论文标题
从离散的外观演算的角度来解释Trisk型方案
An interpretation of TRiSK-type schemes from a discrete exterior calculus perspective
论文作者
论文摘要
储层计算是预测湍流的有力工具，其简单的架构具有处理大型系统的计算效率。然而，其实现通常需要完整的状态向量测量和系统非线性知识。我们使用非线性投影函数将系统测量扩展到高维空间，然后将其输入到储层中以获得预测。我们展示了这种储层计算网络在时空混沌系统上的应用，该系统模拟了湍流的若干特征。我们表明，使用径向基函数作为非线性投影器，即使只有部分观测并且不知道控制方程，也能稳健地捕捉复杂的系统非线性。最后，我们表明，当测量稀疏、不完整且带有噪声，甚至控制方程变得不准确时，我们的网络仍然可以产生相当准确的预测，从而为实际湍流系统的无模型预测铺平了道路。
TRiSK-type numerical schemes are widely used in both atmospheric and oceanic dynamical cores, due to their discrete analogues of important properties such as energy conservation and steady geostrophic modes. In this work, we show that these numerical methods are best understood as a discrete exterior calculus (DEC) scheme applied to a Hamiltonian formulation of the rotating shallow water equations based on split exterior calculus. This comprehensive description of the differential geometric underpinnings of TRiSK-type schemes completes the one started in \cite{Thuburn2012,Eldred2017}, and provides a new understanding of certain operators in TRiSK-type schemes as discrete wedge products and topological pairings from split exterior calculus. All known TRiSK-type schemes in the literature are shown to fit inside this general framework, by identifying the (implicit) choices made for various DEC operators by the different schemes. In doing so, unexplored choices and combinations are identified that might offer the possibility of fixing known issues with TRiSK-type schemes such as operator accuracy and Hollingsworth instability.
