  In this paper, we introduce a high-order tensor-train (TT) finite volume
method for the Shallow Water Equations (SWEs). We present the implementation of
the $3^{rd}$ order Upwind and the $5^{th}$ order Upwind and WENO reconstruction
schemes in the TT format. It is shown in detail that the linear upwind schemes
can be implemented by directly manipulating the TT cores while the WENO scheme
requires the use of TT cross interpolation for the nonlinear reconstruction. In
the development of numerical fluxes, we directly compute the flux for the
linear SWEs without using TT rounding or cross interpolation. For the nonlinear
SWEs where the TT reciprocal of the shallow water layer thickness is needed for
fluxes, we develop an approximation algorithm using Taylor series to compute
the TT reciprocal. The performance of the TT finite volume solver with linear
and nonlinear reconstruction options is investigated under a physically
relevant set of validation problems. In all test cases, the TT finite volume
method maintains the formal high-order accuracy of the corresponding
traditional finite volume method. In terms of speed, the TT solver achieves up
to 124x acceleration of the traditional full-tensor scheme.
