Shock-Capturing Schemes in Computational MHD
The purpose of the present review is to present and discuss some introductory aspects relevant to computational compressible magnetohydrodynamics (MHD). The shock-capturing framework developed for the Euler equations of gasdynamics is extended to MHD by i
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INAF Osservatorio Astronomico di Torino, 10025 Pino Torinese, Italy, [email protected] Dipartimento di Fisica Generale dell’Universit` a, Via Pietro Giuria 1, I-10125 Torino, Italy, [email protected]
Abstract. The purpose of the present review is to present and discuss some introductory aspects relevant to computational compressible magnetohydrodynamics (MHD). The shock-capturing framework developed for the Euler equations of gasdynamics is extended to MHD by illustrating differences and additional complexities introduced by the presence of magnetic fields. In particular, we focus our attention on the characteristic structure of the equations by investigating the nature of different MHD waves, the solution to the Riemann problem and last, but not least, various computational strategies to control the divergence-free condition of magnetic fields.
Keywords Magnetohydrodynamics (MHD) · Methods: numerical · Shock waves · Waves
1 Introduction Astrophysical plasmas can often be described by means of the ideal compressible magnetohydrodynamic (MHD) equations. A far from exhaustive list includes jets, accretion disks, stellar or galactic atmospheres, and the interstellar medium. In many instances, one has to deal with flows with shocks and discontinuities, and, in such situations, the numerical methods used in the simulations are based on the shock-capturing framework developed for the Euler equations of gasdynamics. The extension of such framework to MHD has proven, however, to be nontrivial because of several properties of the MHD system that makes it different from the Euler counterpart. A first example of the problems encountered when moving from gasdynamic to MHD is nonstrict hyperbolicity. This has been addressed by [5] and [40], and following this and other advancements, several second-order upwind codes were then constructed and tested mainly for the one-dimensional case, see for example [2, 42, 50] and [10]. New problems have to be considered for the
Mignone, A., Bodo, G.: Shock-Capturing Schemes in Computational MHD. Lect. Notes Phys. 754, 71–101 (2008) c Springer-Verlag Berlin Heidelberg 2008 DOI 10.1007/978-3-540-76967-5 2
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A. Mignone and G. Bodo
multidimensional case, in particular MHD equations are supplemented by the condition of null divergence of the magnetic field that has to be preserved during the evolution. A failure of the numerical scheme to maintain this constraint, as shown by [4], leads to unphysical effects in the solution. Several different methods have then been proposed for dealing with this issue, see for example [3, 4, 12, 17, 19, 29, 36]. In this review, after a presentation of the equations in Sect. 2, we will discuss their characteristic structure and the nature of the waves with particular reference to the peculiarities of the MHD system in Sect. 3. The solution to the Riemann problem is one of the main building blocks of shock capturing methods, and our discussion will be focused on approximate solvers, in particular, of the HLL class. In the last section, we will deal with the ot
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