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An Entropy-Stable Discontinuous Galerkin Discretization of the Ideal Multi-Ion Magnetohydrodynamics System

Andrés M Rueda-Ramírez, Aleksey Sikstel, Gregor J Gassner

TL;DR

This work tackles the challenge of designing entropy-stable, high-order discretizations for the ideal multi-ion MHD system by introducing a thermodynamically consistent reformulation and integrating GLM divergence cleaning. It develops both low-order entropy-conservative FV and high-order entropy-stable DG schemes using SBP operators, ensuring the second law of thermodynamics at the semi-discrete level and providing a framework consistent with single-fluid MHD when $N_i=1$. The authors derive EC and ES fluxes that decompose into Euler, MHD, and GLM components, and implement a dissipative surface flux via a symmetric, SPD entropy-dissipation matrix, with 2D validation including convergence tests and a magnetized Kelvin-Helmholtz instability. Numerical results show high-order convergence, entropy dissipation where appropriate, and enhanced robustness when combining ES discretization with GLM divergence cleaning, including realistic multi-ion turbulence scenarios. The methods are validated on 1D and 2D problems and are implemented in an open-source pipeline, enabling extension to 3D and unstructured grids in future work.

Abstract

In this paper, we present an entropy-stable (ES) discretization using a nodal discontinuous Galerkin (DG) method for the ideal multi-ion magneto-hydrodynamics (MHD) equations. We start by performing a continuous entropy analysis of the ideal multi-ion MHD system, described by, e.g., Toth (2010) [Multi-Ion Magnetohydrodynamics], which describes the motion of multi-ion plasmas with independent momentum and energy equations for each ion species. Following the continuous entropy analysis, we propose an algebraic manipulation to the multi-ion MHD system, such that entropy consistency can be transferred from the continuous analysis to its discrete approximation. Moreover, we augment the system of equations with a generalized Lagrange multiplier (GLM) technique to have an additional cleaning mechanism of the magnetic field divergence error. We first derive robust entropy-conservative (EC) fluxes for the alternative formulation of the multi-ion GLM-MHD system that satisfy a Tadmor-type condition and are consistent with existing EC fluxes for single-fluid GLM-MHD equations. Using these numerical two-point fluxes, we construct high-order EC and ES DG discretizations of the ideal multi-ion MHD system using collocated Legendre--Gauss--Lobatto summation-by-parts (SBP) operators. The resulting nodal DG schemes satisfy the second-law of thermodynamics at the semi-discrete level, while maintaining high-order convergence and local node-wise conservation properties. We demonstrate the high-order convergence, and the EC and ES properties of our scheme with numerical validation experiments. Moreover, we demonstrate the importance of the GLM divergence technique and the ES discretization to improve the robustness properties of a DG discretization of the multi-ion MHD system by solving a challenging magnetized Kelvin-Helmholtz instability problem that exhibits MHD turbulence.

An Entropy-Stable Discontinuous Galerkin Discretization of the Ideal Multi-Ion Magnetohydrodynamics System

TL;DR

This work tackles the challenge of designing entropy-stable, high-order discretizations for the ideal multi-ion MHD system by introducing a thermodynamically consistent reformulation and integrating GLM divergence cleaning. It develops both low-order entropy-conservative FV and high-order entropy-stable DG schemes using SBP operators, ensuring the second law of thermodynamics at the semi-discrete level and providing a framework consistent with single-fluid MHD when . The authors derive EC and ES fluxes that decompose into Euler, MHD, and GLM components, and implement a dissipative surface flux via a symmetric, SPD entropy-dissipation matrix, with 2D validation including convergence tests and a magnetized Kelvin-Helmholtz instability. Numerical results show high-order convergence, entropy dissipation where appropriate, and enhanced robustness when combining ES discretization with GLM divergence cleaning, including realistic multi-ion turbulence scenarios. The methods are validated on 1D and 2D problems and are implemented in an open-source pipeline, enabling extension to 3D and unstructured grids in future work.

