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Discontinuous Galerkin Methods for Compressible and Incompressible Flows on Space–Time Adaptive Meshes: Toward a Novel Family of Efficient Numerical Methods for Fluid Dynamics

  • Francesco FambriEmail author
Original Paper
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Abstract

In this work the numerical discretization of the partial differential governing equations for compressible and incompressible flows is dealt within the discontinuous Galerkin (DG) framework along space–time adaptive meshes. Two main numerical frameworks can be distinguished: (1) fully explicit ADER-DG methods on collocated grids for compressible fluids (2) spectral semi-implicit and spectral space–time DG methods on edge-based staggered grids for the incompressible Navier–Stokes equations. In this work, the high-resolution properties of the aforementioned numerical methods are significantly enhanced within a 'cell-by-cell' Adaptive Mesh Refinement (AMR) implementation together with time accurate local time stepping (LTS). 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. In this work the nonlinear stabilization of the scheme is sequentially and locally introduced only for troubled cells on the basis of a multidimensional optimal order detection (MOOD) criterion. ADER-DG is a novel, communication-avoiding family of algorithms, which achieves high order of accuracy in time not via the standard multi-stage Runge–Kutta (RK) time discretization like most other DG schemes, but at the aid of an element-local predictor stage. In practice the method first produces a so-called candidate solution by using a high order accurate unlimited DG scheme. Then, in those cells where at least one of the chosen admissibility criteria 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 on a suitable sub-grid in order to preserve the natural sub-cell resolution of the DG scheme. Then, 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 sub-grid averages within that troubled cell. Finally, a high order DG polynomial is reconstructed back from the evolved sub-cell averages. In the ADER-DG framework several PDE system are investigated, ranging from the Euler equations of compressible gas dynamics, over the viscous and resistive magneto-hydrodynamics (MHD), to special and general relativistic MHD. Indeed, the adopted formalism is quite general, leading to a novel family of adaptive ADER-DG schemes suitable for hyperbolic systems of partial differential equations in which the numerical fluxes also depend on the gradient of the state vector because of the parabolic nature of diffusive terms. The presented results show clearly that the high-resolution and shock-capturing capabilities of the news schemes are significantly enhanced within the cell-by-cell AMR implementation together with time accurate LTS. A special treatment has been followed for the incompressible Navier–Stokes equations. In fact, the elliptic character of the incompressible Navier–Stokes equations introduces an important difficulty in their numerical solution: whenever the smallest physical or numerical perturbation arises in the fluid flow then it will instantaneously affect the entire computational domain. Thus, a semi-implicit approach has been used. The main advantage of making use of a semi-implicit discretization is that the numerical stability can be obtained for large time-steps without leading to an excessive computational demand. In this context, we derived two new families of spectral semi-implicit and spectral space–time DG methods for the solution of the two and three dimensional Navier–Stokes equations on edge-based adaptive staggered Cartesian grids. The discrete solutions of pressure and velocity are expressed in the form of piecewise polynomials along different meshes. While the pressure is defined on the control volumes of the main grid, the velocity components are defined on edge-based dual control volumes, leading to a spatially staggered mesh. In the first family, high order of accuracy is achieved only in space, while a simple semi-implicit time discretization is derived by introducing an implicitness factor \(\theta \in [0.5,1]\) for the pressure gradient in the momentum equation. The real advantages of the staggering arise after substituting the discrete momentum equation into the weak form of the continuity equation. In fact, the resulting linear system for the pressure is symmetric and positive definite and either block penta-diagonal (in 2D) or block hepta-diagonal (in 3D). As a consequence, the pressure system can be solved very efficiently by means of a classical matrix-free conjugate gradient method. The resulting algorithm is stable, computationally very efficient, and at the same time arbitrary high order accurate in both space and time. This new numerical method has been thoroughly validated for approximation polynomials of degree up to \(N = 12\), using a large set of non-trivial test problems in two and three space dimensions, for which either analytical, numerical or experimental reference solutions exist.

Notes

Acknowledgements

This work has been carried out under the supervision of Prof. Dr.-Ing Michael Dumbser, who is warmly thanked by the author, within the three-years Doctoral Programme in Civil, Environmental and Mechanical Engineering at DICAM Dept. of University of Trento. The author would like to thank also prof. Arturo Hidalgo, prof. Olindo Zanotti, prof. Vincenzo Casulli, prof. Francesco Serra-Capizzano and prof. Luciano Rezzolla for the fruitful collaboration and the valuable discussions. The presented research has been financed mainly by the European Research Council (ERC) under the European Union’s Seventh Framework Programme (FP7/2007-2013) with the research project STiMulUs, ERC Grant Agreement No. 278267, and by the European Unions Horizon 2020 Research and Innovation Programme under the project ExaHyPE, Grant No. 671698 (call FETHPC-1-2014). The simulations were performed on the SuperMUC supercomputer at the LRZ in Garching, Germany, on the LOEWE cluster in CSC in Frankfurt, on the HazelHen supercomputer at the HLRS in Stuttgart, Germany, as well as on the local HPC cluster at the University of Trento. Several numerical simulations have been also tested on the MARCONI supercomputer at at the CINECA HPC center, Italy, by means of the class C project ’THOAst’—Towards High Order methods for Astrophysics—(no. IsC58, ISCRA call). We acknowledge the CINECA award under the ISCRA initiative, for the availability of high performance computing resources and support. In particular, the author is grateful for the technical support provided by Dr. Björn Dick (HLRS) for the strong scaling study performed on the Hazel-Hen supercomputer. Moreover, some tests concerning of ADERDG schemes have been performed also on the ’Swan’ CRAY XC50 system, Access to the Cray XC40 supercomputer ’Swan’ was kindly provided through Cray Inc.’s Marketing Partner Network by means of Dr. Stefan Andersson (CRAY). The author is members of the Italian INdAM Research group GNCS.

Compliance with Ethical Standards

Conflict of interest

The authors declare that they have no conflict of interest.

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© CIMNE, Barcelona, Spain 2019

Authors and Affiliations

  1. 1.Laboratory of Applied Mathematics, Department of Civil, Environmental and Mechanical EngineeringUniversity of TrentoTrentoItaly

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