Abstract

The high-accuracy solution of the MHD equations is of great interest in various fields of physics, mathematics, and engineering. Higher-order DG schemes offer low dissipation and dispersion as well as the ability to model complex geometries, which is very desirable in various applications. Numerical solution of the MHD equations is made challenging by the fact that the PDE system has an involution constraint. Therefore, we construct high-order, globally divergence-free DG schemes for compressible MHD. The modes of the fluid variables are collocated at the zones of the mesh; the magnetic field components and their higher-order modes are collocated at the faces of the mesh. The fluid equations are evolved using classical DG, while the magnetic fields are evolved using a novel DG-like approach, first proposed by Balsara and Käppeli (J. Comput. Phys. 336 (2017), 104–127). This DG-like method ensures the globally divergence-free evolution of the magnetic field.

The method is built around three building blocks. The first building block consists of a divergence-free reconstruction of the magnetic field. The second building block consists of a DG-like formulation of Faraday’s law that provides a weak-form interpretation of Stokes’ law (as opposed to traditional DG, which relies on Gauss’s law). To provide a physically consistent electric field for the update of Faraday’s law, we use the third building block, which consists of a multidimensional Riemann solver that is evaluated at the edges of the mesh. We recognize that the limiting of facial variables makes the design of the MHD limiter very different from the usual DG limiter. As a result, a limiter strategy is presented for DG schemes which retains the traditional DG limiting approach while building into it a positivity-enforcement step and a step that updates the facial modes in a constraint-preserving fashion. This limiter is crucial to the robust and physically consistent operation of our DG scheme for MHD even at high orders.

It is shown that our schemes meet their design accuracies at second, third, and fourth orders on smooth test problems. Several stringent test problems with complex flow features are presented, which are robustly handled by our DG method.

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