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

This paper proposes entropy-stable numerical schemes for the generalized Lagrange multiplier (GLM) reformulation of the Chew, Goldberger & Low (CGL) plasma flow equations, demonstrating through numerical results that this approach significantly improves the control of magnetic field divergence compared to the standard CGL model.

The Gist

Imagine you are trying to simulate a storm of invisible, super-hot gas (plasma) swirling around a star or a planet. This gas is charged, so it carries magnetic fields, like invisible rubber bands that hold the gas together.

Scientists use complex math equations (called the CGL equations) to predict how this plasma moves. However, there's a big problem: in the real world, magnetic field lines must form perfect loops; they can't just start or stop in mid-air. In math terms, the "divergence" of the magnetic field must be zero.

But when computers try to solve these equations, they make tiny mistakes. Over time, these tiny errors add up, and the computer starts thinking magnetic field lines are appearing out of nowhere or vanishing. It's like a video game where the physics engine glitches, and suddenly, gravity stops working in one corner of the map. The simulation becomes garbage.

This paper is about building a better, more stable computer program to fix this glitch while also ensuring the simulation doesn't violate the laws of thermodynamics (entropy).

Here is the breakdown of their solution using simple analogies:

1. The "Glitch" and the "Cleanup Crew" (GLM Technique)

The authors introduce a new tool called the GLM (Generalized Lagrange Multiplier).

• The Analogy: Imagine you are painting a wall, but you keep accidentally dripping paint onto the floor. Instead of just ignoring the mess, you hire a "Cleanup Crew" (the GLM variable, $\Psi$).
• How it works: This crew runs around the simulation, constantly checking for "paint drips" (divergence errors). If they find a spot where the magnetic field is breaking the rules, they instantly send a wave of "cleaning fluid" (a hyperbolic wave) to wash the error away before it spreads.
• The Result: The magnetic field stays clean and looped, just like it should in the real universe.

2. The "Thermodynamic Safety Net" (Entropy Stability)

Computers are great at crunching numbers, but they are terrible at respecting the "Second Law of Thermodynamics." This law basically says that in a closed system, things tend to get more chaotic (entropy increases) and never spontaneously become more ordered.

• The Problem: Standard computer schemes often accidentally make the system too ordered or create energy out of nothing, which is physically impossible. This causes the simulation to explode or give nonsense results.
• The Solution: The authors designed a "Safety Net." They reformulated the math so that the computer is forced to obey the laws of entropy.
• The Analogy: Think of a bank account. Standard schemes might accidentally let you withdraw more money than you have (creating energy). The authors' scheme acts like a strict accountant who ensures you can never spend more than you earn. If the math tries to create energy, the "Safety Net" stops it.

3. The "Re-arranging the Furniture" (Reformulation)

To make the math work with their Safety Net, they had to rearrange the equations.

• The Analogy: Imagine you have a heavy sofa (the equations) that is too big to fit through a door (the computer's logic). Instead of forcing it, they take the sofa apart, move the pieces through the door, and put them back together on the other side in a way that works perfectly.
• The Trick: They took some parts of the equation that looked like "conservative" (staying the same) and treated them as "non-conservative" (changing). This sounds counter-intuitive, but it allowed them to prove that the "Safety Net" (entropy) would never be broken.

4. The "High-Resolution Camera" (Numerical Schemes)

The authors didn't just fix the math; they built a high-definition camera to take pictures of the plasma.

• Low-Order Schemes: These are like taking a photo with a blurry, old camera. You see the general shape of the storm, but the details are fuzzy, and errors pile up quickly.
• High-Order Schemes: The authors created 3rd and 4th-order schemes. These are like 4K or 8K cameras. They capture the tiny, swirling details of the plasma without blurring them out.
• The Proof: They tested their camera on famous "test drives" (like the Brio-Wu shock tube and the Orszag-Tang vortex). In every test, their new method:
  1. Kept the magnetic field clean (no glitches).
  2. Followed the laws of physics (entropy).
  3. Showed much sharper, more accurate details than older methods.

