An Entropy-Stable Discontinuous Galerkin Discretization of the Ideal Multi-Ion Magnetohydrodynamics System
CoRR(2024)
摘要
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.
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