Entropy Stable Split Forms for the Flux Reconstruction High-Order Method: Numerical Validation
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Bibliographic record
Abstract
View Video Presentation: https://doi.org/10.2514/6.2021-1444.vid The flux reconstruction method has gained popularity in the research community as it recovers promising high-order methods through modally filtered correction fields, such as the Discontinuous Galerkin (DG) method, on unstructured grids over complex geometries. The attraction of the method follows with its stability proofs for the linear advection problem on linear elements, under a class of energy stable flux reconstruction (ESFR) schemes also known as Vincent-Castonguay-Jameson-Huynh (VCJH) schemes. Additionally, split forms have gained popularity in the research community due to the proofs of robustness for nonlinear, unsteady problems on unstructured grids, albeit only having been proved for the strong form DG scheme and numerically shown for the collocated strong form ESFR scheme for the Euler equations. This paper follows a new approach for implementing ESFR schemes on nonlinear split forms, which recovers the classical ESFR scheme on linear problems. The new ESFR approach is proven to not change the order of accuracy, and allows for proofs of stability for both the ESFR strong and weak forms, on unstructured and curvilinear grids. The methodology is demonstrated for Burgers' equation with numerical results demonstrating stability for all ESFR schemes.
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