Projection-based reduced-order modeling has been used successfully to develop efficient models of fluid flows. However, a majority of the applications are limited to small perturbations about a nominal flow condition and do not typically address strong nonlinearities. In the present work, we assess the viability of reduced-order modeling to the problem of unstart in high-speed engine inlets. A complicating factor in this application is the presence of strong shock waves. Models based on a linearized flow assumption fail to capture the large shock motions associated with unstart. Projection-based, Proper Orthogonal Decomposition (POD) models can -in theory -account for such nonlinearities, but at a very high cost. Advances have been made recently in developing techniques to further accelerate projection-based models. One such method, the Discrete Empirical Interpolation Method (DEIM), is applied in this study to two nozzle flow cases: a fully-expanded nozzle and a nozzle with a normal shock present. In both examples, the model is based on a quasi-one-dimensional inviscid flow assumption. This study highlights the lack of robustness of the DEIM and proposes an alternative acceleration method based on L2-norm minimization. For both test cases, instances are found where DEIM is unstable but L2-norm minimization is not, motivating further work in nonlinear acceleration techniques employing optimization, including L2-norm minimization and compressed sensing.