High-order Tensor-Train Finite Volume Method for Shallow Water Equations

TL;DR

This work develops a high-order tensor-train finite volume framework for solving both linear and nonlinear shallow water equations, leveraging TT decompositions to achieve substantial computational speedups. It implements $3^{rd}$-order Upwind3, $5^{th}$-order Upwind5, and $5^{th}$-order WENO5 reconstructions in TT format, using direct TT-core manipulation for linear reconstructions and TT cross interpolation for nonlinear WENO. For nonlinear fluxes, it computes the TT reciprocal $1/h_{TT}$ via a Taylor-series approximation (with AMEn as an alternative) and uses an LLF-type flux in TT, achieving speedups up to $124\times$ in linear tests while maintaining the formal accuracy of the underlying discretization; nonlinear tests show reduced acceleration due to increased TT operations. Numerical experiments on Coastal Kelvin Waves, Inertia-Gravity Waves, Barotropic Tide, and a Manufactured Solution validate high-order convergence and demonstrate the method’s potential to accelerate large-scale geophysical simulations, motivating future extension to spherical geometries and primitive equations on modern hardware.

Abstract

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.

Figures (9)

• Figure 1: Step-by-step $3^{rd}$ order reconstruction from cell averages in the $x$-direction
• Figure 2: Exact and TT solutions of surface elevation for Coastal Kelvin Wave at $T=3$ hours. The TT solution is obtained with Upwind5 on $1280\times1280$ grid.
• Figure 3: Speed-up and $L_2$ error of surface elevation for Coastal Kelvin Wave at $T=3$ hours
• Figure 4: Exact and TT solutions of surface elevation for Inertia-Gravity Wave at $T=3$ hours. The TT solution is obtained with Upwind5 on $1280\times1280$ grid.
• Figure 5: Speed-up and $L_2$ errors of surface elevation for Inertia-Gravity Wave at $T=3$ hours