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

Lukas Einkemmer, Jingwei Hu, Shiheng Zhang

TL;DR

This paper tackles the prohibitive cost of solving the Boltzmann equation by introducing a dynamical low-rank (DLR) framework that represents the distribution function $f(x,v,t)$ with low-rank factors. The XL and sXL integrators advance the low-rank factors in a way that decouples advection and collision, achieving asymptotic-preserving behavior in the stiff, small-$\varepsilon$ regime without requiring explicit Maxwellian evaluations or fully implicit Boltzmann solves. Key contributions include the general XL/sXL schemes, an AP analysis for a special BGK case, and comprehensive 1+2D numerical experiments showing accuracy comparable to full-tensor methods with drastically reduced collision-solver calls $\mathcal{O}(r^2)$. The approach offers a practical and scalable route to simulate kinetic-to-fluid transitions in high-dimensional phase space, with potential extensions to higher-order time integration.

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).

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

TL;DR

This paper tackles the prohibitive cost of solving the Boltzmann equation by introducing a dynamical low-rank (DLR) framework that represents the distribution function with low-rank factors. The XL and sXL integrators advance the low-rank factors in a way that decouples advection and collision, achieving asymptotic-preserving behavior in the stiff, small- regime without requiring explicit Maxwellian evaluations or fully implicit Boltzmann solves. Key contributions include the general XL/sXL schemes, an AP analysis for a special BGK case, and comprehensive 1+2D numerical experiments showing accuracy comparable to full-tensor methods with drastically reduced collision-solver calls . The approach offers a practical and scalable route to simulate kinetic-to-fluid transitions in high-dimensional phase space, with potential extensions to higher-order time integration.

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 times, where , 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).

Paper Structure

This paper contains 9 sections, 2 theorems, 86 equations, 6 figures.

Table of Contents

  1. Introduction
  2. The general XL and sXL integrators
  3. Dynamical low-rank method for the non-stiff Boltzmann equation
  4. Asymptotic-preserving dynamical low-rank method for the stiff Boltzmann equation
  5. The AP property for a special BGK equation
  6. Numerical results
  7. Sine-type initial data
  8. Shock tube problem
  9. Conclusions

Key Result

Theorem 2.2

Let $A(t) \in \mathbb{R}^{m_1 \times m_2}$ have rank $r$ for all $t \in [t^n, t^{n+1}]$, then it admits the factorization $A(t) = X(t) L(t)^{T} = X(t) S(t) V(t)^{T}.$ Assume that $V(t^{n+1})^{T} V(t^{n})$ is invertible. With $Y^{n} = A(t^{n})$, the XL integrator for matrix-eq is exact, that is, $Y^{

Figures (6)

  • Figure 1: Comparison of density, bulk velocity (first component), and temperature for the sine problem with $\Delta t = 10^{-3}$, $\varepsilon = 1$, $r = 6$ and $\tilde{r} = 6$ for DLR-XL and DLR-sXL-1. The $\tilde{r}$ of DLR-sXL-2 is shown in \ref{['sin1d']} with $\text{tol} = 1$. Solid line: full tensor method, red circle: DLR-XL, green star: DLR-sXL-1, blue diamond: DLR-sXL-2.
  • Figure 2: Comparison of density, bulk velocity (first component), and temperature for the sine problem with $\Delta t = 10^{-3}$, $\varepsilon = 10^{-6}$, $r = 10$ and $\tilde{r} = 10$ for DLR-XL and DLR-sXL-1. The $\tilde{r}$ of DLR-sXL-2 is shown in \ref{['sin1-6d']} with $\text{tol} = 0.05$. Solid line: full tensor method, red circle: DLR-XL, green star: DLR-sXL-1, blue diamond: DLR-sXL-2.
  • Figure 3: Comparison of DLR-sXL-2 with rank $r = 10$ under three different tolerances, $\text{tol} \in \{0.2, 0.1, 0.05\}$, for $\varepsilon = 10^{-6}$. Solid line: full tensor method, red circle: DLR-sXL-2 with $\text{tol} = 0.2$, green star: DLR-sXL-2 with $\text{tol} = 0.1$, blue diamond: DLR-sXL-2 with $\text{tol} = 0.05$. In Figure \ref{['sin1-6-3c']}, one can observe a slight discrepancy between DLR-sXL-2-0.2 and others. The additional ranks refers to the number of extra basis functions introduced during the augmentation steps.
  • Figure 4: Comparison of density, bulk velocity (first component), and temperature for the shock tube problem with $\Delta t = 10^{-4}$, $\varepsilon = 10^{-3}$, $r = 14$ and $\tilde{r} = 14$ for DLR-XL and DLR-sXL-1. The $\tilde{r}$ of DLR-sXL-2 is shown in \ref{['shock1-3d']} with $\text{tol} = 1$. Solid line: full tensor method, red circle: DLR-XL, green star: DLR-sXL-1, blue diamond: DLR-sXL-2.
  • Figure 5: Comparison of density, bulk velocity (first component), and temperature for the shock tube problem with $\Delta t = 10^{-4}$, $\varepsilon = 10^{-6}$, $r = 20$ and $\tilde{r} = 20$ for DLR-XL and DLR-sXL-1. The $\tilde{r}$ of DLR-sXL-2 is shown in \ref{['shock1-6d']} with $\text{tol} = 0.01$. Solid line: full tensor method, red circle: DLR-XL, green star: DLR-sXL-1, blue diamond: DLR-sXL-2.
  • ...and 1 more figures

Theorems & Definitions (8)

  • Remark 2.1
  • Theorem 2.2: Exactness property
  • proof
  • Remark 2.3
  • Remark 4.1
  • Remark 4.2
  • Theorem 5.1: AP property
  • proof