Construction of reduced-order models (ROMs) for hyperbolic conservation laws is notoriously challenging mainly due to the translational property and nonlinearity of the governing equations. While the Lagrangian framework for ROM construction resolves the translational issue, it is valid only before a shock forms. Once that occurs, characteristic lines cross each other and projection from a high-fidelity model space onto a ROM space distorts a moving grid, resulting in numerical instabilities. We address this grid distortion issue by developing a physics-aware dynamic mode decomposition (DMD) method based on hodograph transformation. The latter provides a map between the original nonlinear system and its linear counterpart, which coincides with the Koopman operator. This strategy is consistent with the spirit of physics-aware DMDs in that it retains information about shock dynamics. Several numerical examples are presented to validate the proposed physics-aware DMD approach to constructing accurate ROMs.
Structure-Informed Data-Driven Reduced-Order Modeling of Scalar Hyperbolic Conservation Laws via Kinetic Defect Measure
Oden Institute for Computational Engineering and Sciences, The University of Texas at Austin（德克萨斯大学奥斯汀分校奥登计算工程与科学研究所）；Department of Aerospace Engineering and Engineering Mechanics, The University of Texas at Austin（德克萨斯大学奥斯汀分校航空航天工程与应用力学系）
Kinetic relations for undercompressive shock waves. Physical, mathematical, and numerical issues
Laboratoire Jacques-Louis Lions & Centre National de la Recherche Scientifique, Université Pierre et Marie Curie (Paris 6)（雅克-路易·利翁斯实验室与法国国家科学研究中心，皮埃尔和玛丽居里大学（巴黎第六大学））
Beijing International Center for Mathematical Research, Peking University（北京大学北京国际数学研究中心）；Center for Quantitative Biology, Center for Machine Learning Research, Peking University（北京大学定量生物学中心、机器学习研究中心）；Academy for Multidisciplinary Studies, Capital Normal University（首都师范大学跨学科研究院）；Beijing National Center for Applied Mathematics（北京应用数学中心）
Macro-micro decomposition for consistent and conservative model order reduction of hyperbolic shallow water moment equations: A study using POD-Galerkin and dynamical low rank approximation
Haar method, averaged matrix, wave cancellations, and L1 stability for hyperbolic systems
Laboratoire Jacques-Louis Lions & Centre National de la Recherche Scientifique (CNRS), Université Pierre et Marie Curie (Paris 6)（雅克-路易·利翁斯实验室 & 法国国家科学研究中心，皮埃尔和玛丽·居里大学）
Numerical methods with controlled dissipation for small-scale dependent shocks
Laboratoire Jacques-Louis Lions and Centre National de la Recherche Scientifique Université Pierre et Marie Curie（雅克-路易·利翁斯实验室与法国国家科学研究中心皮埃尔和玛丽·居里大学）；Seminar for Applied Mathematics D-Math Eidgenössische Technische Hochschule HG（苏黎世联邦理工学院应用数学研讨会）
