An adaptive, training-free reduced-order model for convection-dominated problems based on hybrid snapshots

The vast majority of reduced-order models (ROMs) first obtain a low dimensional representation of the problem from high-dimensional model (HDM) training data which is afterwards used to obtain a system of reduced complexity. Unfortunately, convection-dominated problems generally have a slowly decaying Kolmogorov n-width, which makes obtaining an accurate ROM built solely from training data very challenging. The accuracy of a ROM can be improved through enrichment with HDM solutions; however, due to the large computational expense of HDM evaluations for complex problems, they can only be used parsimoniously to obtain relevant computational savings. In this work, we exploit the local spatial and temporal coherence often exhibited by these problems to derive an accurate, cost-efficient approach that repeatedly combines HDM and ROM evaluations without a separate training phase. Our approach obtains solutions at a given time step by either fully solving the HDM or by combining partial HDM and ROM solves. A dynamic sampling procedure identifies regions that require the HDM solution for global accuracy and the reminder of the flow is reconstructed using the ROM. Moreover, solutions combining both HDM and ROM solves use spatial filtering to eliminate potential spurious oscillations that may develop. We test the proposed method on inviscid compressible flow problems and demonstrate speedups up to an order of magnitude.Comment: 29 pages, 13 figure

Calibration of projection-based reduced-order models for unsteady compressible flows

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. In order to reduce the computational costs of the LSPG models, an accelerated greedy missing point estimation (MPE) algorithm is employed for hyper-reduction. 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 are 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