We propose an automated nonlinear model reduction and mesh adaptation framework for rapid and reliable solution of parameterized advection-dominated problems, with emphasis on compressible flows. The key features of our approach are threefold: (i) a metric-based mesh adaptation technique to generate an accurate mesh for a range of parameters, (ii) a general (i.e., independent of the underlying equations) registration procedure for the computation of a mapping $\Phi$ that tracks moving features of the solution field, and (iii) an hyper-reduced least-square Petrov-Galerkin reduced-order model for the rapid and reliable estimation of the mapped solution. We discuss a general paradigm -- which mimics the refinement loop considered in mesh adaptation -- to simultaneously construct the high-fidelity and the reduced-order approximations, and we discuss actionable strategies to accelerate the offline phase. We present extensive numerical investigations for a quasi-1D nozzle problem and for a two-dimensional inviscid flow past a Gaussian bump to display the many features of the methodology and to assess the performance for problems with discontinuous solutions.

翻译：我们提出了一种自动非线性降阶与网格自适应框架，用于快速可靠地求解参数化对流主导问题，重点针对可压缩流动。该方法具有三个关键特征：(i) 基于度量的网格自适应技术，为参数范围生成精确网格；(ii) 通用（即与底层方程无关的）配准过程，用于计算追踪解场运动特征的映射$\Phi$；(iii) 超缩减最小二乘Petrov-Galerkin降阶模型，用于快速可靠估计映射解。我们讨论了一种通用范式——模仿网格自适应中的细化循环——以同时构建高保真与降阶近似，并提出了加速离线阶段的可操作策略。通过准一维喷管问题和二维高斯凸起无粘流动的广泛数值研究，展示了该方法的多重特性，并评估了其针对间断解问题的性能表现。