Entropy Stable DGSEM Schemes of Gauss Points Based on Subcell Limiting
<p>Shu–Osher problem, <math display="inline"><semantics> <mrow> <mi>D</mi> <mi>O</mi> <mi>F</mi> <mi>s</mi> <mspace width="3.33333pt"/> <mo>=</mo> <mspace width="3.33333pt"/> <mn>640</mn> </mrow> </semantics></math>. (<b>a</b>) Global density distribution; (<b>b</b>) local density distribution.</p> "> Figure 2
<p>Using the HLL flux vorticity field. (<b>a</b>) DGSEM-LGL; (<b>b</b>) ESDGSEM-LGL; (<b>c</b>) ESDGSEM-LG.</p> "> Figure 3
<p>Using the LLF flux vorticity field. (<b>a</b>) DGSEM-LGL; (<b>b</b>) ESDGSEM-LGL; (<b>c</b>) ESDGSEM-LG.</p> "> Figure 4
<p>Density distribution and total entropy evolution. (<b>a</b>) DGSEM-LGL; (<b>b</b>) ESDGSEM-LGL; (<b>c</b>) ESDGSEM-LG; (<b>d</b>) ESDGSEM-LG-SL; (<b>e</b>) troubled cell distribution; (<b>f</b>) total entropy evolution.</p> "> Figure 5
<p>Density contour and troubled cell distribution for the double Mach reflection problem. (<b>a</b>) Density; (<b>b</b>) troubled cell distribution.</p> "> Figure 6
<p>The results of the NACA0012 airfoil. (<b>a</b>) Pressure coefficient; (<b>b</b>) density; (<b>c</b>) Mach number; (<b>d</b>) pressure; (<b>e</b>) troubled cell distribution; (<b>f</b>) ratio of troubled cells.</p> "> Figure 6 Cont.
<p>The results of the NACA0012 airfoil. (<b>a</b>) Pressure coefficient; (<b>b</b>) density; (<b>c</b>) Mach number; (<b>d</b>) pressure; (<b>e</b>) troubled cell distribution; (<b>f</b>) ratio of troubled cells.</p> ">
Abstract
:1. Introduction
- High-order entropy-stable schemes based on different solution points are discussed. Although high-order schemes have been shown to satisfy the discrete entropy condition. They have different stability properties for the same test case. The stability and resolution of the high-order schemes are analysed in the context of the Kelvin–Helmholtz instability problem and under-resolved vortical flows.
- A provably entropy-stable high-order scheme based on subcell limiting is proposed for hyperbolic conservation laws. First, we use an indicator considering the modal decay of the polynomial representation based on an extended stencil to detect troubled cells. Second, the cells to be solved are divided into troubled and smooth cells [34,35]. To ensure that the hybrid scheme is entropy-stable, we choose the same entropy-stable or entropy conservative numerical flux. The hybrid scheme can be implemented according to the magnitude of the troubled cell indicator to mediate the change from a smooth region to a discontinuous region.
2. A Brief Review of the Basic Theory
2.1. Conservation Laws
2.2. Entropy Inequality
3. The Hybrid Scheme
3.1. ESDGSEM Schemes
3.2. Entropy-Stable Scheme Based on the Subcell Limiting Technique
3.2.1. Conservation of the Hybrid Scheme
3.2.2. Entropy Stability of the Hybrid Scheme
4. Numerical Validation
4.1. Shu–Osher
4.2. Isentropic Vortex
4.3. Steady Shock Wave
4.4. Under-Resolved Vortical Flows
4.5. Kelvin–Helmholtz Instability
4.6. Double Mach Reflection
4.7. Transonic Flows around the NACA0012 Airfoil
5. Conclusions
- The DGSEM methods based on the entropy-stable strategy have better robustness when solving under-resolved flows than the traditional DGSEM method.
- The ESDGSEM-LG-SL scheme can stably calculate high-Mach-number problems, indicating that the scheme has good stability. Furthermore, the results of the double Mach reflection problem and transonic flows around the NACA0012 airfoil case also show that this scheme has a better shock-capturing ability than the other schemes.
