In the case of hyperbolic conservation laws, high-order methods, such as the classical DG method, experience the phenomenon of unwanted high-frequency oscillations in the vicinity of a shock. Shock-capturing methods such as artificial dissipation, solution, flux, or TVD limiting are generally used to eliminate non-physical oscillations and provide bounds on physical quantities. For entropy-stable schemes, the additional objective would be to retain provable entropy dissipation guarantees of the underlying scheme, i.e. subcell limiting or entropy filtering [1, 2, 3, 4]. The nonlinearly-stable flux reconstruction (NSFR) semi-discretization given in Eq. 7 with a suitable flux reconstruction scheme has been demonstrated to mitigate spurious oscillations in the presence of shock discontinuities and at CFL values substantially larger than the DG variant of the NSFR scheme whilst retaining the property of entropy stability [5]. NSFR schemes achieve this by introducing an alternative lifting operator for surface numerical flux penalization, albeit at the expense of accuracy. In this technical note, we present an adaptive approach to the choice of the lifting operator employed, which maintains higher accuracy and allows for larger CFL values while retaining the underlying provable attributes of the scheme. While it cannot eliminate oscillations such as the aforementioned shock-capturing methods, together with a positivity-preserving limiter, the scheme provides for solutions that are essentially oscillation-free.

翻译：在双曲守恒律情形下，高阶方法（如经典间断伽辽金方法）会在激波附近出现非期望的高频振荡现象。通常采用激波捕捉方法，如人工耗散、解限制、通量限制或总变差递减限制，以消除非物理振荡并提供物理量的有界性。对于熵稳定格式，额外目标在于保持底层格式可证明的熵耗散保证，例如子单元限制或熵滤波[1, 2, 3, 4]。公式7给出的非线性稳定通量重构半离散化方案，配合适当的通量重构方法，已被证明能在激波间断存在时抑制伪振荡，并在保持熵稳定性[5]的同时，允许比NSFR方案的间断伽辽金变体显著更大的CFL值。NSFR方案通过引入替代的提升算子进行表面数值通量惩罚来实现这一目标，但代价是牺牲了精度。在本技术报告中，我们提出了一种自适应选择提升算子的方法，该方法在保持更高精度、允许更大CFL值的同时，保留了底层方案的可证明属性。尽管该方法无法完全消除如前述激波捕捉方法所针对的振荡，但结合保正限制器，该方案能够提供基本无振荡的解。