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An implicit Monte Carlo method for rarefied gas dynamics. I: The space homogeneous case. (English) Zbl 0953.76075

Summary: 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 method, using the implicit formulation for the time evolution. Computational simulations of spatially homogeneous problems are presented for the Kac model and for the variable hard sphere model (including Maxwell molecules). Comparison to exact solutions and to direct simulation Monte Carlo computations shows the robustness and the efficiency of the new method.

MSC:

76M35 Stochastic analysis applied to problems in fluid mechanics
76P05 Rarefied gas flows, Boltzmann equation in fluid mechanics
65C05 Monte Carlo methods
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