Author Contributions
Funding
Institutional Review Board Statement
Informed Consent Statement
Data Availability Statement
Acknowledgments
Conflicts of Interest
Appendix A
References
- Williams, D.M.; Jameson, A. Energy Stable Flux Reconstruction Schemes for Advection–Diffusion Problems on Tetrahedra. J. Sci. Comput. 2014, 59, 721–759. [Google Scholar] [CrossRef]
- Cockburn, B.; Shu, C.-W. TVB runge-kutta local projection discontinuous Galerkin finite elementmethod for scalar conservation laws III: One dimensional systems. J. Comput. Phys. 1989, 84, 90–113. [Google Scholar] [CrossRef] [Green Version]
- Cockburn, B.; Shu, C.-W. The Runge-Kutta local projection discontinuous Galerkin finite element method for scalar conservation laws V: Multidimensional systems. J. Comput. Phys. 1998, 141, 199–224. [Google Scholar] [CrossRef]
- Cockburn, B.; Shu, C.W. The Runge-Kutta local projection P1 Discontinuous Galerkin finite element method for scalar conservation laws. Math. Comput. 1990, 54, 545–581. [Google Scholar]
- Wang, Z.J.; Fidkowski, K.; Abgrall, R.; Bassi, F.; Caraeni, D.; Cary, A.; Deconinck, H.; Hartmann, R.; Hillewaert, K.; Huynh, H.T.; et al. High-order CFD methods: Current status and perspective. Int. J. Numer. Methods Fluids 2013, 72, 811–845. [Google Scholar] [CrossRef]
- Persson, P.O.; Peraire, J. Subcell shock capturing for discontinuous Galerkin methods. In Proceedings of the 44th AIAA Aerospace Science Meeting and Exhibit, Reno, NV, USA, 9–12 January 2006; pp. 9–12. [Google Scholar]
- Discacciati, N.; Hesthaven, J.S.; Ray, D. Cotrolling oscillations in high-order discontinuous Galerkin schemes using artificial viscosity tuned by neural networks. J. Comput. Phys. 2020, 409, 109304. [Google Scholar] [CrossRef] [Green Version]
- Hesthaven, J.S.; Warburton, T. Nodal Discontinuous Galerkin Methods; Springer: Berlin/Heidelberg, Germany, 2008; Volume 11, pp. 731–754. [Google Scholar]
- Kirby, R.M.; Karniadakis, G.E. De-aliasing on non-uniform grids: Algorithms and applications. J. Comput. Phys. 2003, 191, 249–264. [Google Scholar] [CrossRef] [Green Version]
- Gassner, G.J.; Winters, A.R.; Kopriva, D.A. Split form nodal discontinuous Galerkin schemes with summation-by-parts property for the compressible Euler equations. J. Comput. Phys. 2016, 327, 39–66. [Google Scholar] [CrossRef] [Green Version]
- Chan, J.; Ranocha, H.; Warburton, T. On the entropy projection and the robustness of high order entropy stable discontinuous Galerkin schemes for under-resolved flows. arXiv 2022, arXiv:2203.10238. [Google Scholar] [CrossRef]
- Sonntag, M.; Munz, C.-D. Shock capturing for discontinuous Galerkin methods using finite volume subcells. In Finite Volumes for Complex Applications VII-Elliptic, Parabolic and Hyperbolic Problems; Springer: Berlin/Heidelberg, Germany, 2014; Volume 32, pp. 945–953. [Google Scholar]
- Zhong, X.; Shu, C.-W. A simple weighted essentially nonoscillatory limiter for Runge–Kutta discontinuous Galerkin methods. J. Comput. Phys. 2013, 232, 397–415. [Google Scholar] [CrossRef]
