  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.
