An Implicit Monte Carlo Method for Rarefied Gas Dynamics: I. the Space Homogeneous Case

Lorenzo Pareschi, Russel E. Caflisch

Research output: Contribution to journalArticlepeer-review


For the space homogeneous Boltzmann equation, we formulate a hybrid Monte Carlo method that is robust in the fluid dynamic limit. This method is based on an analytic representation of the solution over a single time step and involves implicit time differencing derived from a suitable power series expansion of the solution (a generalized Wild expansion). A class of implicit, yet explicitly implementable, numerical schemes is obtained by substituting a Maxwellian distribution in place of the high order terms in the expansion. The numerical solution is represented as a convex combination of a non-equilibrium particle distribution and a Maxwellian. The hybrid distribution is then evolved by Monte Carlo using the implicit formulation for the time evolution. Computational simulations of spatially homogeneous problems by our method are presented here for the Kac model and for the variable hard sphere model (including Maxwell molecules). Comparison to exact solutions and to direct simulation Monte Carlo (DSMC) computations shows the robustness and the efficiency of the new method.

Original languageEnglish (US)
Pages (from-to)90-116
Number of pages27
JournalJournal of Computational Physics
Issue number1
StatePublished - Sep 1 1999


  • Boltzmann equation
  • Fluid dynamic limit
  • Implicit time discretizations
  • Monte Carlo methods

ASJC Scopus subject areas

  • Numerical Analysis
  • Modeling and Simulation
  • Physics and Astronomy (miscellaneous)
  • General Physics and Astronomy
  • Computer Science Applications
  • Computational Mathematics
  • Applied Mathematics


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