- Dafermos, C.M. Hyperbolic Conservation Laws in Continuum Physics, Grundlehrender Mathematischen Wissenschaften; Springer: Berlin/Heidelberg, Germany, 2010; Volume 3, p. 325. [Google Scholar]
- Godlewski, E.; Raviart, P.-A. Numerical Approximation of Hyperbolic Systems of Conservation Laws, Applied Mathematical Sciences; Springer: Berlin/Heidelberg, Germany, 2020; Volume 3, p. 30. [Google Scholar]
- Osher, S.; Tadmor, E. On the convergence of difference approximations to scalar conservation laws. Math. Comput. 1988, 50, 19–51. [Google Scholar] [CrossRef]
- Tadmor, E. The numerical viscosity of entropy stable schemes for systems of conservation laws—I. Math. Comput. 1987, 49, 91–103. [Google Scholar] [CrossRef]
- Chan, J. On discretely entropy conservative and entropy stable discontinuous Galerkin methods. J. Comput. Phys. 2018, 362, 346–374. [Google Scholar] [CrossRef] [Green Version]
- Chan, J.; Wilcox, L.C. Discretely entropy stable weight-adjusted discontinuous Galerkin methods on curvilinear meshes. J. Comput. Phys. 2019, 378, 366–393. [Google Scholar] [CrossRef] [Green Version]
- Chen, T.; Shu, C.W. Entropy stable high order discontinuous Galerkin methods with suitable quadrature rules for hyperbolic conservation laws. J. Comput. Phys. 2017, 345, 427–461. [Google Scholar] [CrossRef]
- Chan, J.; Del Rey Fernández, D.C.; Carpenter, M.H. Efficient entropy stable Gauss collocation methods. SIAM J. Sci. Comput. 2019, 41, A2938–A2966. [Google Scholar] [CrossRef]
- Fernández, D.C.D.R.; Boom, P.D.; Zingg, D.W. A generalized framework for nodal first derivative summation-by-parts operators. J. Comput. Phys. 2014, 266, 214–239. [Google Scholar] [CrossRef]
- Baumann, C.E.; Oden, J.T. A discontinuous hp finite element method for the Euler and Navier-Stokes equations. Int. J. Numer. Methods Fluids 1999, 31, 79–95. [Google Scholar] [CrossRef]
- Burbeau, A.; Sagaut, P.; Bruneau, C.H. A problem-independent limiter for high-order Runge-Kutta discontinuous Galerkin methods. J. Comput. Phys. 2001, 169, 111–150. [Google Scholar] [CrossRef]
- Dumbser, M.; Zanotti, O.; Loubere, R.; Diot, S. A posteriori subcell limiting of the discontinuous Galerkin finite element method for hyperbolic conservation laws. J. Comput. Phys. 2014, 278, 47–75. [Google Scholar] [CrossRef] [Green Version]
- Dumbser, M.; Loubere, R. A simple robust and accurate a posteriori sub-cell finite volume limiter for the discontinuous Galerkin method on unstructured meshes. J. Comput. Phys. 2016, 319, 163–199. [Google Scholar] [CrossRef] [Green Version]
- Sonntag, M.; Munz, C.D. Efficient parallelization of a shock capturing for discontinuous Galerkin methods using finite volume sub-cells. J. Sci. Comput. 2017, 70, 1262–1289. [Google Scholar] [CrossRef]
- Krais, N.; Beck, A.; Bolemann, T. FLEXI: A high order discontinuous Galerkin framework for hyperbolic-parabolic conservation laws. Comput. Math. Appl. 2021, 81, 186–219. [Google Scholar] [CrossRef]
- Zhu, H.; Liu, H.; Yan, Z. Shock capturing schemes based on nonuniform nonlinear weighted interpolation for conservation laws and their application as subcell limiters for FR/CPR. arXiv 2021, arXiv:2107.06471. [Google Scholar]
