Provable nonlinear stability bounds the discrete approximation and ensures that the discretization does not diverge. For high-order methods, discrete nonlinear stability and entropy stability, have been successfully implemented for discontinuous Galerkin (DG) and residual distribution schemes, where the stability proofs depend on properties of L2-norms. In this paper, nonlinearly stable flux reconstruction (NSFR) schemes are developed for three-dimensional compressible flow in curvilinear coordinates. NSFR is derived by merging the energy stable FR (ESFR) framework with entropy stable DG schemes. NSFR is demonstrated to use larger time-steps than DG due to the ESFR correction functions. 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 also prove that discrete kinetic energy stability cannot be preserved to machine precision for quadrature rules where the surface quadrature is not a subset of the volume quadrature. This paper also presents the NSFR modified mass matrix in a weight-adjusted form. This form reduces the computational cost in curvilinear coordinates through sum-fcatorization and low-storage techniques. The nonlinear stability properties of the scheme are verified on a nonsymmetric curvilinear grid for the inviscid Taylor-Green vortex problem and the correct orders of convergence were obtained for a manufactured solution. 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.

翻译：可证明的非线性稳定性限制了离散近似，并确保离散化不会发散。对于高阶方法，离散非线性稳定性和熵稳定性已成功应用于间断伽辽金（DG）和残差分布格式，其稳定性证明依赖于L2范数的性质。本文针对曲线坐标系中的三维可压缩流，发展了非线性稳定通量重构（NSFR）格式。NSFR通过融合能量稳定FR（ESFR）框架与熵稳定DG格式推导得到。研究表明，由于ESFR修正函数的存在，NSFR可采用比DG更大的时间步长。NSFR与文献中的ESFR格式不同之处在于，它通过使用修正质量矩阵将FR修正函数应用于体积项。我们还证明，对于表面求积规则不属于体积求积子集的情况，离散动能稳定性无法在机器精度内保持。本文还以加权调整形式给出了NSFR修正质量矩阵。这种形式通过求和分解和低存储技术降低了曲线坐标系中的计算成本。通过非对称曲线网格上的无黏泰勒-格林涡问题验证了格式的非线性稳定性，并对人造解获得了正确的收敛阶数。最后，我们对守恒型DG、过积分DG以及提出的熵守恒NSFR格式进行了计算成本比较，发现提出的熵守恒NSFR格式在计算效率上与守恒型DG格式具有竞争力。