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Implicit Update of the Moment Equations for a Multi-Species, Homogeneous BGK Model

Evan Habbershaw, Cory D. Hauck, Steven M. Wise

TL;DR

This work develops a Gauss-Seidel-type (GST) iterative solver to implicitly update the moment equations in a multi-species BGK model with state-dependent collision frequencies within an IMEX/backward-Euler framework. It proves a contraction mapping under a mild time-step restriction that is independent of the stiff parameter $\varepsilon$, enabling time steps governed by advection rather than collisions. The approach is validated numerically on Sod-type and Ar-Kr-Xe mixtures, showing accurate hydrodynamic-limit behavior and a dramatic reduction in time steps compared to fully explicit schemes. The results suggest robust applicability to state-dependent relaxation systems and potential extensions to ES-BGK and Fokker-Planck-type models.

Abstract

A simple iterative approach for solving a set of implicit kinetic moment equations is proposed. This implicit solve is a key component in the IMEX discretization of the multi-species Bhatnagar-Gross-Krook (M-BGK) model with nontrivial collision frequencies depending on individual species temperatures. We prove that under mild time step restrictions, the iterative method generates a contraction mapping. Numerical simulations are provided to illustrate results of the IMEX scheme using the implicit moment solver.

Implicit Update of the Moment Equations for a Multi-Species, Homogeneous BGK Model

TL;DR

This work develops a Gauss-Seidel-type (GST) iterative solver to implicitly update the moment equations in a multi-species BGK model with state-dependent collision frequencies within an IMEX/backward-Euler framework. It proves a contraction mapping under a mild time-step restriction that is independent of the stiff parameter , enabling time steps governed by advection rather than collisions. The approach is validated numerically on Sod-type and Ar-Kr-Xe mixtures, showing accurate hydrodynamic-limit behavior and a dramatic reduction in time steps compared to fully explicit schemes. The results suggest robust applicability to state-dependent relaxation systems and potential extensions to ES-BGK and Fokker-Planck-type models.

Abstract

A simple iterative approach for solving a set of implicit kinetic moment equations is proposed. This implicit solve is a key component in the IMEX discretization of the multi-species Bhatnagar-Gross-Krook (M-BGK) model with nontrivial collision frequencies depending on individual species temperatures. We prove that under mild time step restrictions, the iterative method generates a contraction mapping. Numerical simulations are provided to illustrate results of the IMEX scheme using the implicit moment solver.
Paper Structure (27 sections, 43 theorems, 125 equations, 2 figures)

This paper contains 27 sections, 43 theorems, 125 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Background
  3. Models and equations
  4. IMEX strategy and backward Euler step
  5. Iterative Solver
  6. Definition and Main Result
  7. Basic properties of the GST method
  8. Analysis of Velocity Iterates
  9. Velocity Iterates on Null Space
  10. Velocity Iterates on Range Space and Initial Norm Bounds
  11. Summary of Velocity Term Analysis
  12. Analysis of Temperature Iterations
  13. Temperature Iterates on Null Space
  14. Bounding Source Terms in the Temperature Equation
  15. Temperature Iterates on Range Space
  16. ...and 12 more sections

Key Result

Proposition 2.2

If scheme eq:IMEX_BE is applied for the kinetic densities, then, for each $i\in\{1,\cdots,N\}$, where $s_i^\mathfrak{n} = |\boldsymbol{u}_i^\mathfrak{n}|^2$ and the ${N\times N}$ matrices $A$, $B$, $D$, $F$, and $S$, at time step $t^{\mathfrak{n}+1}$ are given by

Figures (2)

  • Figure 1: The total density, total bulk velocity, and total temperature moments are compared to the solution of the Euler equations at time $t=0.2$, with $\varepsilon=10^{-4}$.
  • Figure 2: Noble gas mixture of Ar, Kr, and Xe: Moment plots of species number density, bulk velocity, and temperature, are given at $t_{\textnormal{F}}=0.1$$\mu$s, for a mixture with $\varepsilon \approx 0.00103$. M-BGK uses the GST algorithm for the moment solve. M-BGK$_{\textnormal{EX}}$ uses a fully explicit method.

Theorems & Definitions (85)

  • Definition 2.1
  • Proposition 2.2: Backward Euler
  • Remark 2.3
  • Proposition 2.4: Discrete Conservation Laws
  • proof
  • Proposition 2.5
  • proof
  • Definition 3.1: GST method
  • Proposition 3.2: GST Method for $\boldsymbol{W}$ and $\boldsymbol{\eta}$
  • Definition 3.3
  • ...and 75 more