Galerkin discontinuous approximation of the MHD equations

In this report, we address several aspects of the approximation of the MHD equations by a Galerkin Discontinuous finite volume schemes. This work has been initiated during a CEMRACS project in July and August 2008 in Luminy. The project was entitled GADMHD (for GAlerkin Discontinuous approximation for the Magneto-Hydro-Dynamics). It has been supported by the INRIA CALVI project. 1 Some properties of the MHD system 1.1 Equations The Magneto-Hydro-Dynamics (MHD) equation are a useful model for describing the behavior of a compressible conductive fluid. The unknowns are the fluid density ρ, the velocity u ∈ R, the internal energy e, the pressure p and the magnetic field B ∈ R. All the unknowns depend on the space variable x ∈ R and the time variable t. The equations read   ρ ρu B Q   t +∇ ·   ρu ρu⊗ u+ (p+ B·B 2 )I−B⊗B u⊗B−B⊗ u (Q+ p+ B·B 2 )u− (B · u)B   = 0, Q = e+ u · u 2 . (1) The notation I stands for the 3× 3 identity matrix. The pressure is related to the internal energy e and the density ρ by a pressure law. In this document, we shall only consider the perfect gas law with a constant polytropic exponent γ. It reads p = P (ρ, e) = (γ − 1)ρe, γ > 1. (2) The previous equations are supplemented by the following divergence condition on the magnetic field ∇ ·B = 0. (3) The divergence free condition on the magnetic field is very important for physical reasons: it ensures that there is no magnetic charge. This condition is difficult to express on the numerical side. Therefore some authors [7], [4] have suggested to extend the ideal MHD system in the following way   ρ ρu B Q ψ   t +∇ ·   ρu ρu⊗ u+ (p+ B·B 2 )I−B⊗B u⊗B−B⊗ u+ ψI (Q+ p+ B ·B 2 )u− (B · u)B ch∇ ·B   = 0, Q = e+ u · u 2 . (4) We have added a new unknown ψ whose role is to ”clean” the divergence of the solution. Actually, the divergence perturbations are convected in the 2 ha l-0 03 37 06 3, v er si on 2 15 D ec 2 00 8 computational domain at the constant velocity ch. With adequate boundary conditions, the perturbation will be damped. The velocity ch can be chosen arbitrarily. In practice, it has to be higher than the highest wave speed of the original MHD system. We observe that if ∇ ·B = 0 and ψ =Cst, then the modified system (1) is equivalent to the MHD system (4). The two above systems can be put in a conservative form (with the space dimension d = 3) wt + d ∑