Tensor methods for the Boltzmann-BGK equation

TL;DR

This paper introduces a tensor-decomposition approach to efficiently solve the six-dimensional Boltzmann transport equation, enabling scalable steady-state and transient simulations with potential applications beyond BTE.

Contribution

A novel tensor-decomposition method that reduces the high-dimensional BTE to manageable one-dimensional problems using ALS and Fourier transforms.

Findings

Achieves N log N computational scaling.

Successfully predicts equilibrium and non-equilibrium distributions.

Demonstrates general applicability to other high-dimensional systems.

Abstract

We present a tensor-decomposition method to solve the Boltzmann transport equation (BTE) in the Bhatnagar-Gross-Krook approximation. The method represents the six-dimensional BTE as a set of six one-dimensional problems, which are solved with the alternating least-squares algorithm and the discrete Fourier transform at $N$ collocation points. We use this method to predict the equilibrium distribution (stead-state simulation) and a non-equilibrium distribution returning to the equilibrium (transient simulation). Our numerical experiments demonstrate $N \log N$ scaling. Unlike many BTE-specific numerical techniques, the numerical tensor-decomposition method we propose is a general technique that can be applied to other high-dimensional systems.