FLEXI: A high order discontinuous Galerkin framework for hyperbolic–parabolic conservation laws

Krais, Nico; Beck, Andrea; Bolemann, Thomas; Frank, Hannes; Flad, David; Gassner, Gregor J.; Hindenlang, Florian; Hoffmann, M.; Kuhn, Thomas; Sonntag, Matthias; Munz, Claus‐Dieter

doi:10.1016/j.camwa.2020.05.004

Cited by 95 publications

(75 citation statements)

References 78 publications

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“…For the examples of this section, we use the 2-register SSP-RK of fourth order and five stages of Spiteri and Ruuth [13], where the time step is computed with the typical CFL condition for the DGSEM [9] with CFL = 0.5. To control the positivity conditions (9), we choose the value β = 0.1.…”

Section: Numerical Resultsmentioning

confidence: 99%

A Subcell Finite Volume Positivity-Preserving Limiter for DGSEM Discretizations of the Euler Equations

Rueda-Ramírez¹,

Gassner²

2021

14th WCCM-ECCOMAS Congress

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In this paper, we present a positivity-preserving limiter for nodal Discontinuous Galerkin disctretizations of the compressible Euler equations. We use a Legendre-Gauss-Lobatto (LGL) Discontinuous Galerkin Spectral Element Method (DGSEM) and blend it locally with a consistent LGL-subcell Finite Volume (FV) discretization using a hybrid FV/DGSEM scheme that was recently proposed for entropy stable shock capturing. We show that our strategy is able to ensure robust simulations with positive density and pressure when using the standard and the split-form DGSEM. Furthermore, we show the applicability of our FV positivity limiter in extremely under-resolved vortex dominated simulations and in problems with shocks.

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Section: Numerical Resultsmentioning

confidence: 99%

A Subcell Finite Volume Positivity-Preserving Limiter for DGSEM Discretizations of the Euler Equations

Rueda-Ramírez¹,

Gassner²

2021

14th WCCM-ECCOMAS Congress

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“…In this section, we present a selection of three 2D Riemann problems [39,41,52]. The domain for all simulations is set to Ω = [−0.5, 0.5] 2 with a uniform grid resolution of 64 2 elements, respectively, and 512 2 DOF.…”

Section: D Riemann Problemsmentioning

confidence: 99%

A Sub-element Adaptive Shock Capturing Approach for Discontinuous Galerkin Methods

Markert

Gassner

Walch

2021

Commun. Appl. Math. Comput.

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In this paper, a new strategy for a sub-element-based shock capturing for discontinuous Galerkin (DG) approximations is presented. The idea is to interpret a DG element as a collection of data and construct a hierarchy of low-to-high-order discretizations on this set of data, including a first-order finite volume scheme up to the full-order DG scheme. The different DG discretizations are then blended according to sub-element troubled cell indicators, resulting in a final discretization that adaptively blends from low to high order within a single DG element. The goal is to retain as much high-order accuracy as possible, even in simulations with very strong shocks, as, e.g., presented in the Sedov test. The framework retains the locality of the standard DG scheme and is hence well suited for a combination with adaptive mesh refinement and parallel computing. The numerical tests demonstrate the sub-element adaptive behavior of the new shock capturing approach and its high accuracy.

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“…The distance between the noisy predictionŶ ′ and the standard predictionŶ then yields the stability loss in Eq. (6).…”

Section: Stability Trainingmentioning

confidence: 99%

“…For the a posteriori assessment of the model performance, the trained ANN are used as predictions for the closure terms during actual LES. To this end, the filtered flow state of the blind test run at t = 1.402 is used as initial state for an LES with the discontinuous Galerkin flow solver FLEXI [6]. The solution is advanced in time by integrating Eq.…”

Section: A Posteriori Analysismentioning

confidence: 99%

Investigating Model-Data Inconsistency in Data-Informed Turbulence Closure Terms

Kurz¹,

Beck²

2021

14th WCCM-ECCOMAS Congress

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In the present work, we investigate the stability of turbulence closure predictions from neural network models and highlight the role of model-data-inconsistency during inference. We quantify this inconsistency by applying the Mahalanobis distance and demonstrate that the instability of the model predictions in practical large eddy simulations (LES) correlates with a deviation of the input data between the training dataset and actual simulation data. Moreover, the method of "stability training" is applied to increase the robustness of recurrent artificial neural networks (ANN) against small perturbations in the input, which are typically unavoidable in any practical scenario. We show that this method can increase the stability of simulations with ANN-based closure term predictions significantly. The models also achieve good accuracy on the blind testing set in comparison to the baseline model trained without stability training. The work presented here can thus be seen as a building block towards long-term stable data-driven models for dynamical systems and highlights methods to detect and counter model-data-inconsistencies.

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FLEXI: A high order discontinuous Galerkin framework for hyperbolic–parabolic conservation laws

Cited by 95 publications

References 78 publications

A Subcell Finite Volume Positivity-Preserving Limiter for DGSEM Discretizations of the Euler Equations

A Subcell Finite Volume Positivity-Preserving Limiter for DGSEM Discretizations of the Euler Equations

A Sub-element Adaptive Shock Capturing Approach for Discontinuous Galerkin Methods

Investigating Model-Data Inconsistency in Data-Informed Turbulence Closure Terms

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