Abstract

In this paper, we present an entropy-stable (ES) discretization using a nodal discontinuous Galerkin (DG) method for the ideal multi-ion magneto-hydrodynamics (MHD) equations. We start by performing a continuous entropy analysis of the ideal multi-ion MHD system, described by, e.g., Toth (2010) [Multi-Ion Magnetohydrodynamics], which describes the motion of multi-ion plasmas with independent momentum and energy equations for each ion species. Following the continuous entropy analysis, we propose an algebraic manipulation to the multi-ion MHD system, such that entropy consistency can be transferred from the continuous analysis to its discrete approximation. Moreover, we augment the system of equations with a generalized Lagrange multiplier (GLM) technique to have an additional cleaning mechanism of the magnetic field divergence error. We first derive robust entropy-conservative (EC) fluxes for the alternative formulation of the multi-ion GLM-MHD system that satisfy a Tadmor-type condition and are consistent with existing EC fluxes for single-fluid GLM-MHD equations. Using these numerical two-point fluxes, we construct high-order EC and ES DG discretizations of the ideal multi-ion MHD system using collocated Legendre--Gauss--Lobatto summation-by-parts (SBP) operators. The resulting nodal DG schemes satisfy the second-law of thermodynamics at the semi-discrete level, while maintaining high-order convergence and local node-wise conservation properties. We demonstrate the high-order convergence, and the EC and ES properties of our scheme with numerical validation experiments. Moreover, we demonstrate the importance of the GLM divergence technique and the ES discretization to improve the robustness properties of a DG discretization of the multi-ion MHD system by solving a challenging magnetized Kelvin-Helmholtz instability problem that exhibits MHD turbulence.
Paper Structure (22 sections, 1 theorem, 155 equations, 7 figures, 10 tables)

This paper contains 22 sections, 1 theorem, 155 equations, 7 figures, 10 tables.

Table of Contents

  1. Introduction
  2. Governing Equations
  3. The Ideal Multi-Ion MHD System and its Thermodynamic Properties
  4. Thermodynamic properties of the system:
  5. Alternative Formulation of the Ideal Multi-Ion MHD System and its Thermodynamic Properties
  6. GLM Formulation and its Thermodynamic Properties
  7. "Quasi-conservative" Form of the Alternative Formulation of the Multi-Ion MHD System
  8. One-dimensional Version of the Multi-Ion GLM-MHD System
  9. Entropy-Stable Numerical Schemes
  10. Low-Order Entropy-Conservative Finite Volume Discretization
  11. High-Order Entropy-Stable Discontinuous Galerkin Discretization
  12. Towards an Entropy-Stable Formulation
  13. Standard Rusanov Numerical Flux
  14. A Note on the Maximum Wave Speed:
  15. Numerical Results
  16. ...and 7 more sections

Key Result

Lemma 1

The matrix $\hat{\underline{\bm{\mathcal{H}}}}$ defined in eq:dissipationMatrix is symmetric positive definite (SPD).

Figures (7)

  • Figure 1: Sparsity pattern of the matrix $\hat{\underline{\bm{\mathcal{H}}}}$ for the case of four ion species, $N_i=4$.
  • Figure 2: Contours of $\rho_1$ (top) and $\rho_2$ (bottom) for the weak blast wave simulation at $t=0.4$ with the ES solver and polynomial degree $N=3$. We show results for the coarse resolution of the test, $16 \times 16$ elements (left), and a finer resolution, $128 \times 128$ elements (right).
  • Figure 3: Log-log plots of the total entropy change and the maximum entropy production rate over all time steps of the simulation, $-\max_{t \in [t, t_f]} \dot{S}_{\Omega} (t) = \min_{t \in [t, t_f]} (-\dot{S}_{\Omega} (t))$, as a function of the CFL number for three different schemes: ((i) the entropy-conservative (EC) scheme, (ii) the provably entropy-stable scheme (ES), and (iii) the scheme that uses entropy-conservative fluxes in the volume integral together with the standard LLF scheme at the element interfaces (LLF).
  • Figure 4: Evolution of the magnetized Kelvin-Helmholtz instability problem using a fourth-order ES discretization (ii) of the multi-ion MHD model at times $t=5$ (top panel), $t=8$ (second panel), $t=12$ (third panel) and $t=20$ (bottom panel) with the highest resolution ($256 \times 512$ DOFs). We show the densities of the ion species, $H^+$ and $H^+_2$, and the ratio of the poloidal field, $B_p$, to the toroidal field, $B_t := B_3$.
  • Figure 5: Evolution of the magnetized Kelvin-Helmholtz instability problem using a fourth-order EC+LLF (iii) discretization of the multi-ion MHD model at times $t=5$ (top panel), $t=8$ (second panel), $t=12$ (third panel) and $t=20$ (bottom panel) with the highest resolution ($256 \times 512$ DOFs). We show the densities of the ion species, $H^+$ and $H^+_2$, and the ratio of the poloidal field, $B_p$, to the toroidal field, $B_t := B_3$.
  • ...and 2 more figures

Theorems & Definitions (10)

  • Remark 1
  • Remark 2
  • Remark 3
  • Remark 4
  • Remark 5
  • Lemma 1
  • proof
  • Remark 6
  • proof
  • Remark 7