Nonlinear Stability Meets High-Order Flux Reconstruction

Ensuring provable nonlinear stability provides bounds on the discrete solution and guarantees that the numerical scheme remains convergent. In the context of compressible flows, nonlinear stability is achieved by enforcing a secondary conservation principle—the second law of thermodynamics. For high-order numerical methods, such as discontinuous Galerkin (DG), discrete nonlinear and entropy stability have been effectively realized. These stability guarantees are typically derived using properties of the L2-norm. We have developed a nonlinearly stable flux reconstruction (NSFR) scheme for three-dimensional compressible flow in curvilinear coordinates. NSFR is derived by merging the energy stable flux reconstruction (ESFR) framework with entropy stable DG schemes. NSFR is demonstrated to continue to benefit from the use of larger time-steps than DG due to the ESFR correction functions while preserving discrete nonlinear stability. NSFR differs from ESFR schemes in the literature since it incorporates the FR correction functions on the volume terms through the use of a modified mass matrix. We employ a modified mass matrix in a weight-adjusted form that reduces the computational cost in curvilinear coordinates through a precomputed projection operator that approximates the dense matrix inversion and the inverse of a diagonal matrix on-the-fly and exploits the tensor product basis functions to utilize sum factorization. A novel, fully-discrete, nonlinearly stable flux reconstruction (FD-NSFR) framework is introduced, which guarantees entropy stability in both spatial and temporal domains for high-order methods via the relaxation Runge-Kutta scheme. We developed an FD-NSFR scheme that prevents a 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. Through the use of a bound-preserving limiter, positivity of thermodynamic quantities is preserved and enables the extension of this scheme to hyperbolic conservation laws. Lastly, we perform a computational cost comparison between conservative DG, overintegrated DG, and our proposed entropy-conserving NSFR scheme and find that our proposed entropy-conserving NSFR scheme is computationally competitive with the conservative DG scheme.