Non-linearly stable reduced-order models for incompressible flow with energy-conserving finite volume methods

Abstract

A novel reduced-order model (ROM) formulation for incompressible flows is presented with the key property that it exhibits non-linearly 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 for the full order model 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 constrained singular value decomposition is employed which enforces global momentum conservation. The resulting 'velocity-only' ROM is thus globally conserving mass, momentum and kinetic energy. For non-homogeneous boundary conditions, a (onetime) Poisson equation is solved that accounts for the boundary contribution. The stability of the proposed ROM is demonstrated in several test cases. Furthermore, it is shown that explicit Runge-Kutta methods can be used as a practical alternative to implicit time integration at a slight loss in energy conservation.

Reduced order models for the incompressible Navier‐Stokes equations on collocated grids using a ‘discretize‐then‐project’ approach

“…projecting the fully discrete system, simplifies the treatment of the velocity boundary conditions. A recent study on ROMs on a staggered grid [44] demonstrated that the boundary conditions of the discrete FOM can be inherited by the ROM via the projection of the boundary vectors. With this approach, no additional boundary control method, such as the commonly applied penalty function [51,21,52,53] or lifting function methods [51,54,55,56], is needed to handle the BCs at the ROM level.…”

“…We employ explicit time integration methods instead of implicit ones at the FOM and the ROM level in order to ease the derivations [57]. We base our approach on the recent progression on ROMs on staggered grids [44]. First of all, we project the fully discrete system, i.e.…”

“…Several works on POD-Galerkin reduced order models have shown that the pressure gradient term disappears from the reduced set of momentum equations when the RB for the velocity field is (discretely) divergence-free. [47][48][49] However, it is (in contrast to a staggered grid) not straightforward to derive a stable 'velocity-only' ROM on a collocated grid, since the compatibility relation between divergence and gradient operators is not satisfied. 31,33 Typically, a combination of Rhie-Chow interpolation at the level of the FOM 50 and pressure stabilization on the ROM level is required to obtain stable solutions.…”

“…We base our approach on reduced order modeling approaches that operate at the discrete level 68,69 and the recent progression on ROMs on staggered grids. 49 We derive the reduced order model via projection of the fully discrete system, that is, we project the discrete FOM operators and boundary vectors onto the POD basis spaces. This 'discretize-then-project' is not the same as the 'discrete projection' approach for which a semi-discrete representation of the FOM is projected onto the POD modes in a discrete inner product.…”

“…63 Also the projection itself is not discrete. 70 By using this approach, the ROM inherits the boundary conditions of the discrete FOM via the projection of the boundary vectors, 49 which simplifies the treatment of the boundary conditions. 63 In that way, no additional boundary control method is needed to impose the BCs at the ROM level.…”