A Variance Reduction Approach for Monte Carlo Solutions of the Non-Linear Boltzmann Equation

We show that simple and efficient Monte Carlo-based solution methods for the Boltzmann equation for low-speed applications can be constructed by using appropriate variance reduction techniques. More specifically, we show that evaluation of the collision integral by sampling a representative number of collisions can be significantly accelerated by considering only the deviation from equilibrium, since this allows one to avoid considering a large number of collisions with no net effect. As the deviation from equilibrium decreases, the degree of variance reduction increases, leading to a signal to noise ratio that remains approximately constant. Thus, unlike particle techniques in which statistical sampling results in computational cost that is inversely proportional to the square of the Mach number, the approach presented here exhibits computational cost which is almost independent of the Mach number in the small Mach number limit. This is verified by numerical experiments at Mach numbers as low as O(10⁻⁵). We validate this approach by comparing its results with analytical and direct Monte Carlo simulations of the Boltzmann equation.