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Gradient-Robust Hybrid DG Discretizations for the Compressible Stokes Equations.

P L Lederer¹, C Merdon²

¹Department of Applied Mathematics, University of Twente, Hallenweg 19, 7522NH Enschede, Netherlands.

Journal of Scientific Computing

|July 8, 2024

Summary

This study presents two hybrid discontinuous Galerkin (HDG) methods for compressible Stokes equations. One method ensures convergence, non-negativity, and gradient-robustness for accurate fluid simulations.

Keywords:

Compressible Stokes equations Gradient-robustness Hybrid discontinuous Galerkin methods Well-balanced schemes

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Area of Science:

Computational fluid dynamics
Numerical analysis
Partial differential equations

Background:

Compressible Stokes equations model fluid flow with density variations.
Accurate numerical methods are crucial for simulating well-balanced fluid states.
Gradient-robustness improves accuracy in hydrostatic balance scenarios.

Purpose of the Study:

To investigate two hybrid discontinuous Galerkin (HDG) discretizations for compressible Stokes equations.
To evaluate methods based on convergence, density non-negativity, mass constraints, and gradient-robustness.
To demonstrate the effectiveness of these methods for well-balanced and non-hydrostatic states.

Main Methods:

Development and analysis of two HDG schemes for the velocity-density formulation.
One scheme utilizes a -conforming velocity ansatz space.
The other scheme employs a fully discontinuous approach.
Higher-order extensions of both schemes are presented.

Main Results:

The -conforming HDG scheme satisfies all desired properties, including gradient-robustness.
The fully discontinuous HDG scheme meets all properties except gradient-robustness.
Numerical benchmarks validate the performance of both higher-order schemes.
The importance of gradient-robustness for non-hydrostatic well-balanced states is shown.

Conclusions:

The -conforming HDG method provides a robust and accurate approach for compressible fluid flow.
Gradient-robustness is essential for accurately capturing well-balanced fluid dynamics.
The presented methods are applicable to both Stokes and Navier-Stokes equations.