The discontinuous Galerkin spectral element method (DGSEM) is a nodal discontinuous Galerkin collocation scheme that uses a set of Legendre-Gauss or Legendre-Gauss-Lobatto nodes to represent the solution with Lagrange interpolating polynomials and to compute integrals with a quadrature rule. <\/p>\n
It was recently shown [1] that the DGSEM semi-discretization of a conservation law using Legendre-Gauss nodes can be written in finite-volume form as
\n\\begin{equation*}
\nJ_{j} \\dot{\\mathbf{u}}^{DG}_{j} =
\n\\frac{1}{\\omega_j}
\n\\left(
\n \\hat{\\mathbf{f}}^{DG}_{(j-1,j)}
\n– \\hat{\\mathbf{f}}^{DG}_{(j,j+1)}
\n\\right),
\n\\end{equation*}
\nwhere $J_j$ is the mapping Jacobian, $\\omega_j$ are the Legendre-Gauss quadrature weights, and $\\hat{\\mathbf{f}}^{DG}_{(\\cdot,\\cdot)}$ are high-order DGSEM fluxes between adjacent nodes.<\/p>\n
The existence of a FV form for the Legendre–Gauss DGSEM enables the use of the subcell limiting strategies described in [2]. In these methods, a hybrid DGSEM\/FV method is used, where the semi-discretization at each node reads
\n\\begin{equation*}
\nJ_{j} \\dot{\\mathbf{u}}_{j} =
\n\\frac{1}{\\omega_j}
\n\\left(
\n \\hat{\\mathbf{f}}_{(j-1,j)}
\n– \\hat{\\mathbf{f}}_{(j,j+1)}
\n\\right),
\n\\end{equation*}
\nand the local fluxes are obtained as a convex combination of high-order DGSEM and low-order FV fluxes,
\n\\begin{equation*}
\n \\hat{\\mathbf{f}}_{(i,j)} = (1-\\alpha_{(i,j)}) \\hat{\\mathbf{f}}^{DG}_{(i,j)}
\n +
\n \\alpha_{(i,j)}
\n \\hat{\\mathbf{f}}^{DG}_{(i,j)},
\n ~~~~
\n \\alpha_{(i,j)} \\in [0,1].
\n\\end{equation*}<\/p>\n
To illustrate the shock-capturing capacity of the hybrid DGSEM\/FV method, we simulate a Sedov blast problem describing the evolution of a blast wave expanding from an initial concentration of density and pressure, and adjust the blending coefficient $\\alpha_{(i,j)}$ to avoid non-physical density oscillations in the vicinity of shocks.<\/p>\n
The video shows the distribution of density and the blending coefficient obtained for entropy-stable variants of the Legendre-Gauss and Legendre-Gauss-Lobatto DGSEM methods.<\/p>\n