Asymptotic-Preserving Dynamical Low-Rank Method for the Stiff Nonlinear Boltzmann Equation

TL;DR

This paper introduces an asymptotic-preserving dynamical low-rank method with a novel XL integrator for efficiently solving the stiff nonlinear Boltzmann equation, reducing computational cost while maintaining accuracy across regimes.

Contribution

The paper develops a new low-rank integrator, XL, and its specialized version sXL, for the Boltzmann equation, improving efficiency and stability in stiff regimes.

Findings

The proposed methods are asymptotic-preserving, capturing fluid limits in stiff regimes.

Numerical experiments show high accuracy and efficiency across kinetic to fluid regimes.

The sXL integrator simplifies computations by solving only one differential equation.

Abstract

In kinetic theory, numerically solving the full Boltzmann equation is extremely expensive. This is because the Boltzmann collision operator involves a high-dimensional, nonlinear integral that must be evaluated at each spatial grid point and every time step. The challenge becomes even more pronounced in the fluid (strong collisionality) regime, where the collision operator exhibits strong stiffness, causing explicit time integrators to impose severe stability restrictions. In this paper, we propose addressing this problem through a dynamical low-rank (DLR) approximation. The resulting algorithm requires evaluating the Boltzmann collision operator only $r^2$ times, where $r$, the rank of the approximation, is much smaller than the number of spatial grid points. We propose a novel DLR integrator, called the XL integrator, which reduces the number of steps compared to the available alternatives (such as the projector splitting or basis update & Galerkin (BUG) integrator). For a class of problems including the Boltzmann collision operator which enjoys a separation property between physical and velocity space, we further propose a specialized version of the XL integrator, called the sXL integrator. This version requires solving only one differential equation to update the low-rank factors. Furthermore, the proposed low-rank schemes are asymptotic-preserving, meaning they can capture the asymptotic fluid limit in the case of strong collisionality. Our numerical experiments demonstrate the efficiency and accuracy of the proposed methods across a wide range of regimes, from non-stiff (kinetic) to stiff (fluid).