Non-linearly stable reduced-order models for incompressible flow with energy-conserving finite volume methods
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A novel reduced order model (ROM) formulation for incompressible flows is presented with the key property that it possesses non-linear stability, independent of the mesh (of the full order model), the time step, the viscosity, and the number of modes. The two essential elements to non-linear stability are: (1) first discretise the full order model, and then project the discretised equations, and (2) use spatial and temporal discretisation schemes that are globally energy-conserving (in the limit of vanishing viscosity). For this purpose, as full order model a staggered-grid finite volume method in conjunction with an implicit Runge-Kutta method is employed. In addition, a new constrained singular value decomposition is proposed which enforces global momentum conservation. The resulting ROM is thus globally conserving mass, momentum and kinetic energy. For non-homogeneous boundary conditions, a (one-time) Poisson equation is solved that accounts for the boundary contribution. The stability of the proposed ROM is demonstrated in several test cases.
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