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

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 ([Formula: see text]) 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. [Formula: see text] enforces a dimension-wise superconsistent condition compatible with tensor structures, achieving extremely low compression ratios, such as [Formula: see text], while preserving spectral accuracy. Numerical experiments on linear problems demonstrate that [Formula: see text] 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.