Entropy Stable Discontinuous Galerkin Schemes on Moving Meshes with Summation-by-Parts Property for Hyperbolic Conservation Laws

Schnücke, Gero; Krais, Nico; Bolemann, Thomas; Gassner, Gregor J.

doi:10.48550/arxiv.1812.09093

Cited by 1 publication

(3 citation statements)

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“…Again, a mesh as depicted in Figure 2(b) with refinement level l is used. Sine-like mesh deformations are commonly used to verify high-order ALE-DG implementations, see for example [9,10,12]. In these works, however, the sine functions are defined in a way that the boundaries are not moving.…”

Section: Temporal and Spatial Convergence Behaviormentioning

confidence: 99%

“…In a second set of experiments, the inviscid Taylor-Green vortex problem is studied, which is considered one of the most challenging benchmark examples to test the robustness of a flow solver for turbulent flows due to the absence of viscous dissipation, see for example [12]. Although the test case is academic, it can be expected that if a numerical method is robust for the inviscid Taylor-Green problem, it can also be successfully applied to practical, engineering problems.…”

Section: Robustness For Under-resolved Turbulent Flowsmentioning

confidence: 99%

“…It has later been developed for finite element discretizations of the compressible Navier-Stokes equations in [4,2], of the incompressible Navier-Stokes equations using linear elements in [5] and spectral element discretizations in [6], and also for finite volume discretizations, see for example [7]. In the context of discontinuous Galerkin (DG) discretizations, this technique has first been applied to the compressible Navier-Stokes equations being solved on the deforming domain [8,9,10], or solving transformed equations on a reference domain [11,12]. For the incompressible of the GCL and discretizations that formally exhibit arbitrarily high order of accuracy in space can also be achieved with a classical method-of-lines approach.…”

Section: Introductionmentioning

confidence: 99%

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High-order arbitrary Lagrangian-Eulerian discontinuous Galerkin methods for the incompressible Navier-Stokes equations

Fehn,

Heinz,

Wall

et al. 2020

Preprint

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This paper presents robust discontinuous Galerkin methods for the incompressible Navier-Stokes equations on moving meshes. High-order accurate arbitrary Lagrangian-Eulerian formulations are proposed in a unified framework for both monolithic as well as projection or splitting-type Navier-Stokes solvers. The framework is flexible, allows implicit and explicit formulations of the convective term, and adaptive time-stepping. The Navier-Stokes equations with ALE transport term are solved on the deformed geometry storing one instance of the mesh that is updated from one time step to the next. Discretization in space is applied to the time discrete equations so that all weak forms and mass matrices are evaluated at the end of the current time step. This design ensures that the proposed formulations fulfill the geometric conservation law automatically, as is shown theoretically and demonstrated numerically by the example of the free-stream preservation test. We discuss the peculiarities related to the imposition of boundary conditions in intermediate steps of projectiontype methods and the ingredients needed to preserve high-order accuracy. We show numerically that the formulations proposed in this work maintain the formal order of accuracy of the Navier-Stokes solvers. Moreover, we demonstrate robustness and accuracy for under-resolved turbulent flows.

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Section: Temporal and Spatial Convergence Behaviormentioning

confidence: 99%

Section: Robustness For Under-resolved Turbulent Flowsmentioning

confidence: 99%