Abstract

We present an entropy-stable discontinuous Galerkin (DG) scheme for the multi-ion magneto-hydrodynamics (MHD) equations and efficient subcell limiting strategies to improve its robustness for challenging plasma simulations.

We start by performing a continuous entropy analysis of the multi-ion MHD system described by, e.g., Toth et al. [1], 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 a modification to the multi-ion MHD system, such that entropy consistency is guaranteed at the continuous level. Moreover, we augment the system of equations with a generalized Lagrange multiplier (GLM) technique to enforce the divergence-free condition on the magnetic field.

We derive robust entropy-conservative and entropy-stable finite volume (FV) fluxes for the modified multi-ion GLM-MHD system, and extend them to a high-order DG framework using collocated Legendre-Gauss-Lobatto SBP operators. These FV and DG schemes are consistent with the EC and ES schemes for the single-fluid GLM-MHD equations of Derigs et al. [2]. The resulting schemes guarantee the fulfillment of the second law of thermodynamics at the semi-discrete level, while maintaining local node-wise conservation properties. Finally, we extend the family of subcell limiting strategies for high-order discontinuous Galerkin spectral element methods (DGSEM) presented in [3] to the multi-ion GLM-MHD system.

We numerically validate the high-order convergence and entropic properties of our scheme and use it for challenging multi-species MHD simulations.

References

[1] Toth, G., Glocer, A., Ma, Y., Najib, D., & Gombosi, T. (2010). Multi-ion magnetohydrodynamics. In Numerical Modeling of Space Plasma Flows, Astronum-2009 (Vol. 429, p. 213).

[2] Derigs, D., Winters, A. R., Gassner, G. J., Walch, S., & Bohm, M. (2018). Ideal GLM-MHD: about the entropy consistent nine-wave magnetic field divergence diminishing ideal magnetohydrodynamics equations. Journal of Computational Physics.

[3] Rueda-Ramírez, A. M., Pazner, W., & Gassner, G. J. (2022). Subcell limiting strategies for discontinuous Galerkin spectral element methods. Computers & Fluids.