Abstract
There are two main strategies for improving the projection-based reduced order model (ROM) accuracy—(i) improving the ROM, that is, adding new terms to the standard ROM; and (ii) improving the ROM basis, that is, constructing ROM bases that yield more accurate ROMs. In this paper, we use the latter. We propose two new Lagrangian inner products that we use together with Eulerian and Lagrangian data to construct two new Lagrangian ROMs, which we denote α-ROM and λ-ROM. We show that both Lagrangian ROMs are more accurate than the standard Eulerian ROMs, that is, ROMs that use standard Eulerian inner product and data to construct the ROM basis. Specifically, for the quasi-geostrophic equations, we show that the new Lagrangian ROMs are more accurate than the standard Eulerian ROMs in approximating not only Lagrangian fields (e.g., the finite time Lyapunov exponent (FTLE)), but also Eulerian fields (e.g., the streamfunction). In particular, the α-ROM can be orders of magnitude more accurate than the standard Eulerian ROMs. We emphasize that the new Lagrangian ROMs do not employ any closure modeling to model the effect of discarded modes (which is standard procedure for low-dimensional ROMs of complex nonlinear systems). Thus, the dramatic increase in the new Lagrangian ROMs’ accuracy is entirely due to the novel Lagrangian inner products used to build the Lagrangian ROM basis.
Highlights
Projection-based reduced order models (ROMs) have been successful in the numerical simulation of fluid flows [1,2,3,4,5,6]
In the new Lagrangian inner products, Lagrangian data steers the resulting Lagrangian ROM basis toward an accurate approximation of Lagrangian quantities, whereas Eulerian data helps the Lagrangian ROM basis yield an accurate approximation of Eulerian quantities
To separate the ROM closure modeling from the ROM basis generation, we investigate the two new Lagrangian ROMs without any closure model or stabilization mechanism
Summary
Projection-based reduced order models (ROMs) have been successful in the numerical simulation of fluid flows [1,2,3,4,5,6]. In the numerical simulation of the quasi-geostrophic equations [26,27,28,29] (which model large scale ocean circulation), the new Lagrangian ROMs are orders of magnitude more accurate than standard Eulerian ROMs (i.e., ROMs that use standard Eulerian data and inner products to build the ROM bases). The rest of the paper is organized as follows: In Section 2, we propose the novel Lagrangian inner products and construct the new Lagrangian ROMs. In Section 3, for the quasi-geostrophic equations, we show that the new Lagrangian ROMs increase the numerical accuracy of standard Eulerian ROMs by orders of magnitude.
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