Ideal GLM-MHD: About the entropy consistent nine-wave magnetic field divergence diminishing ideal magnetohydrodynamics equations

Abstract

The paper presents two contributions in the context of the numerical simulation of magnetized fluid dynamics. First, we show how to extend the ideal magnetohydrodynamics (MHD) equations with an inbuilt magnetic field divergence cleaning mechanism in such a way that the resulting model is consistent with the second law of thermodynamics. As a byproduct of these derivations, we show that not all of the commonly used divergence cleaning extensions of the ideal MHD equations are thermodynamically consistent. Secondly, we present a numerical scheme obtained by constructing a specific finite volume discretization that is consistent with the discrete thermodynamic entropy. It includes a mechanism to control the discrete divergence error of the magnetic field by construction and is Galilean invariant. We implement the new high-order MHD solver in the adaptive mesh refinement code FLASH where we compare the divergence cleaning efficiency to the constrained transport solver available in FLASH (unsplit staggered mesh scheme).The conservative formulation, i.e. Υ 0 := 0, of the ideal MHD equations is not entropy consistent as demonstrated by Godunov in the 1970s [6] B), all of the non-conservative terms mentioned are interchangeable in an entropic sense. That is, they all ensure

High-order magnetohydrodynamics for astrophysics with an adaptive mesh refinement discontinuous Galerkin scheme

“…The Powell method results in divergence levels about one order of magnitude greater than hyperbolic cleaning, in broad overall agreement with existing comparisons across different test problems (see e.g. Tricco & Price 2012;Hopkins & Raives 2016;Derigs et al 2018).…”

“…16, and like for the Orszag-Tang vortex, we find that the AMR solutions are in very good agreement with each other, as well as with the Cartesian solution. This solution can also be compared to Figures 6 and 7 from Derigs et al (2018), which also display an AMR solution for this problem spanning the same resolution range. Their "no GLM" method is a Powell-type method (extra source terms, no divergence cleaning), which produces divergence artefacts (see their Figure 7); we observe no such damage in our Powell solution.…”

“…Susanto (2014). Approaches have been developed to address positivity issues with divergence cleaning, either by ensuring that the magnetic energy can only be decreased by the scheme (Tricco & Price 2012), or through an entropy-stable formulation with entropy variables (Derigs et al 2018). We have not investigated these directions in this work, but leave them as potential future improvements.…”

Entropy Stable Space–Time Discontinuous Galerkin Schemes with Summation-by-Parts Property for Hyperbolic Conservation Laws

“…This non-conservative term is an important aspect of the spatial entropy analysis particularly when #» ∇x · #» B = 0 see, e.g. [2,9,13,34]. We address how this non-conservative term proportional to #» ∇x · #» B affects the spatial part of the DG approximation in Appendix B.2.…”

A geometric multigrid library for quadtree/octree AMR grids coupled to MPI-AMRVAC

“…The extra terms can have a parabolic (diffusive) character, as in the 'diffusive' method described in [34] which only adds a diffusion term to the induction equation. When using an extended version of the MHD equations with a variable that links to ∇ · B error damping and transport, the method can also have a hyperbolic character, as in the Generalized Lagrange multiplier (GLM) method described in [35] and the recently derived ideal GLM-MHD scheme presented in [36].…”