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A stable multiplicative dynamical low-rank discretization for the linear Boltzmann-BGK equation

Lena Baumann, Lukas Einkemmer, Christian Klingenberg, Jonas Kusch

Abstract

The numerical method of dynamical low-rank approximation (DLRA) has recently been applied to various kinetic equations showing a significant reduction of the computational effort. In this paper, we apply this concept to the linear Boltzmann-Bhatnagar-Gross-Krook (Boltzmann-BGK) equation which due its high dimensionality is challenging to solve. Inspired by the special structure of the non-linear Boltzmann-BGK problem, we consider a multiplicative splitting of the distribution function. We propose a rank-adaptive DLRA scheme making use of the basis update & Galerkin integrator and combine it with an additional basis augmentation to ensure numerical stability, for which an analytical proof is given and a classical hyperbolic Courant-Friedrichs-Lewy (CFL) condition is derived. This allows for a further acceleration of computational times and a better understanding of the underlying problem in finding a suitable discretization of the system. Numerical results of a series of different test examples confirm the accuracy and efficiency of the proposed method compared to the numerical solution of the full system.

A stable multiplicative dynamical low-rank discretization for the linear Boltzmann-BGK equation

Abstract

The numerical method of dynamical low-rank approximation (DLRA) has recently been applied to various kinetic equations showing a significant reduction of the computational effort. In this paper, we apply this concept to the linear Boltzmann-Bhatnagar-Gross-Krook (Boltzmann-BGK) equation which due its high dimensionality is challenging to solve. Inspired by the special structure of the non-linear Boltzmann-BGK problem, we consider a multiplicative splitting of the distribution function. We propose a rank-adaptive DLRA scheme making use of the basis update & Galerkin integrator and combine it with an additional basis augmentation to ensure numerical stability, for which an analytical proof is given and a classical hyperbolic Courant-Friedrichs-Lewy (CFL) condition is derived. This allows for a further acceleration of computational times and a better understanding of the underlying problem in finding a suitable discretization of the system. Numerical results of a series of different test examples confirm the accuracy and efficiency of the proposed method compared to the numerical solution of the full system.

Paper Structure

This paper contains 17 sections, 6 theorems, 71 equations, 8 figures.

Table of Contents

  1. Introduction
  2. Linear Boltzmann-BGK
  3. Discretization of the $Mg$ system
  4. Discretization in velocity and in space
  5. Time discretization
  6. Numerical stability
  7. Naive discretization
  8. Stable discretization
  9. Dynamical low-rank approximation for the stable $Mg$ system
  10. Background on dynamical low-rank approximation
  11. DLRA algorithm for multiplicative linear Boltzmann-BGK
  12. Stability of the proposed low-rank scheme
  13. Numerical results
  14. 1D Plane source
  15. 2D Plane source
  16. ...and 2 more sections

Key Result

Lemma 1

Let $y,z \in \mathbb{R}^{N_x}$ with indices $i,j=1,...,N_x$. Then it holds Moreover, let $\mathbf D^{+}\in\mathbb{R}^{N_x \times N_x}$ be defined as Then, $\sum_{i,j =1}^{N_x} z_j D_{ji}^{xx} z_i = - \sum_{j=1}^{N_x} \left(\sum_{i=1}^{N_x} D_{ji}^+ z_i\right)^2$.

Figures (8)

  • Figure 1: Flowchart of the (simplified) stable DLRA scheme \ref{['alg:DLRA']}.
  • Figure 2: Numerical results for the solution $f(t,x,v)$ of the 1D plane source analogue at time $t=0$ (first column), $t=2$ (second column), $t=4$ (third column), and $t=6$ (fourth column), computed with the full solver (Mg (reference)) (first row), the reduced DLRA scheme (Mg DLRA) (second row) and the basis augmented DLRA scheme (Mg DLRA BasisAug) (third row).
  • Figure 3: Numerical results for the density $\rho(t, x)$ of the 1D plane source analogue at time $t=0$, $t=2$, $t=4$, and $t=6$, computed with the full solver (Mg (reference)), the reduced DLRA scheme (Mg DLRA) and the basis augmented DLRA scheme (Mg DLRA BasisAug).
  • Figure 4: Left: Evolution of the rank in time for the 1D plane source analogue for the reduced DLRA scheme (Mg DLRA) and the basis augmented DLRA scheme (Mg DLRA BasisAug). Middle: Evolution of the $\mathscr{H}$-norm in time for the full solver (Mg (reference)), the reduced DLRA scheme (Mg DLRA) and the basis augmented DLRA scheme (Mg DLRA BasisAug). Right: Evolution of $\kappa^\pm$ in time for the full solver (Mg (reference)), the reduced DLRA scheme (Mg DLRA) and the basis augmented DLRA scheme (Mg DLRA BasisAug). The red line has the constant value $1$. The deviations of the DLRA schemes from $1$ are of order $\mathcal{O}\left(10^{-11}\right)$.
  • Figure 5: Numerical results for the density $\rho(t, \mathbf{x})$ of the 2D plane source analogue at time $t=0$ (first column), $t=1$ (second column), $t=2$ (third column, and $t=3$ (fourth column), computed with the full solver (Mg (reference)) (first row) and the reduced DLRA scheme (Mg DLRA) (second row).
  • ...and 3 more figures

Theorems & Definitions (14)

  • Lemma 1: Summation by parts
  • Definition 1: Fully discrete solution and Maxwellian
  • Theorem 1
  • proof
  • Definition 2: Stability norm
  • Lemma 2
  • proof
  • Lemma 3
  • proof
  • Theorem 2
  • ...and 4 more