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摘要
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 (Dumbser et al., 2014, Zanotti et al., 2015 [40,411). 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'. In the present work, a nonlinear stabilization of the scheme is sequentially and locally introduced only for troubled cells on the basis of a novel a posteriori detection criterion, i.e. the MOOD approach. 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. 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 criteria 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 criteria is violated the computed candidate solution is detected as troubled and is locally rejected. 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 distinguished high-resolution properties of the presented numerical scheme stand out against a wide number of non-trivial test cases both for the compressible Navier Stokes and the viscous and resistive magnetohydrodynamics equations. The present results show clearly that the shock-capturing capability of the news schemes is significantly enhanced within a cell by-cell Adaptive Mesh Refinement (AMR) implementation together with time accurate local time stepping (LTS).
- 出版日期2017-11
