In this study, we introduce a tensor-train (TT) finite difference WENO method for solving compressible Euler equations. In a step-by-step manner, the tensorization of the governing equations is demonstrated. We also introduce \emph{LF-cross} and \emph{WENO-cross} methods to compute numerical fluxes and the WENO reconstruction using the cross interpolation technique. A tensor-train approach is developed for boundary condition types commonly encountered in Computational Fluid Dynamics (CFD). The performance of the proposed WENO-TT solver is investigated in a rich set of numerical experiments. We demonstrate that the WENO-TT method achieves the theoretical $\text{5}^{\text{th}}$-order accuracy of the classical WENO scheme in smooth problems while successfully capturing complicated shock structures. In an effort to avoid the growth of TT ranks, we propose a dynamic method to estimate the TT approximation error that governs the ranks and overall truncation error of the WENO-TT scheme. Finally, we show that the traditional WENO scheme can be accelerated up to 1000 times in the TT format, and the memory requirements can be significantly decreased for low-rank problems, demonstrating the potential of tensor-train approach for future CFD application. This paper is the first study that develops a finite difference WENO scheme using the tensor-train approach for compressible flows. It is also the first comprehensive work that provides a detailed perspective into the relationship between rank, truncation error, and the TT approximation error for compressible WENO solvers.

翻译：本文提出了一种张量列（TT）有限差分WENO方法，用于求解可压缩欧拉方程。我们逐步展示了控制方程的张量化过程，并引入了\emph{LF-cross}和\emph{WENO-cross}方法，利用交叉插值技术计算数值通量和WENO重构。针对计算流体力学（CFD）中常见的边界条件类型，我们开发了张量列处理方法。通过丰富的数值实验，研究了所提出的WENO-TT求解器的性能。结果表明，WENO-TT方法在光滑问题中达到了经典WENO格式的理论五阶精度，同时成功捕捉了复杂的激波结构。为避免TT秩的增长，我们提出了一种动态方法来估计控制WENO-TT格式秩与整体截断误差的TT近似误差。最后，我们证明传统WENO格式在TT格式下可加速高达1000倍，且对于低秩问题，内存需求显著降低，展示了张量列方法在未来CFD应用中的潜力。本文是首个利用张量列方法开发可压缩流动有限差分WENO格式的研究，也是首部全面探讨可压缩WENO求解器中秩、截断误差与TT近似误差之间关系的综合性工作。