Hybridisable Discontinuous Galerkin Formulation of Compressible Flows

This work presents a review of high-order hybridisable discontinuous Galerkin (HDG) methods in the context of compressible flows. Moreover, an original unified framework for the derivation of Riemann solvers in hybridised formulations is proposed. This framework includes, for the first time in an HDG context, the HLL and HLLEM Riemann solvers as well as the traditional Lax-Friedrichs and Roe solvers. HLL-type Riemann solvers demonstrate their superiority with respect to Roe in supersonic cases due to their positivity preserving properties. In addition, HLLEM specifically outstands in the approximation of boundary layers because of its shear preservation, which confers it an increased accuracy with respect to HLL and Lax-Friedrichs. A comprehensive set of relevant numerical benchmarks of viscous and inviscid compressible flows is presented. The test cases are used to evaluate the competitiveness of the resulting high-order HDG scheme with the aforementioned Riemann solvers and equipped with a shock treatment technique based on artificial viscosity.Comment: 60 pages, 31 figures. arXiv admin note: substantial text overlap with arXiv:1912.0004

Lagrangian ADER-WENO Finite Volume Schemes on Unstructured Triangular Meshes Based On Genuinely Multidimensional HLL Riemann Solvers

In this paper we use the genuinely multidimensional HLL Riemann solvers recently developed by Balsara et al. to construct a new class of computationally efficient high order Lagrangian ADER-WENO one-step ALE finite volume schemes on unstructured triangular meshes. A nonlinear WENO reconstruction operator allows the algorithm to achieve high order of accuracy in space, while high order of accuracy in time is obtained by the use of an ADER time-stepping technique based on a local space-time Galerkin predictor. The multidimensional HLL and HLLC Riemann solvers operate at each vertex of the grid, considering the entire Voronoi neighborhood of each node and allows for larger time steps than conventional one-dimensional Riemann solvers. The results produced by the multidimensional Riemann solver are then used twice in our one-step ALE algorithm: first, as a node solver that assigns a unique velocity vector to each vertex, in order to preserve the continuity of the computational mesh; second, as a building block for genuinely multidimensional numerical flux evaluation that allows the scheme to run with larger time steps compared to conventional finite volume schemes that use classical one-dimensional Riemann solvers in normal direction. A rezoning step may be necessary in order to overcome element overlapping or crossing-over. We apply the method presented in this article to two systems of hyperbolic conservation laws, namely the Euler equations of compressible gas dynamics and the equations of ideal classical magneto-hydrodynamics (MHD). Convergence studies up to fourth order of accuracy in space and time have been carried out. Several numerical test problems have been solved to validate the new approach

Solving the system of radiation magnetohydrodynamics for solar physical simulations in 3d

In this study we present a finite-volumen scheme for solving the equations of radiation magnetohydrodynamics in two and three space dimensions. Among other applications this system is used to model the plasma in the solar convection zone and in the solar photosphere. It is a non--linear system of balance laws derived from the Euler equations of gas dynamics and the Maxwell equations; the energy transport through radiation is also included in the model. The starting point of our presentation is a standard explicit first and second order finite-volume scheme on both structured and unstructured grids. We first study the convergence of a finite-volume scheme applied to a scalar model problem for the full system of radiation magnetohydrodynamics. We then present modifications of the base scheme. These make it possible to approximate the system of magnetohydrodynamics with an arbitrary equation of state; they reduce errors due to a violation of the divergence constraint on the magnetic field, and they lead to an improved accuracy in the approximation of solution near an equilibrium state. These modifications significantly increase the robustness of the scheme and are essential for an accurate simulation of processes in the solar atmosphere ...thesi