Monte Carlo method for the Spatially Homogenous Boltzmann equation
This work presents a Monte Carlo computational approach for solving the spatially homo-
geneous nonlinear Boltzmann equation through a modified “particle-in-cells” model. Building upon the
Belotserkovski-Yanitski and Kac models, the paper introduces a stochastic simulation scheme that synthe-
sizes the principle of splitting operators with branching Markov processes. The method constructs unbiased
estimators with small variance to compute molecular velocities and particle distributions, providing higher
numerical accuracy and physical realism compared to traditional linear models. The developed algorithm
enables efficient simulation of rarefied gas dynamics and demonstrates stable results across varying particle
densities. Computational experiments, including shock wave structure modeling, show that this approach
achieves improved convergence and fidelity in approximating spatially heterogeneous Boltzmann behavior.
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