APTT: ACCURACY-PRESERVED TENSOR-TRAIN METHOD FOR THE BOLTZMANN-BGK EQUATION

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 BoltzmannBGK 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.