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

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 (T2S2) 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. T2S2 enforces a dimension-wise superconsistent condition compatible with tensor structures, achieving extremely low compression ratios, such as 10^(-12), while preserving spectral accuracy. Numerical experiments on linear problems demonstrate that T2S2 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.