  Solving the Boltzmann-BGK equation with traditional numerical methods suffers
from high computational and memory costs due to the curse of dimensionality. In
this paper, we propose a novel accuracy-preserved tensor-train (APTT) method to
efficiently solve the Boltzmann-BGK equation. A second-order finite difference
scheme is applied to discretize the Boltzmann-BGK equation, resulting in a
tensor algebraic system at each time step. Based on the low-rank TT
representation, the tensor algebraic system is then approximated as a TT-based
low-rank system, which is efficiently solved using the TT-modified alternating
least-squares (TT-MALS) solver. Thanks to the low-rank TT representation, the
APTT method can significantly reduce the computational and memory costs
compared to traditional numerical methods. Theoretical analysis demonstrates
that the APTT method maintains the same convergence rate as that of the finite
difference scheme. The convergence rate and efficiency of the APTT method are
validated by several benchmark test cases.
