论文标题
动态模式分解用于构建减少双曲线问题的降级模型
Dynamic Mode Decomposition for Construction of Reduced-Order Models of Hyperbolic Problems with Shocks
论文作者
论文摘要
众所周知，用于双曲线保护法的减少阶模型（ROM）主要是由于管理方程的转化属性和非线性而挑战。尽管Lagrangian用于ROM Construction的框架可以解决转化问题，但仅在冲击形式之前才有效。一旦发生这种情况，特征线相互交叉并从高保真模型空间投影到ROM空间会扭曲移动的网格，从而导致数值不稳定性。我们通过开发基于Hodograph Transformation的物理感知动态模式分解（DMD）方法来解决这个网格失真问题。后者提供了原始非线性系统与其线性对应物之间的地图，与Koopman操作员一致。该策略与物理感知的DMD的精神一致，因为它保留了有关冲击动态的信息。提出了几个数值示例，以验证提出的物理意识的DMD方法来构建准确的ROM。
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.
