and Propagation of Exponential Moments in the Boltzmann Equation , 2013 .

[29]  Irene M. Gamba,et al.  High performance computing with a conservative spectral Boltzmann solver , 2012, 1211.0540.

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

[31]  C. Cercignani.  The Boltzmann equation and its applications , 1988 .

[32]  M. E. Galassi,et al.  GNU SCIENTI C LIBRARY REFERENCE MANUAL , 2005 .

[33]  Sergej Rjasanow,et al.  Numerical solution of the Boltzmann Equation using fully conservative difference scheme based on the Fast Fourier Transform , 2000 .

[34]  L. C. Pitchford,et al.  A Numerical Solution of the Boltzmann Equation , 1983 .

[35]  Mohammed Seaïd,et al.  Discrete-Velocity Models and Relaxation Schemes for Traffic Flows , 2006, SIAM J. Sci. Comput..

[36]  Yingda Cheng,et al.  A Discontinuous Galerkin Solver for Full-Band Boltzmann-Poisson Models , 2009, 2009 13th International Workshop on Computational Electronics.

[37]  L. Mieussens.  Discrete-Velocity Models and Numerical Schemes for the Boltzmann-BGK Equation in Plane and Axisymmetric Geometries , 2000 .

[38]  G. Toscani,et al.  Relaxation Schemes for Nonlinear Kinetic Equations , 1997 .

[39]  Francis Filbet,et al.  High order numerical methods for the space non-homogeneous Boltzmann equation , 2003 .

[40]  D. B. Goldstein,et al.  Monte Carlo solution of the Boltzmann equation via a discrete velocity model , 2011, J. Comput. Phys..

[41]  Carlo Cercignani,et al.  Shock Waves for a Discrete Velocity Gas Mixture , 2000 .

[42]  Henning Struchtrup,et al.  A linearization of Mieussens's discrete velocity model for kinetic equations , 2007 .

[43]  David B. Goldstein,et al.  Improvement of a Discrete Velocity Boltzmann Equation Solver With Arbitrary Post‐Collision Velocities , 2009 .

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

[45]  Irene M. Gamba,et al.  A conservative spectral method for the Boltzmann equation with anisotropic scattering and the grazing collisions limit , 2013, J. Comput. Phys..

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

[47]  Xiangxiong Zhang,et al.  On positivity-preserving high order discontinuous Galerkin schemes for compressible Euler equations on rectangular meshes , 2010, J. Comput. Phys..

[48]  Shuichi Kawashima.  Global solution of the initial value problem for a discrete velocity model of the Boltzmann equation , 1981 .

[49]  David B. Goldstein,et al.  Variance Reduction For A Discrete Velocity Gas , 2011 .

[50]  P. J. Morrison,et al.  A discontinuous Galerkin method for the Vlasov-Poisson system , 2010, J. Comput. Phys..

[51]  Irene M. Gamba,et al.  Moment Inequalities and High-Energy Tails for Boltzmann Equations with Inelastic Interactions , 2004 .

[52]  Sergej Rjasanow,et al.  Numerical solution of the Boltzmann equation on the uniform grid , 2002, Computing.

[53]  Carlo Cercignani,et al.  Discrete Velocity Models Without Nonphysical Invariants , 1999 .

[54]  Irene M. Gamba,et al.  Convergence and Error Estimates for the Lagrangian-Based Conservative Spectral Method for Boltzmann Equations , 2016, SIAM J. Numer. Anal..