- Hennemann, S.; Rueda-Ramírez, A.M.; Hindenlang, F.J.; Gassner, G.J. A provably entropy stable subcell shock capturing approach for high order split form DG for the compressible Euler equations. J. Comput. Phys. 2021, 426, 109935. [Google Scholar] [CrossRef]
- Gassner, G.J.; Beck, A.D. On the accuracy of high-order discretizations for underresolved turbulence simulations. Theor. Comput. Fluid Dyn. 2013, 27, 221–237. [Google Scholar] [CrossRef]
- Winters, A.R.; Moura, R.C.; Mengaldo, G. A comparative study on polynomial dealiasing and split form discontinuous Galerkin schemes for under-resolved turbulence computations. J. Comput. Phys. 2018, 372, 1–21. [Google Scholar] [CrossRef]
- Renac, F. Entropy stable DGSEM for nonlinear hyperbolic systems in nonconservative form with application to two-phase flows. arXiv 2018, arXiv:1806.10880. [Google Scholar] [CrossRef] [Green Version]
- Zhu, H.; Liu, H.; Yan, Z.; Shi, G.; Deng, X. A priori subcell limiting based on compact nonuniform nonlinear weighted schemes of high-order cpr method for hyperbolic conservation laws. Comput. Fluids Accept 2022, 241, 105456. [Google Scholar] [CrossRef]
- Zhu, H.; Yan, Z.; Liu, H.; Mao, M.; Deng, X. High-order hybrid WCNS-CPR schemes on hybrid meshes with curved edges for conservation law I: Spatial accuracy and geometric conservation laws. Commun. Comput. Phys. 2018, 23, 1355–1392. [Google Scholar] [CrossRef]
- Shi, G.; Zhu, H.; Yan, Z.G. A priori subcell limiting approach for the FR/CPR method on unstructured meshes. Commun. Comput. Phys. 2022, 31, 1215–1241. [Google Scholar] [CrossRef]
- Hu, C.; Shu, C.W. Weighted essentially non-oscillatory schemes on triangular meshes. J. Comput. Phys. 1999, 150, 97–127. [Google Scholar] [CrossRef] [Green Version]
- Rueda-Ramírez, A.M.; Gassner, G.J. A subcell finite volume positivity-preserving limiter for dgsem discretizations of the euler equations. arXiv 2021, arXiv:2102.06017. [Google Scholar]
- Shu, C.W.; Osher, S. Efficient implementation of essentially non-oscillatory shock-capturing schemes—II. J. Comput. Phys. 1989, 83, 32–78. [Google Scholar] [CrossRef]
- Kitamura, K.; Roe, P.L. An Evaluation of Euler Fluxes for Hypersonic Flow Computations. In Proceedings of the AIAA Computational Fluid Dynamics Conference, Miami, FL, USA, 25–28 June 2007. [Google Scholar]
- Moura, R.C.; Mengaldo, G.; Peiro, J.; Sherwin, S.J. On the eddy-resolving capability of high-order discontinuous Galerkin approaches to implicit LES/under-resolved DNS of Euler turbulence. J. Comput. Phys. 2017, 330, 615–623. [Google Scholar] [CrossRef] [Green Version]
- Hughes, T.J.R.; Franca, L.P.; Mallet, M. A new finite element formulation for computational fluid dynamics: I. Symmetric forms of the compressible Euler and Navier-Stokes equations and the second law of thermodynamics. Comput. Methods Appl. Mech. Eng. 1986, 54, 223–234. [Google Scholar] [CrossRef]
