In this work we propose a novel method to ensure important entropy inequalities are satisfied semi-discretely when constructing reduced order models (ROMs) on nonlinear reduced manifolds. We are in particular interested in ROMs of systems of nonlinear hyperbolic conservation laws. The so-called entropy stability property endows the semi-discrete ROMs with physically admissible behaviour. The method generalizes earlier results on entropy-stable ROMs constructed on linear spaces. The ROM works by evaluating the projected system on a well-chosen approximation of the state that ensures entropy stability. To ensure accuracy of the ROM after this approximation we locally enrich the tangent space of the reduced manifold with important quantities. Using numerical experiments on some well-known equations (the inviscid Burgers equation, shallow water equations and compressible Euler equations) we show the improved structure-preserving properties of our ROM compared to standard approaches and that our approximations have minimal impact on the accuracy of the ROM. We additionally generalize the recently proposed polynomial reduced manifolds to rational polynomial manifolds and show that this leads to an increase in accuracy for our experiments.

翻译：本文提出了一种新颖方法，用于在非线性降维流形上构建降阶模型时，确保重要熵不等式在半离散层面得到满足。我们特别关注非线性双曲守恒律系统的降阶建模。所谓的熵稳定性赋予了半离散降阶模型物理上可容许的行为特性。该方法推广了先前在线性空间上构建熵稳定降阶模型的研究成果。该降阶模型通过在一个精心选择的、能保证熵稳定的状态近似上进行投影系统求值来实现。为确保近似处理后的降阶模型精度，我们在降维流形的切空间中局部增补了关键物理量。通过对若干经典方程（无粘性Burgers方程、浅水方程和可压缩Euler方程）的数值实验，我们证明了相较于标准方法，本降阶模型具有更优的结构保持特性，且所采用的近似处理对模型精度影响极小。此外，我们将近期提出的多项式降维流形推广至有理多项式流形，并通过实验验证了该推广能有效提升模型精度。