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arXiv · 2506.04732

A Fast, Accurate and Oscillation-free Spectral Collocation Solver for High-dimensional Transport Problems

Nicola Cavallini·Gianmarco Manzini·Daniele Funaro·Andrea Favalli
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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 (T${^2}$S${^2}$) 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. T${^2}$S${^2}$ enforces a dimension-wise superconsistent condition compatible with tensor structures, achieving extremely low compression ratios, in the order of $(10^{-12})$, while preserving spectral accuracy. Numerical experiments on linear problems demonstrate that T${^2}$S${^2}$ 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.

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Nicola Cavallini, Gianmarco Manzini, Daniele Funaro, Andrea Favalli. 2025-06-05. A Fast, Accurate and Oscillation-free Spectral Collocation Solver for High-dimensional Transport Problems. https://arxiv.org/abs/2506.04732

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