Tensor-Train WENO Scheme for Compressible Flows
<|reference_start|>Tensor-Train WENO Scheme for Compressible Flows: 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.<|reference_end|>
@article{danis2024tensor-train,
    title={Tensor-Train WENO Scheme for Compressible Flows},
    author={Mustafa Engin Danis and Duc Truong and Ismael Boureima and Oleg
  Korobkin and Kim Rasmussen and Boian Alexandrov},
    journal={arXiv preprint arXiv:2405.12301},
    year={2024},
    number={LA-UR-24-24954},
    archivePrefix={arXiv},
    eprint={2405.12301},
    primaryClass={math.NA cs.NA}
}