Summary: Why does this matter?

Before this paper, simulating complex plasma (like in fusion reactors or solar flares) was risky. The computer might crash, or the results might be physically impossible because of magnetic field glitches or energy errors.

This paper provides a robust, high-definition toolkit that ensures:

1. No Magnetic Glitches: The "Cleanup Crew" keeps the magnetic field lines perfect.
2. No Physics Violations: The "Safety Net" ensures the simulation respects the laws of thermodynamics.
3. High Precision: The "High-Res Camera" captures the beautiful, chaotic details of plasma flows without blurring them.

In short, they built a better engine for the "video game" of the universe, ensuring the physics never breaks, even when the action gets intense.

Technical Summary

1. Problem Statement

The paper addresses the numerical simulation of plasma flows where the assumption of local thermodynamic equilibrium (LTE) is invalid. In such collisionless or weakly collisional plasmas, the pressure tensor is anisotropic (gyrotropic), described by two scalar components: parallel ($p_\parallel$) and perpendicular ($p_\perp$) to the magnetic field. These flows are modeled by the Chew, Goldberger, and Low (CGL) equations.

The CGL system presents two major challenges for numerical simulation:

1. Non-conservative Products: The equations contain non-conservative terms (products of discontinuous variables and derivatives), which complicate the definition of weak solutions and require specific path choices in numerical schemes.
2. Magnetic Divergence Error: Like Magnetohydrodynamics (MHD), the CGL equations require the magnetic field to be divergence-free ($\nabla \cdot \mathbf{B} = 0$). Standard numerical discretizations often fail to preserve this constraint, leading to unphysical results and instability.
3. Entropy Stability: For hyperbolic systems with discontinuities, numerical schemes must satisfy an entropy inequality to ensure physical admissibility and stability. Existing schemes for CGL often lack rigorous entropy stability guarantees, especially when combined with divergence cleaning techniques.

2. Methodology

The authors propose a comprehensive framework to construct entropy-stable numerical schemes for the CGL equations that also control magnetic divergence errors. The methodology proceeds in four main stages:

A. GLM-CGL Reformulation

The authors adopt the Generalized Lagrange Multiplier (GLM) technique, originally developed for MHD, to control the divergence of the magnetic field.

• An auxiliary scalar field $\Psi$ is introduced to evolve the divergence error.
• The CGL system is extended to a GLM-CGL system by adding transport equations for $\Psi$ and modifying the magnetic field and energy equations to include $\Psi$ and a cleaning speed $c_h$.
• The total energy is redefined to include the energy of the $\Psi$ field ($E = e + \frac{1}{2}\Psi^2$).

B. System Reformulation for Entropy Stability

To make the system suitable for entropy-stable discretization, the authors perform a specific reformulation:

• Non-conservative Splitting: Certain terms that are naturally conservative are treated as non-conservative products. This is done specifically so that the new non-conservative terms do not contribute to entropy production (i.e., $\mathbf{V}^T \cdot \boldsymbol{\eta}_{NC} = 0$, where $\mathbf{V}$ are entropy variables).
• Symmetrization: The conservative part of the system is symmetrized using Godunov's process. This involves defining an entropy potential and ensuring the Jacobian matrices are symmetric with respect to the entropy variables.
• Entropy Analysis: It is proven that the smooth solutions of the reformulated GLM-CGL system satisfy the entropy conservation law, and non-smooth solutions satisfy the entropy inequality, provided the divergence of $\mathbf{B}$ is zero (or controlled).

