An adaptive dynamical low rank method for the nonlinear Boltzmann equation

TL;DR

This paper introduces an adaptive low rank numerical method combining Fourier spectral techniques for efficiently solving the high-dimensional nonlinear Boltzmann equation, especially for steady state computations.

Contribution

It presents a novel adaptive dynamical low rank approach that improves efficiency and accuracy in solving the Boltzmann equation by incorporating boundary information and controlling computational rank.

Findings

Demonstrates high efficiency and accuracy in 1D and 2D benchmark tests.

Outperforms full tensor grid approaches in computational speed.

Effectively computes steady states of the Boltzmann equation.

Abstract

Efficient and accurate numerical approximation of the full Boltzmann equation has been a longstanding challenging problem in kinetic theory. This is mainly due to the high dimensionality of the problem and the complicated collision operator. In this work, we propose a highly efficient adaptive low rank method for the Boltzmann equation, concerning in particular the steady state computation. This method employs the fast Fourier spectral method (for the collision operator) and the dynamical low rank method to obtain computational efficiency. An adaptive strategy is introduced to incorporate the boundary information and control the computational rank in an appropriate way. Using a series of benchmark tests in 1D and 2D, we demonstrate the efficiency and accuracy of the proposed method in comparison to the full tensor grid approach.