- Woodward, P.; Colella, P. The numerical simulation of two-dimensional fluid flow with strong shocks. J. Comput. Phys. 1984, 54, 115–173. [Google Scholar] [CrossRef]
- Park, J.S.; Yoon, S.H.; Kim, C. Multi-dimensional limiting process for hyperbolic conservation laws on unstructured grids. J. Comput. Phys. 2010, 229, 788–812. [Google Scholar] [CrossRef]
- Ismail, F.; Roe, P.L. Affordable, entropy-consistent Euler flux functions II: Entropy production at shocks. J. Comput. Phys. 2009, 228, 5410–5436. [Google Scholar] [CrossRef]
Error | Order | Error | Order | Error | Order | |
---|---|---|---|---|---|---|
10 | 10 | 10 | ||||
10 | 2.99 | 10 | 2.98 | 10 | 2.92 | |
10 | 3.02 | 10 | 3.00 | 10 | 2.98 | |
10 | 3.01 | 10 | 3.00 | 10 | 3.00 |
Error | Order | Error | Order | Error | Order | |
---|---|---|---|---|---|---|
10 | 10 | 10 | ||||
10 | 2.99 | 10 | 2.98 | 10 | 2.92 | |
10 | 3.02 | 10 | 3.00 | 10 | 2.98 | |
10 | 3.01 | 10 | 3.00 | 10 | 3.00 |
Error | Order | Error | Order | Error | Order | |
---|---|---|---|---|---|---|
10 | 10 | 10 | ||||
10 | 3.02 | 10 | 3.02 | 10 | 2.98 | |
10 | 3.01 | 10 | 3.00 | 10 | 2.99 | |
10 | 3.00 | 10 | 3.00 | 10 | 3.00 |
Case | (0, 0) | (0, 1) | (1, 0) | (1, 1) |
---|---|---|---|---|
Number | 0 | 1 | 2 | 3 |
Computational Convergence | Position of the Shock Wave Is Not Shifted |
---|---|
Yes: 1 | Yes: 1 |
No: 0 | No: 0 |
Schemes | 0.0 | 0.2 | 0.4 | 0.6 | 0.8 | 1.0 |
---|---|---|---|---|---|---|
ESDGSEM-LGL | 3 | 3 | 3 | 3 | 3 | 3 |
ESDGSEM-LG | 1 | 1 | 1 | 1 | 1 | 1 |
ESDGSEM-LG-SL | 3 | 3 | 3 | 3 | 3 | 3 |
Schemes | 1.0 | 2.0 | 4.0 | 6.0 | 8.0 | 10.0 |
---|---|---|---|---|---|---|
ESDGSEM-LGL | 3 | 3 | 3 | 3 | 0 | 0 |
ESDGSEM-LG | 3 | 1 | 1 | 1 | 1 | 1 |
ESDGSEM-LG-SL | 3 | 3 | 3 | 3 | 3 | 3 |
Schemes | LLF | HLL |
---|---|---|
DGSEM-LGL | 23.6 | 150 |
ESDGSEM-LGL | 150 | 150 |
ESDGSEM-LG | 150 | 150 |
ESDGSEM-LG-SL | 150 | 150 |
Schemes | LLF | HLL |
---|---|---|
DGSEM-LGL | 3.16 | 4.72 |
ESDGSEM-LGL | 6.26 | 5.02 |
ESDGSEM-LG | 25.0 | 25.0 |
ESDGSEM-LG-SL | 25.0 | 25.0 |
Disclaimer/Publisher’s Note: The statements, opinions and data contained in all publications are solely those of the individual author(s) and contributor(s) and not of MDPI and/or the editor(s). MDPI and/or the editor(s) disclaim responsibility for any injury to people or property resulting from any ideas, methods, instructions or products referred to in the content. |
© 2023 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https://creativecommons.org/licenses/by/4.0/).
Share and Cite
Liu, Y.; Zhu, H.; Yan, Z.-G.; Jia, F.; Feng, X. Entropy Stable DGSEM Schemes of Gauss Points Based on Subcell Limiting. Entropy 2023, 25, 911. https://doi.org/10.3390/e25060911
Liu Y, Zhu H, Yan Z-G, Jia F, Feng X. Entropy Stable DGSEM Schemes of Gauss Points Based on Subcell Limiting. Entropy. 2023; 25(6):911. https://doi.org/10.3390/e25060911
Chicago/Turabian StyleLiu, Yang, Huajun Zhu, Zhen-Guo Yan, Feiran Jia, and Xinlong Feng. 2023. "Entropy Stable DGSEM Schemes of Gauss Points Based on Subcell Limiting" Entropy 25, no. 6: 911. https://doi.org/10.3390/e25060911