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

TL;DR

This paper introduces a new reduced-order modeling approach for incompressible flows that guarantees non-linear stability and energy conservation, regardless of mesh, time step, viscosity, or modes, using energy-conserving finite volume discretizations.

Contribution

The paper presents a novel ROM formulation that ensures non-linear stability and energy conservation for incompressible flows, independent of discretization parameters, using a specific discretization and projection strategy.

Findings

The ROM is globally conserving mass, momentum, and kinetic energy.

Stability is demonstrated across multiple test cases.

Explicit Runge-Kutta methods offer a practical alternative with slight energy conservation loss.

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 (one-time) 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.