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.

翻译：求解Boltzmann-BGK方程的传统数值方法因维度灾难而面临高昂的计算与内存开销。本文提出一种新型保精度张量列车（APTT）方法，用于高效求解Boltzmann-BGK方程。采用二阶有限差分格式对方程进行离散，在每个时间步生成张量代数系统。基于低秩TT表示，该张量代数系统可近似为基于TT的低秩系统，并通过TT修正交替最小二乘法（TT-MALS）求解器高效求解。得益于低秩TT表示，APTT方法相较于传统数值方法能大幅降低计算与内存成本。理论分析表明，APTT方法保持了与有限差分格式相同的收敛速率。通过多个基准测试案例验证了APTT方法的收敛速率与计算效率。