C. Numerical Discretization

The authors design high-order finite difference schemes on a uniform mesh:

• Entropy Conservative Fluxes: They derive numerical fluxes that satisfy the jump relations for entropy conservation. These fluxes are constructed using logarithmic averages and specific entropy-scaled eigenvectors.
• Entropy Stable Fluxes: To ensure stability for discontinuous solutions, a numerical diffusion term is added to the conservative flux. This diffusion uses Rusanov-type matrices constructed from entropy-scaled right eigenvectors.
• High-Order Reconstruction: To achieve higher-order accuracy (3rd and 4th order), the authors employ ENO (Essentially Non-Oscillatory) reconstruction with sign-preserving properties (using MinMod limiters for 2nd order) on the scaled entropy variables.
• Non-conservative Term Approximation: Derivatives in the non-conservative terms are approximated using central difference schemes of appropriate order.

D. Time Integration

• Explicit Schemes (Anisotropic): For standard CGL simulations, explicit Strong Stability Preserving Runge-Kutta (SSP-RK) methods are used.
• IMEX Schemes (Isotropic): To simulate the isotropic limit (recovering standard MHD behavior), a source term is added to relax $p_\parallel$ and $p_\perp$. Since this source term makes the system stiff, ARK-IMEX (Additive Runge-Kutta Implicit-Explicit) schemes are used, treating the stiff source term implicitly and the hyperbolic part explicitly.

3. Key Contributions

1. GLM-CGL System: Introduction and rigorous derivation of the GLM-CGL system, proving its consistency with the entropy condition and its ability to control magnetic divergence.
2. Entropy-Stable Reformulation: A novel reformulation of the CGL equations where non-conservative terms are strategically defined to vanish in the entropy evolution equation, enabling the construction of provably entropy-stable schemes.
3. High-Order Schemes: Development of 2nd, 3rd, and 4th-order accurate finite difference schemes that are simultaneously entropy stable and divergence diminishing.
4. Comprehensive Validation: Extensive numerical testing across 1D and 2D problems, comparing anisotropic (CGL) and isotropic (MHD-like) solutions, and demonstrating the superiority of the GLM approach in controlling divergence errors.

4. Numerical Results

The paper validates the schemes using several benchmark test cases:

• Accuracy Tests: 1D and 2D advection problems demonstrate that the schemes achieve their theoretical order of convergence (2nd, 3rd, and 4th order) for density and magnetic field variables.
• Divergence Control (Artificial Divergence): In a test with an initial non-zero magnetic divergence, the standard CGL scheme preserves the error, while the GLM-CGL scheme successfully damps the divergence to zero over time.
• Shock Tube (Brio-Wu): The schemes successfully resolve complex shock structures. The GLM-CGL solutions are slightly more diffusive than standard CGL but remain stable and accurate.
• Orszag-Tang Vortex & Rotor Problem: These 2D tests with complex interactions show that GLM-CGL significantly reduces the $L_1$ and $L_2$ norms of the magnetic divergence error (often by a factor of 3 or more) compared to standard CGL. The isotropic GLM-CGL solutions match standard MHD results closely.
• Field Loop Advection & Riemann Problems: These tests confirm that higher-order schemes are less diffusive and that the GLM approach consistently maintains lower divergence errors across various flow regimes.

5. Significance

This work bridges a critical gap in computational plasma physics by providing a mathematically rigorous, entropy-stable framework for simulating anisotropic plasmas (CGL) that also handles the notoriously difficult magnetic divergence constraint.

• Physical Fidelity: By ensuring entropy stability, the schemes prevent non-physical oscillations and instabilities that often plague high-order simulations of hyperbolic systems with non-conservative terms.
• Robustness: The GLM technique ensures that the fundamental physical constraint $\nabla \cdot \mathbf{B} = 0$ is maintained numerically, which is crucial for long-time simulations of astrophysical and fusion plasmas.
• Versatility: The framework supports both anisotropic (collisionless) and isotropic (collisional/MHD) regimes, making it a versatile tool for multi-scale plasma modeling.
• High-Order Capability: The ability to run 3rd and 4th-order schemes with these stability guarantees allows for more accurate and efficient simulations compared to traditional lower-order methods.

In conclusion, the paper presents a significant advancement in numerical methods for plasma physics, offering a robust, high-order, and physically consistent solver for the complex dynamics of anisotropic magnetized plasmas.