Fully-Discretely Nonlinearly-Stable Flux Reconstruction Methods for Compressible Flows
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866913487873114112 |
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| author | Pethrick, Carolyn M V Nadarajah, Siva |
| author_facet | Pethrick, Carolyn M V Nadarajah, Siva |
| contents | A fully-discrete, nonlinearly-stable flux reconstruction (FD-NSFR) scheme is developed, which ensures robustness through entropy stability in both space and time for high-order flux reconstruction schemes. We extend the entropy-stable flux reconstruction semidiscretization of Cicchino et al. [1,2,3] with the relaxation Runge Kutta method to construct the FD-NSFR scheme. We focus our study on entropy-stable flux reconstruction methods, which allow a larger time step size than discontinuous Galerkin. In this work, we develop an FD-NSFR scheme that prevents temporal numerical entropy change in the broken Sobolev norm if the governing equations admit a convex entropy function that can be expressed in inner-product form. For governing equations with a general convex numerical entropy function, temporal entropy change in the physical $L_2$ norm is prevented. As a result, for general convex numerical entropy, the FD-NSFR scheme achieves fully-discrete entropy stability only when the DG correction function is employed. We use entropy-conserving and entropy-stable test cases for the Burgers', Euler, and Navier-Stokes equations to demonstrate that the FD-NSFR scheme prevents temporal numerical entropy change. The FD-NSFR scheme therefore allows for a larger time step size while maintaining the robustness offered by entropy-stable schemes. We find that the FD-NSFR scheme is able to recover both integrated quantities and solution contours at a higher target time-step size than the semi-discretely entropy-stable scheme, suggesting a robustness advantage for low-Mach turbulence simulations. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2409_00208 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Fully-Discretely Nonlinearly-Stable Flux Reconstruction Methods for Compressible Flows Pethrick, Carolyn M V Nadarajah, Siva Numerical Analysis A fully-discrete, nonlinearly-stable flux reconstruction (FD-NSFR) scheme is developed, which ensures robustness through entropy stability in both space and time for high-order flux reconstruction schemes. We extend the entropy-stable flux reconstruction semidiscretization of Cicchino et al. [1,2,3] with the relaxation Runge Kutta method to construct the FD-NSFR scheme. We focus our study on entropy-stable flux reconstruction methods, which allow a larger time step size than discontinuous Galerkin. In this work, we develop an FD-NSFR scheme that prevents temporal numerical entropy change in the broken Sobolev norm if the governing equations admit a convex entropy function that can be expressed in inner-product form. For governing equations with a general convex numerical entropy function, temporal entropy change in the physical $L_2$ norm is prevented. As a result, for general convex numerical entropy, the FD-NSFR scheme achieves fully-discrete entropy stability only when the DG correction function is employed. We use entropy-conserving and entropy-stable test cases for the Burgers', Euler, and Navier-Stokes equations to demonstrate that the FD-NSFR scheme prevents temporal numerical entropy change. The FD-NSFR scheme therefore allows for a larger time step size while maintaining the robustness offered by entropy-stable schemes. We find that the FD-NSFR scheme is able to recover both integrated quantities and solution contours at a higher target time-step size than the semi-discretely entropy-stable scheme, suggesting a robustness advantage for low-Mach turbulence simulations. |
| title | Fully-Discretely Nonlinearly-Stable Flux Reconstruction Methods for Compressible Flows |
| topic | Numerical Analysis |
| url | https://arxiv.org/abs/2409.00208 |