Model Reduction for Transport-Dominated Problems via Cross-Correlation Based Snapshot Registration

Abstract: Traditional linear approximation methods, such as proper orthogonal decomposition and the reduced basis method, are ill-suited for transport-dominated problems due to the slow decay of the Kolmogorov $n$-width, leading to inefficient and inaccurate reduced-order models. In this work, we propose a model reduction approach for transport-dominated problems by employing cross-correlation based snapshot registration to accelerate the Kolmogorov $n$-width decay, thereby enabling the construction of efficient and accurate reduced-order models using linear approximation methods. We propose a complete framework comprising offline-online stages for the development of reduced order models using the cross-correlation based snapshots registration. The effectiveness of the proposed approach is demonstrated using two test cases: 1D travelling waves and the higher-order methods benchmark test case, 2D isentropic convective vortex.