We solve the multi-component magneto-hydrodynamics (MHD) equations using a hybrid finite volume (FV)\/discontinuous Galerkin (DG) method [1], which combines the two discretization approaches at the node level. To test the robustness of the hybrid FV\/DG method, we compute a modification of the Orszag-Tang vortex problem [2], where we use two ion species and initialize the flow with a sharp interface at $y=0$:
\n\\[
\n\\begin{array}{rlrl}
\n\\rho^1(x,y,t=0) &= \\begin{cases}
\n{25}\/{36 \\pi} & y \\ge 0 \\\\
\n10^{-8} & y < 0
\n\\end{cases} ,
\n&\\rho^2(x,y,t=0) &= \\begin{cases}
\n10^{-8} & y < 0\\\\
\n{25}\/{36 \\pi} & y \\ge 0
\n\\end{cases} , \\\\
\nv_1 (x,y,t=0) &= – \\sin (2 \\pi y),
\n&v_2 (x,y,t=0) &= \\sin (2 \\pi x), \\\\
\nB_1 (x,y,t=0) &= -\\frac{1}{\\sqrt{4 \\pi}} \\sin (2 \\pi y),
\n&B_2 (x,y,t=0) &= -\\frac{1}{\\sqrt{4 \\pi}} \\sin (4 \\pi x). \\\\
\np (x,y,t=0) &= \\frac{5}{12 \\pi},
\n& & \\\\
\n\\end{array}
\n\\]
\nwhere $\\rho^1$ is the density of the first ion species, $\\rho^2$ is the density of the second ion species,$\\vec{v}=(v_1,v_2)$ is the plasma velocity, $\\vec{B}=(B_1,B_2)$ is the magnetic field, and $p$ is the plasma pressure (without magnetic pressure).
\n
\nThe fourth-order DG method delivers high accuracy, but fails to describe shocks and material interfaces. Therefore, we use a modal indicator [3] to detect the elements where the solution changes abruptly. In those elements, we combine the DG method with a robust lower order FV scheme locally (at the node level) to impose TVD-like conditions on the density of the ion species,
\n\\[
\n\\min_{k \\in \\mathcal{N} (j)} {\\rho}^{i,\\mathrm{FV}}_{k}
\n\\le \\rho^i_{j} \\le
\n\\max_{k \\in \\mathcal{N} (j)} {\\rho}^{i,\\mathrm{FV}}_{k},
\n\\]
\nand a local minimum principle on the specific entropy,
\n\\[
\ns({\\mathbf{u}}_{j}) \\ge \\min_{k \\in \\mathcal{N} (j)} s(\\mathbf{u}^{\\mathrm{FV}}_{k}),.
\n\\]
\nwhere $\\mathcal{N} (j)$ denotes the collection of nodes in the low-order stencil of each node $j$.
\n
\nThe video shows the total density ($\\rho=\\rho^1+\\rho^2$), the density of the first ion species ($\\rho^1$), and the so-called blending coefficient ($\\alpha$) for the combination of a fourth-order accurate entropy stable DG method with first- and second-order accurate FV methods (Rusanov solver) and $1024^2$ degrees of freedom. A blending coefficient $\\alpha=0$ means that only the DG is being used, and a blending coefficient $\\alpha=1$ means that only the FV method is being used. The example shows that the hybrid FV\/DG scheme is able to capture the shocks of the simulation correctly, deal with near-vacuum conditions and sharp interfaces.<\/p>\n
[1] Rueda-Ram\u00edrez, A. M.; Pazner, W., & Gassner, G. J. (2022). Subcell limiting strategies for discontinuous Galerkin spectral element methods. Computers & Fluids. arXiv:2202.00576<\/a>.
\n[2] Orszag, S.; Tang, C (1979). Small-scale structure of two-dimensional magnetohydrodynamic turbulence. Journal of Fluid Mechanics.
\n[3] Persson, P. O.; Peraire, J. (2006). Sub-cell shock capturing for discontinuous Galerkin methods. In 44th AIAA Aerospace Sciences Meeting and Exhibit (p. 112).<\/p>\n