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Entropy stable numerical schemes for divergence diminishing Chew, Goldberger & Low equations for plasma flows

Chetan Singh, Harish Kumar, Deepak Bhoriya, Dinshaw S. Balsara

Abstract

Chew, Goldberger & Low (CGL) equations are a set of hyperbolic PDEs with non-conservative products used to model the plasma flows, when the assumption of local thermodynamic equilibrium is not valid, and the pressure tensor is assumed to be rotated by the magnetic field. This results in the pressure tensor, which is described by the two scalar components. As the magnetic field also evolves, controlling the divergence of the magnetic field is important. In this work, we consider the generalized Lagrange multiplier (GLM) technique for the CGL model. The resulting model is referred to as the GLM-CGL system. To make the system suitable for entropy-stable schemes, we reformulate the GLM-CGL system by treating some conservative terms as non-conservative. The resulting system has a non-conservative part that does not affect entropy evolution. We then propose entropy stable numerical methods for the GLM-CGL model. The numerical results for the GLM-CGL system are then compared with the CGL system without the GLM divergence diminishing approach to demonstrate that the GLM approach indeed leads to significant improvement in the magnetic field divergence diminishing.

Entropy stable numerical schemes for divergence diminishing Chew, Goldberger & Low equations for plasma flows

Abstract

Chew, Goldberger & Low (CGL) equations are a set of hyperbolic PDEs with non-conservative products used to model the plasma flows, when the assumption of local thermodynamic equilibrium is not valid, and the pressure tensor is assumed to be rotated by the magnetic field. This results in the pressure tensor, which is described by the two scalar components. As the magnetic field also evolves, controlling the divergence of the magnetic field is important. In this work, we consider the generalized Lagrange multiplier (GLM) technique for the CGL model. The resulting model is referred to as the GLM-CGL system. To make the system suitable for entropy-stable schemes, we reformulate the GLM-CGL system by treating some conservative terms as non-conservative. The resulting system has a non-conservative part that does not affect entropy evolution. We then propose entropy stable numerical methods for the GLM-CGL model. The numerical results for the GLM-CGL system are then compared with the CGL system without the GLM divergence diminishing approach to demonstrate that the GLM approach indeed leads to significant improvement in the magnetic field divergence diminishing.
Paper Structure (28 sections, 2 theorems, 115 equations, 18 figures, 3 tables)

This paper contains 28 sections, 2 theorems, 115 equations, 18 figures, 3 tables.

Table of Contents

  1. Introduction
  2. CGL Model
  3. GLM-CGL system
  4. Entropy analysis
  5. Reformulation of the GLM-CGL system
  6. Symmetrization
  7. Semi-discrete numerical schemes
  8. Entropy conservative schemes
  9. Entropy stable schemes
  10. Time discretization
  11. Numerical results
  12. One-dimensional accuracy test
  13. One-dimensional artificial non-zero magnetic field divergence test
  14. Brio-Wu shock tube problem
  15. Two-dimensional accuracy test
  16. ...and 13 more sections

Key Result

Lemma 3.1

The smooth solutions of eq:glm_cgl_con satisfies which implies if $\nabla\cdot\boldsymbol{B}=0$.

Figures (18)

  • Figure 1: \ref{['test:1d_artificial']}: Plots of magnetic field evolution for $\textbf{O2}^\textbf{es}_\textbf{exp}$, $\textbf{O3}^\textbf{es}_\textbf{exp}$ and $\textbf{O4}^\textbf{es}_\textbf{exp}$ schemes for CGL and GLM-CGL at four different time steps.
  • Figure 2: \ref{['test:bw']}: Plots of density, parallel and perpendicular pressure components for explicit schemes and IMEX schemes using $2000$ cells at final time $t = 0.2$.
  • Figure 3: \ref{['test:ot']}: Plots of density and $|(\nabla\cdot\boldsymbol{B})_{i,j}|$ for $\textbf{O2}^\textbf{es}_\textbf{imex}$, $\textbf{O3}^\textbf{es}_\textbf{imex}$ and ${\textbf{O4}^\textbf{es}_\textbf{imex}}$ schemes for isotropic CGL and isotropic GLM-CGL at time $t=0.5$.
  • Figure 4: \ref{['test:ot']}: Evolution of the magnetic field divergence constraint errors till time $t=0.5$.
  • Figure 5: \ref{['test:ot']}: Cut plots of pressure components along $y = 0.3125$ at time $t=0.5$.
  • ...and 13 more figures

Theorems & Definitions (4)

  • Lemma 3.1
  • proof
  • Remark 3.1
  • Theorem 4.1: see singh2024entropy