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