arXiv:2408.02350v3 Announce Type: replace Abstract: In this paper, we present high-performance computing for the BGK model of the Boltzmann equation with a mesh-free method. For the numerical simulation of the BGK equation, we use an Arbitrary-Lagrangian-Eulerian (ALE) method developed in previous work, where the approximation of spatial derivatives and the reconstruction of a function are based on the weighted least squares method. A Graphics Processing Unit (GPU) is used to accelerate the code. The result is compared with sequential and parallel versions of the CPU code. Two- and three-dimensional driven cavity problems are solved, where a speed-up of several orders of magnitude is obtained compared to a sequential CPU simulation. The transport step is implemented as a node-parallel CUDA kernel for both two- and three-dimensional physical space: one GPU thread advances one particle-velocity-node pair by tracing its characteristic foot point and reconstructing the distribution there with a register-resident weighted least squares solve. For the three-dimensional case, no velocity reduction is applicable: a single distribution function $f$ is evolved on the full $N_v^3$ velocity grid, so the transport in every coordinate direction follows the resolved velocity node. The device memory footprint is kept at exactly twice the distribution storage by a structure-of-arrays layout with a single transport buffer; particle management runs on the host, so no device-side scratch copies of the cloud are required.