Stabilization Techniques for High-Order Flux Reconstruction Schemes

  • Cicchino, Alexander (McGill University)
  • Nadarajah, Siva (McGill University)

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Entropy stable and nonlinearly stable high-order methods have gained popularity in the research field as they guarantee robustness on extremely coarse meshes. Following from the entropy conservative, two-point flux formulation by Tadmor (Tadmor 1987) for finite volume schemes, the approach has been extended using the Summation-by-Parts property for general, modal, high-order methods such as discontinuous Galerkin (Chan 2018) and Flux Reconstruction (Cicchino et al. 2022). Unfortunately, as proved by Gassner and coauthors (Gassner et al. 2022), the nonlinearly stable schemes do not satisfy a local energy stability condition, and thus result in the wrong solution. Gassner and coauthors (Gassner et al. 2022) demonstrated that the eigenvalues of the residual's Jacobian must have real parts less than or equal to 0 to satisfy the local energy stability criteria. In this presentation, for both Burgers' and Euler's equations, we present a slight modification to the entropy conservative two-point flux that provably ensures both energy stability, entropy stability, and linear stability--that the eigenvalues of the residual's Jacobian have negative semi-definite real parts. Therefore, we will present the numerical scheme that provably solves the issues raised by Gassner and coauthors (Gassner et al. 2022). Our proposed method is verified by numerically showing the eigenvalues having negative semi-definite real parts and that the scheme can indefinitely run long time integration for both the Burgers' test case and the Euler density wave presented by Gassner and coauthors (Gassner et al. 2022).