Space–time adaptive ADER-DG schemes for dissipative flows: Compressible Navier–Stokes and resistive MHD equations

Abstract

This paper presents an arbitrary high-order accurate ADER Discontinuous Galerkin (DG) method on space-time adaptive meshes (AMR) for the solution of two important families of non-linear time dependent partial differential equations for compressible dissipative flows: the compressible Navier-Stokes equations and the equations of viscous and resistive magnetohydrodynamics in two and three space-dimensions.The work continues a recent series of papers concerning the development and application of a proper a posteriori subcell finite volume limiting procedure suitable for discontinuous Galerkin methods [49,112,111]. It is a well known fact that a major weakness of high order DG methods lies in the difficulty of limiting discontinuous solutions, which generate spurious oscillations, namely the so-called 'Gibbs phenomenon'. Over the years, several attempts have been made to cope with this problem and different kinds of limiters have been proposed. Among them, a rather intriguing paradigm has been defined in the work of [23], in which the nonlinear stabilization of the scheme is sequentially and locally introduced only for troubled cells on the basis of a novel a posteriori detection criterion (MOOD approach). In the present work the main benefits of the MOOD paradigm, i.e. the computational robustness even in the presence of strong shocks, are preserved and the numerical diffusion is considerably reduced also for the limited cells by resorting to a proper sub-grid. An important feature of our new scheme is its ability to cure even floating point errors (NaN) that may occur during a simulation, for example when taking real roots of negative numbers or after divisions by zero. In practice the method first produces a so-called candidate solution by using a high order accurate unlimited DG scheme. Then, a set of numerical and physical detection critera is applied to the candidate solution, namely: positivity of pressure and density, absence of floating point errors and satisfaction of a discrete maximum principle in the sense of polynomials. Furthermore, in those cells where at least one of these critera is violated the computed candidate solution is detected as troubled and is locally rejected. Next, the numerical solution of the previous time step is scattered onto cell averages of a suitable sub-grid in order to preserve the natural sub-cell resolution of the DG scheme. Subsequently, a more reliable numerical solution is recomputed a posteriori by employing a more robust but still very accurate ADER-WENO finite volume scheme on the subgrid averages within that troubled cell. Finally, a high order DG polynomial is reconstructed back from the evolved subcell averages.We apply the whole approach for the first time to the equations of compressible gas dynamics and magnetohydrodynamics in the presence of viscosity, thermal conductivity and magnetic resistivity, therefore extending our family of adaptive ADER-DG schemes to cases for which the numerical fluxes also depend on the gradient of the state vector.The disting...

A simple diffuse interface approach for compressible flows around moving solids of arbitrary shape based on a reduced Baer–Nunziato model

“…The corrector step evolves the discrete solution u h in time and takes into account the information coming from the cell neighbors via classical numerical flux functions (approximate Riemann solvers). We close the section by describing our a posteriori sub-cell finite volume limiter [45,141,40,47,9], which assures the robustness of the scheme even in the presence of discontinuities, but keeping at the same time also the high resolution of the underlying DG scheme.…”

“…Furthermore, in order to increase the resolution in the areas of interest, the ADER-DG scheme described above has been implemented on space-time adaptive Cartesian meshes, with a cell-by-cell refinement approach; for all the details we refer to [44,38,141,47,46,13,140,113].…”

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High‐order divergence‐free velocity reconstruction for free surface flows on unstructured Voronoi meshes

“…As a consequence, the method is fully conservative for mass but not for momentum. 13 Semi-implicit methods have been extended to discontinuous Galerkin (DG) schemes [14][15][16][17][18][19][20] for viscous and inviscid hydrodynamics and magnetohydrodynamics, whereas, in the works of Dumbser et al, 21,22 they are used for the simulation of compressible flows in tubes. 12 The efficient use of subgrid resolution has also been included in the same framework in the work of Casulli and Stelling.…”

“…12 The efficient use of subgrid resolution has also been included in the same framework in the work of Casulli and Stelling. 13 Semi-implicit methods have been extended to discontinuous Galerkin (DG) schemes [14][15][16][17][18][19][20] for viscous and inviscid hydrodynamics and magnetohydrodynamics, whereas, in the works of Dumbser et al, 21,22 they are used for the simulation of compressible flows in tubes. Fully implicit DG methods can be found in related works [23][24][25][26] for the numerical solution of both compressible and incompressible Navier-Stokes equations.…”