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An asymptotic preserving scheme for the M1model of non-local thermal transport for two-dimensional structured and unstructured meshes

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

TL;DR

This work tackles non-local electron thermal transport in laser-plasma settings by developing a two-dimensional asymptotic-preserving scheme based on the Unified Gas Kinetic Scheme (UGKS) for the M1 moment model. The central approach is to apply the M1 closure at the numerical level within the UGKS flux computations, yielding an AP method that correctly recovers the diffusion limit while remaining stable across Knudsen regimes. Key contributions include a detailed entropic M1 closure, a structured-to-unstructured mesh extension with diamond diffusion schemes, second-order spatial and temporal extensions, and robust half-sphere moment computations, all validated on a suite of 1D/2D test problems showing accurate non-local transport capture. The results indicate significant potential for efficient multiscale simulations in inertial confinement fusion and related laser-plasma applications, with demonstrated advantages over standard diffusion schemes and compatibility with complex meshes.

Abstract

The M1 moment model for electronic transport is commonly used to describe non-local thermal transport effects in laser-plasma simulations. In this article, we propose a new asymptotic-preserving scheme based on the Unified Gas Kinetic Scheme (UGKS) for this model in two-dimensional space. This finite volume kinetic scheme follows the same approach as in our previous article and relies on a moment closure, at the numerical scale, of the microscopic flux of UGKS. The method is developed for both structured and unstructured meshes, and several techniques are introduced to ensure accurate fluxes in the diffusion limit. A second-order extension is also proposed. Several test cases validate the different aspects of the scheme and demonstrate its efficiency in multiscale simulations. In particular, the results demonstrate that this method accurately captures non-local thermal effects.

An asymptotic preserving scheme for the M1model of non-local thermal transport for two-dimensional structured and unstructured meshes

TL;DR

This work tackles non-local electron thermal transport in laser-plasma settings by developing a two-dimensional asymptotic-preserving scheme based on the Unified Gas Kinetic Scheme (UGKS) for the M1 moment model. The central approach is to apply the M1 closure at the numerical level within the UGKS flux computations, yielding an AP method that correctly recovers the diffusion limit while remaining stable across Knudsen regimes. Key contributions include a detailed entropic M1 closure, a structured-to-unstructured mesh extension with diamond diffusion schemes, second-order spatial and temporal extensions, and robust half-sphere moment computations, all validated on a suite of 1D/2D test problems showing accurate non-local transport capture. The results indicate significant potential for efficient multiscale simulations in inertial confinement fusion and related laser-plasma applications, with demonstrated advantages over standard diffusion schemes and compatibility with complex meshes.

Abstract

The M1 moment model for electronic transport is commonly used to describe non-local thermal transport effects in laser-plasma simulations. In this article, we propose a new asymptotic-preserving scheme based on the Unified Gas Kinetic Scheme (UGKS) for this model in two-dimensional space. This finite volume kinetic scheme follows the same approach as in our previous article and relies on a moment closure, at the numerical scale, of the microscopic flux of UGKS. The method is developed for both structured and unstructured meshes, and several techniques are introduced to ensure accurate fluxes in the diffusion limit. A second-order extension is also proposed. Several test cases validate the different aspects of the scheme and demonstrate its efficiency in multiscale simulations. In particular, the results demonstrate that this method accurately captures non-local thermal effects.

Paper Structure

This paper contains 42 sections, 3 theorems, 116 equations, 16 figures, 1 table.

Table of Contents

  1. Introduction
  2. The entropic closure for the electron transport equation
  3. The electron transport equation
  4. Asymptotic regimes
  5. The M1 moment closure
  6. A UGKS based numerical scheme for the M1 model for structured meshes
  7. UGKS
  8. A finite volume formulation
  9. A characteristic based approach
  10. Asymptotic behaviour
  11. UGKS-M1
  12. A M1 closure of the UGKS
  13. Speed discretization
  14. Dirichlet boundary condition
  15. Half sphere moments evaluation
  16. ...and 27 more sections

Key Result

Proposition 2.1

Let $\mathbf{U}=^T$ and $u=\frac{||\mathbf{f}_1||}{f_0}$. The moment vector is realizable if and only if $f_0 > 0$ and $u<1$, or $\mathbf{U}=\mathbf{0}$.

Figures (16)

  • Figure 1: Schematic view of two triangular elements of an unstructured mesh
  • Figure 2: Schematic view of two triangular elements of an unstructured mesh and the associated diamond $\mathscr{D}=(\mathbf{x}_K,\mathbf{x}_e^-,\mathbf{x}_L,\mathbf{x}_e^+)$
  • Figure 3: Schematic view of the diamond and of the modified diamond in a unstructured mesh
  • Figure 4: Regular unstructured mesh (used for the 1D test cases) with 4036 triangles and level of angular deformation of triangles in colour scale
  • Figure 5: Deformed unstructured mesh and cut lines (used for the 1D test cases) with 10752 triangles and level of angular deformation of triangles in colour scale
  • ...and 11 more figures

Theorems & Definitions (4)

  • Definition 2.1: Moment realizability
  • Proposition 2.1
  • Proposition 2.2: M1 distribution function
  • Proposition 2.3: M1 closure