Tensor-Train WENO Scheme for Compressible Flows

TL;DR

The paper presents a tensor-train finite difference WENO method for the compressible Euler equations, introducing LF-cross and WENO-cross cross interpolation to compute fluxes and reconstructions within the TT framework. It establishes a dynamic rule for the TT approximation error $\varepsilon_{TT}$ to maintain $5^{\text{th}}$-order accuracy while controlling TT ranks, and demonstrates dramatic speedups (up to $10^3\times$) and memory savings on low-rank problems across a suite of smooth and shock-capturing tests. The approach shows that TT-based solvers can effectively mitigate the curse of dimensionality in CFD when problems exhibit low-rank structure, with a detailed exploration of how TT ranks interact with truncation error and discretization error. The work is a first comprehensive study connecting TT approximations to high-order WENO schemes for compressible flows and highlights the potential for TT methods to enable efficient 3D CFD simulations, while also identifying challenges related to rank growth in complex shock structures.

Abstract

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.

Figures (12)

• Figure 1: Candidate stencils $S_r(i)$ for the WENO5-JS scheme
• Figure 2: TT format of a 3D tensor $\mathcal{X}$, with TT-cores $\mathcal{G}_1,\ \mathcal{G}_2,\ \mathcal{G}_3$, TT-ranks $\mathbf{r} = (r_1,r_2)$, and approximation error $\varepsilon$.
• Figure 3: Cross Interpolation for Numerical Flux and WENO Reconstruction
• Figure 4: Effect of $C_\varepsilon$ on accuracy and ranks of $\rho_{TT}$ for isentropic vortex problem at $T=1$
• Figure 5: Speed-up and memory compression ratio for isentropic vortex problem