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Chew, Goldberger & Low Equations: Eigensystem Analysis and Applications to One-Dimensional Test Problems

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

TL;DR

The paper addresses plasma flows with pressure anisotropy using the CGL equations, a non-conservative hyperbolic system. It delivers a complete eigensystem, including eigenvalues and right eigenvectors, and identifies linear degeneracy in several characteristic fields. Leveraging this, it develops HLL and HLLI Riemann solvers and very high-order 1D AFD-WENO schemes, with stiffness-enabled MHD limits via IMEX integration. The methods are validated on a broad set of 1D test problems—Brio–Wu, Ryu–Jones, reconnection layers, and circular Alfvén waves—showing accurate wave resolution, convergence, and robustness, and outlining pathways to higher dimensions and divergence-control for the magnetic field.

Abstract

Chew, Goldberger & Low (CGL) equations describe one of the simplest plasma flow models that allow anisotropic pressure, i.e., pressure is modeled using a symmetric tensor described by two scalar pressure components, one parallel to the magnetic field, another perpendicular to the magnetic field. The system of equations is a non-conservative hyperbolic system. In this work, we analyze the eigensystem of the CGL equations. We present the eigenvalues and the complete set of right eigenvectors. We also prove the linear degeneracy of some of the characteristic fields. Using the eigensystem for CGL equations, we propose HLL and HLLI Riemann solvers for the CGL system. Furthermore, we present the AFD-WENO schemes up to the seventh order in one dimension and demonstrate the performance of the schemes on several one-dimensional test cases.

Chew, Goldberger & Low Equations: Eigensystem Analysis and Applications to One-Dimensional Test Problems

TL;DR

The paper addresses plasma flows with pressure anisotropy using the CGL equations, a non-conservative hyperbolic system. It delivers a complete eigensystem, including eigenvalues and right eigenvectors, and identifies linear degeneracy in several characteristic fields. Leveraging this, it develops HLL and HLLI Riemann solvers and very high-order 1D AFD-WENO schemes, with stiffness-enabled MHD limits via IMEX integration. The methods are validated on a broad set of 1D test problems—Brio–Wu, Ryu–Jones, reconnection layers, and circular Alfvén waves—showing accurate wave resolution, convergence, and robustness, and outlining pathways to higher dimensions and divergence-control for the magnetic field.

Abstract

Chew, Goldberger & Low (CGL) equations describe one of the simplest plasma flow models that allow anisotropic pressure, i.e., pressure is modeled using a symmetric tensor described by two scalar pressure components, one parallel to the magnetic field, another perpendicular to the magnetic field. The system of equations is a non-conservative hyperbolic system. In this work, we analyze the eigensystem of the CGL equations. We present the eigenvalues and the complete set of right eigenvectors. We also prove the linear degeneracy of some of the characteristic fields. Using the eigensystem for CGL equations, we propose HLL and HLLI Riemann solvers for the CGL system. Furthermore, we present the AFD-WENO schemes up to the seventh order in one dimension and demonstrate the performance of the schemes on several one-dimensional test cases.

Paper Structure

This paper contains 18 sections, 1 theorem, 53 equations, 14 figures, 4 tables.

Table of Contents

  1. Introduction
  2. Chew, Goldberger & Low (CGL) equations for plasma flows
  3. Eigenvalues and Eigenvectors
  4. AFD-WENO Schemes for CGL equations
  5. HLL Riemann Solver for CGL System
  6. HLLI Riemann Solver for CGL System
  7. Implementation
  8. Implementation
  9. Numerical results
  10. Accuracy test
  11. Circularly polarized Alfvén waves
  12. One-Dimensional reconnection layer
  13. Riemann problem 1: Brio and Wu Shock tube problem
  14. Riemann problem 2: Ryu–Jones test problem
  15. Riemann Problem 3
  16. ...and 3 more sections

Key Result

Lemma 2.1

For the CGL system, the characteristic field corresponding to the entropy wave $u_x$, the pressure anisotropy wave $u_x$, and the Alfvén waves$u_x\pm c_a$ are linearly degenerate.

Figures (14)

  • Figure 1: Part of the mesh around zone $i$
  • Figure 2: part of the mesh around zone boundary $i+\frac{1}{2}$
  • Figure 3: \ref{['test:reconnectionlayer']}: Plots of density, velocity in $x,~y$ and $z$ direction, parallel and perpendicular pressure components and magnetic field in $y$ and $z$ direction for 3rd, 5th and 7th order numerical schemes using HLL Riemann solver and 2000 cells at final time t = 3500.
  • Figure 4: \ref{['test:reconnectionlayer']}: Plots of density, velocity in $x,~y$ and $z$ direction, parallel and perpendicular pressure components and magnetic field in $y$ and $z$ direction for 3rd, 5th and 7th order numerical schemes using HLLI Riemann solver and 2000 cells at final time t = 3500.
  • Figure 5: \ref{['test:rp1:brio']}: Plots of density, velocity in $x$-direction and parallel and perpendicular pressure components for 3rd, 5th, and 7th order numerical schemes without and with source term using the HLL Riemann solver and 800 cells at final time t = 0.2.
  • ...and 9 more figures

Theorems & Definitions (3)

  • Definition 2.1.1
  • Lemma 2.1
  • proof