Solving the Boltzmann Equation on GPU

We show how to accelerate the direct solution of the Boltzmann equation using Graphics Processing Units (GPUs). In order to fully exploit the computational power of the GPU, we choose a method of solution which combines a finite difference discretization of the free-streaming term with a Monte Carlo evaluation of the collision integral. The efficiency of the code is demonstrated by solving the two-dimensional driven cavity flow. Computational results show that it is possible to cut down the computing time of the sequential code of two order of magnitudes. This makes the proposed method of solution a viable alternative to particle simulations for studying unsteady low Mach number flows.

💡 Research Summary

The paper presents a novel approach to accelerate the direct numerical solution of the Boltzmann equation by exploiting the massive parallelism of modern Graphics Processing Units (GPUs). Traditional methods for solving the Boltzmann equation fall into three broad categories: particle‑based Direct Simulation Monte Carlo (DSMC), semi‑regular (finite‑difference streaming with Monte Carlo collision evaluation), and regular deterministic schemes. While DSMC is widely used for rarefied gas dynamics, it becomes inefficient for low‑Mach, unsteady micro‑flows because it relies on extensive time averaging and a large number of simulated particles. Semi‑regular and regular methods, on the other hand, suffer from the “curse of dimensionality”: discretizing the six‑dimensional phase space (three spatial and three velocity dimensions) quickly exhausts memory and computational resources, limiting their practical applicability.

The authors adopt a semi‑regular strategy that combines a finite‑difference discretization of the free‑streaming term with a Monte Carlo evaluation of the collision integral. To reduce statistical variance, they reformulate the distribution function (f) as a perturbation around a Maxwellian equilibrium: \