A fast, accurate and oscillation-free spectral collocation solver for high-dimensional transport problems

TL;DR

A new solver called T²S² efficiently and accurately solves complex transport problems in minutes using advanced mathematical techniques.

Contribution

The T²S² solver combines spectral collocation, superconsistency, and tensor train compression for high-dimensional transport problems.

Findings

T²S² achieves exponential convergence and spectral accuracy with minimal data compression.

The solver can handle high-dimensional transport problems on standard hardware in minutes.

T²S² enables modeling of previously intractable transport phenomena with high precision.

Abstract

Transport phenomena—describing the movement of particles, energy, or other physical quantities—are fundamental in various scientific disciplines, including nuclear physics, plasma physics, astrophysics, engineering, and the natural sciences. However, solving the associated seven-dimensional transport equations poses a significant computational challenge due to the curse of dimensionality. We introduce the Tensor Train Superconsistent Spectral (\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$${\hbox {T}}^2{\hbox {S}}^2$$\end{document}) solver to address this challenge, integrating Spectral Collocation for exponential convergence, Superconsistency for stabilization in transport-dominated regimes, and Tensor Train format for substantial data compression. \documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$${\hbox {T}}^2{\hbox {S}}^2$$\end{document} enforces a dimension-wise superconsistent condition compatible with tensor structures, achieving extremely low compression ratios, such as \documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$\mathscr {O}(10^{-12})$$\end{document}, while preserving spectral accuracy. Numerical experiments on linear problems demonstrate that \documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$${\hbox {T}}^2{\hbox {S}}^2$$\end{document} can solve high-dimensional transport problems in minutes on standard hardware, making previously intractable problems computationally feasible. This advancement opens new avenues for efficiently and accurately modeling complex transport phenomena.

Figures