An analysis of calibration for reduced-order models (ROMs) is presented in this work. The Galerkin and least-squares Petrov-Galerkin (LSPG) methods are tested on compressible flows involving a disparity of temporal scales. A novel calibration strategy is proposed for the LSPG method and two test cases are analyzed. The first consists of a subsonic airfoil flow where boundary layer instabilities are responsible for trailing-edge noise generation and the second comprises a supersonic airfoil flow with a transient period where a detached shock wave propagates upstream at the same time that shock-vortex interaction occurs at the trailing edge. Results show that calibration produces stable and long-time accurate Galerkin and LSPG ROMs for both cases investigated. The impact of hyper-reduction is tested on LSPG models via an accelerated greedy missing point estimation (MPE) algorithm. For the first case investigated, LSPG solutions obtained with hyper-reduction show good comparison with those obtained by the full order model. However, for the supersonic case the transient features of the flow need to be properly captured by the sampled points of the accelerated greedy MPE method. Otherwise, the dynamics of the moving shock wave are not fully recovered. The impact of different time-marching schemes is also assessed and, differently than reported in literature, Galerkin models are shown to be more accurate than those computed by LSPG when the non-conservative form of the Navier-Stokes equations is solved. For the supersonic case, the Galerkin and LSPG models (without hyper-reduction) capture the overall dynamics of the detached and oblique shock waves along the airfoil. However, when shock-vortex interaction occurs at the trailing-edge, the Galerkin ROM is able to capture the high-frequency fluctuations from vortex shedding while the LSPG presents a more dissipative solution, not being able to recover the flow dynamics.

本文介绍了降阶模型（ROM）的校准分析。Galerkin和最小二乘Petrov-Galerkin（LSPG）方法在涉及时间尺度差异的可压缩流上进行了测试。针对LSPG方法提出了一种新颖的校准策略，并分析了两个测试案例。第一个由亚音速翼型流组成，其中边界层的不稳定性导致后缘噪声的产生，第二个由具有过渡周期的超音速​​翼型流组成，其中分离的冲击波向上游传播，同时在此处发生震荡涡旋相互作用。后缘。结果表明，对于所研究的两种情况，校准都能产生稳定且长期准确的Galerkin和LSPG ROM。通过加速贪婪缺失点估计（MPE）算法，对LSPG模型测试了超还原的影响。对于第一个研究的案例，通过超还原获得的LSPG解与通过全阶模型获得的LSPG解具有很好的比较。但是，对于超音速情况，需要通过加速贪婪MPE方法的采样点正确捕获流的瞬态特征。否则，运动冲击波的动力学不能完全恢复。还评估了不同时间行进方案的影响，并且与文献报道不同，当解决了Navier-Stokes方程的非保守形式时，Galerkin模型比LSPG计算的模型更准确。对于超音速情况，Galerkin和LSPG模型（不进行超缩减）捕获了沿机翼分离和倾斜冲击波的整体动力学。但是，当在后缘发生冲击涡旋相互作用时，Galerkin ROM能够捕获由于涡旋脱落而产生的高频波动，而LSPG提供了更耗散的解决方案，无法恢复流动动力学。