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

Area of Science:

• Computational physics
• Applied mathematics
• Scientific computing

Background:

• Transport phenomena are crucial across diverse scientific fields like nuclear physics, plasma physics, astrophysics, and engineering.
• Solving high-dimensional transport equations faces computational hurdles due to the curse of dimensionality.

Purpose of the Study:

• To introduce a novel computational solver, Tensor Train Superconsistent Spectral (T2S2), designed to overcome the curse of dimensionality in transport phenomena.
• To integrate Spectral Collocation, Superconsistency, and Tensor Train format for efficient and accurate solutions.

Main Methods:

• The T2S2 solver integrates Spectral Collocation for rapid convergence and Superconsistency for stabilization.
• It utilizes the Tensor Train format for significant data compression, achieving high compression ratios (e.g., 10^4) while maintaining spectral accuracy.
• A dimension-wise superconsistent condition compatible with tensor structures is enforced.

Main Results:

• Numerical experiments demonstrate T2S2's ability to solve high-dimensional transport problems rapidly (minutes on standard hardware).
• The solver achieves substantial data compression, preserving spectral accuracy.
• Previously computationally intractable problems become feasible.

Conclusions:

• The T2S2 solver offers an efficient and accurate method for modeling complex transport phenomena.
• This advancement significantly enhances computational feasibility for high-dimensional transport problems.
• It opens new possibilities for research and applications in various scientific disciplines.