Nonlinearly stable reducedorder models for incompressible flow with energyconserving finite volume methods
Abstract
A novel reducedorder model (ROM) formulation for incompressible flows is presented with the key property that it exhibits nonlinearly 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 nonlinear 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 energyconserving (in the limit of vanishing viscosity). For this purpose, as full order model a staggeredgrid finite volume method in conjunction with an implicit RungeKutta method is employed. In addition, a constrained singular value decomposition is employed which enforces global momentum conservation. The resulting 'velocityonly' ROM is thus globally conserving mass, momentum and kinetic energy. For nonhomogeneous 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 RungeKutta methods can be used as a practical alternative to implicit time integration at a slight loss in energy conservation.
 Publication:

Journal of Computational Physics
 Pub Date:
 November 2020
 DOI:
 10.1016/j.jcp.2020.109736
 arXiv:
 arXiv:1909.11462
 Bibcode:
 2020JCoPh.42109736S
 Keywords:

 Incompressible NavierStokes equations;
 Reducedorder model;
 Energy conservation;
 PODGalerkin;
 Finite volume method;
 Stability;
 Mathematics  Numerical Analysis;
 Physics  Fluid Dynamics
 EPrint:
 Journal of Computational Physics 421, 15 November 2020, 109736