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An asymptotic preserving kinetic scheme for the M1 model of linear transport

Feugeas Jean-Luc, Mathiaud Julien, Mieussens Luc, Vigier Thomas

TL;DR

A new asymptotic preserving scheme for the M1 model of linear transport that works uniformly for any Knudsen number is proposed and a new density reconstruction in time is proposed to obtain moments realizability.

Abstract

Moment models with suitable closure can lead to accurate and computationally efficient solvers for particle transport. Hence, we propose a new asymptotic preserving scheme for the M1 model of linear transport that works uniformly for any Knudsen number. Our idea is to apply the M1 closure at the numerical level to an existing asymptotic preserving scheme for the corresponding kinetic equation, namely the Unified Gas Kinetic scheme (UGKS) originally proposed in [27] and extended to linear transport in [24]. In order to ensure the moments realizability in this new scheme, the UGKS positivity needs to be maintained. We propose a new density reconstruction in time to obtain this property. A second order extension is also suggested and validated. Several test cases show the performances of this new scheme.

An asymptotic preserving kinetic scheme for the M1 model of linear transport

TL;DR

A new asymptotic preserving scheme for the M1 model of linear transport that works uniformly for any Knudsen number is proposed and a new density reconstruction in time is proposed to obtain moments realizability.

Abstract

Moment models with suitable closure can lead to accurate and computationally efficient solvers for particle transport. Hence, we propose a new asymptotic preserving scheme for the M1 model of linear transport that works uniformly for any Knudsen number. Our idea is to apply the M1 closure at the numerical level to an existing asymptotic preserving scheme for the corresponding kinetic equation, namely the Unified Gas Kinetic scheme (UGKS) originally proposed in [27] and extended to linear transport in [24]. In order to ensure the moments realizability in this new scheme, the UGKS positivity needs to be maintained. We propose a new density reconstruction in time to obtain this property. A second order extension is also suggested and validated. Several test cases show the performances of this new scheme.
Paper Structure (27 sections, 5 theorems, 60 equations, 8 figures, 1 table)

This paper contains 27 sections, 5 theorems, 60 equations, 8 figures, 1 table.

Table of Contents

  1. Introduction
  2. The M1 closure for the linear transport
  3. The linear transport equation
  4. Asymptotic regimes
  5. Entropy
  6. The M1 moment closure
  7. A UGKS based numerical scheme for the M1 model
  8. UGKS
  9. A finite volume formulation
  10. A characteristic based approach
  11. Asymptotic behaviour and stability
  12. UGKS-M1
  13. An entropic closure of the UGKS
  14. Definition and realizability of the scheme
  15. Second order in space
  16. ...and 12 more sections

Key Result

Proposition 2.1

Let $\mathbf{U}=^T$ and $u=j/\rho$. The moment vector is realizable if and only if $\rho > 0$ and $|u|<1$, or $\mathbf{U}=\mathbf{0}$.

Figures (8)

  • Figure 1: Structure of the UGKS-M1 scheme
  • Figure 2: Test n°1: mesh convergence study for UGKS-M1 and HLL. Density in the domain at time $t=1.0$.
  • Figure 3: Test n°1: UGKS-M1 density error as a function of the step size.
  • Figure 4: Test n°2: transport regime. Density in the domain at different times for UGKS and M1 (with UGKS-M1 and HLL).
  • Figure 5: Test n°3: intermediate regime. Density in the domain at different times for UGKS and M1 (with UGKS-M1 and HLL).
  • ...and 3 more figures

Theorems & Definitions (10)

  • Definition 2.1: Moment realizability
  • Proposition 2.1
  • proof
  • Proposition 2.2: M1 distribution function
  • proof
  • Proposition 2.3: M1 closure
  • Proposition 2.4: System structure
  • Proposition 2.5: Diffusion limit
  • Remark 4.1
  • Remark 4.2