High order numerical methods for the space non-homogeneous Boltzmann equation. In this paper we present accurate methods for the numerical solution of the Boltzmann equation of rarefied gas. The methods are based on a time splitting technique. The transport is solved by a third order accurate (in space) positive and flux conservative (PFC) method. The collision step is treated by a Fourier approximation of the collision integral, which guarantees spectral accuracy in velocity, coupled with several high order integrators in time. Strang splitting is used to achieve second order accuracy in space and time. Several numerical tests illustrate the properties of the methods.

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  1. Dimarco, Giacomo; Loubère, Raphaël; Narski, Jacek; Rey, Thomas: An efficient numerical method for solving the Boltzmann equation in multidimensions (2018)
  2. Aoki, Kazuo; Baranger, Céline; Hattori, Masanari; Kosuge, Shingo; Martalò, Giorgio; Mathiaud, Julien; Mieussens, Luc: Slip boundary conditions for the compressible Navier-Stokes equations (2017)
  3. Gamba, Irene M.; Haack, Jeffrey R.; Hauck, Cory D.; Hu, Jingwei: A fast spectral method for the Boltzmann collision operator with general collision kernels (2017)
  4. Cho, H.; Venturi, D.; Karniadakis, G.E.: Numerical methods for high-dimensional probability density function equations (2016)
  5. Liu, Chang; Xu, Kun; Sun, Quanhua; Cai, Qingdong: A unified gas-kinetic scheme for continuum and rarefied flows. IV: Full Boltzmann and model equations (2016)
  6. Dimarco, Giacomo; Loubère, Raphaël; Narski, Jacek: Towards an ultra efficient kinetic scheme. part III: high-performance-computing (2015)
  7. Dimarco, Giacomo; Loubère, Raphaël; Rispoli, Vittorio: A multiscale fast semi-Lagrangian method for rarefied gas dynamics (2015)
  8. Alekseenko, A.; Josyula, E.: Deterministic solution of the spatially homogeneous Boltzmann equation using discontinuous Galerkin discretizations in the velocity space (2014)
  9. Dimarco, Giacomo; Loubere, Raphaël: Towards an ultra efficient kinetic scheme. I: Basics on the BGK equation (2013)
  10. Dimarco, Giacomo; Loubere, Raphaël: Towards an ultra efficient kinetic scheme. II: The high order case (2013)
  11. Kudryavtsev, A.N.; Shershnev, A.A.: A numerical method for simulation of microflows by solving directly kinetic equations with WENO schemes (2013)
  12. Wu, Lei; White, Craig; Scanlon, Thomas J.; Reese, Jason M.; Zhang, Yonghao: Deterministic numerical solutions of the Boltzmann equation using the fast spectral method (2013)
  13. Alaia, Alessandro; Puppo, Gabriella: A hybrid method for hydrodynamic-kinetic flow.II: Coupling of hydrodynamic and kinetic models (2012)
  14. Filbet, Francis: On deterministic approximation of the Boltzmann equation in a bounded domain (2012)
  15. Alaia, Alessandro; Puppo, Gabriella: A hybrid method for hydrodynamic-kinetic flow. I: A particle-grid method for reducing stochastic noise in kinetic regimes (2011)
  16. Dimarco, Giacomo; Pareschi, Lorenzo: Exponential Runge-Kutta methods for stiff kinetic equations (2011)
  17. Filbet, Francis; Jin, Shi: An asymptotic preserving scheme for the ES-BGK model of the Boltzmann equation (2011)
  18. Filbet, Francis; Mouhot, Clément: Analysis of spectral methods for the homogeneous Boltzmann equation (2011)
  19. Filbet, Francis; Jin, Shi: A class of asymptotic-preserving schemes for kinetic equations and related problems with stiff sources (2010)
  20. Gamba, Irene M.; Tharkabhushanam, Sri Harsha: Shock and boundary structure formation by spectral-Lagrangian methods for the inhomogeneous Boltzmann transport equation (2010)

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