Entropy Stable Model Reduction of Hyperbolic Systems on Nonlinear Manifolds
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Model reduction of hyperbolic systems is a difficult topic. This is a result of the fact that solutions of such systems typically do not evolve in low-dimensional linear subspaces. This elides the use of well-established linear model reduction techniques. To solve these problems we describe how we can construct novel reduced order models (ROMs) of hyperbolic systems on spaces parameterized by nonlinear functions (see e.g. Lee and Carlberg (2020)). The ROMs are constructed such that important convex entropy functionals are adequately conserved or dissipated depending on the discrete solution's regularity, a concept referred to as entropy stability. The main idea is that we evaluate the projected high-fidelity discretization for an approximation of the current state, assuring entropy stability, a step referred to as an entropy projection. Our entropy projection generalizes the one proposed recently in Chan (2020) to nonlinear model reduction spaces. Accuracy is retained by locally enriching the tangent space of the nonlinear model reduction space with additional information. Using several numerical experiments we demonstrate that our novel nonlinear and entropy-stable ROMs are suitable for model reduction of systems with strong moving discontinuities and also outperform classical linear model reduction methods.
