Tensor-Train Finite Difference Method for Compressible Flows

In this talk, we will present a tensor-train (TT) finite difference WENO method for solving compressible Euler equations. We will describe how governing equations can be transformed into the TT format and discuss the use of TT cross interpolation for the numerical flux computation and WENO reconstruction. The implementation of frequently encountered boundary conditions in Computational Fluid Dynamics (CFD) in the TT format will also be presented. To achieve the best TT solver performance by avoiding the growth of TT ranks, we will also introduce a dynamic method to estimate the TT approximation error that eventually determines the TT ranks and overall numerical error of the WENO-TT scheme. It will be demonstrated that the WENO-TT maintains the traditional fifth-order accuracy in smooth problems and it can also capture complicated shock structures in the TT format. Lastly, we will discuss the speed performance of the WENO-TT scheme compared to the traditional WENO scheme and show that the WENO-TT scheme achieves 1000x speed-up in low-rank problems.