  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.
