Structure-Preserving ROMs for the Incompressible Navier-Stokes Equations

  • Sanderse, Benjamin (Centrum Wiskunde & Informatica, Science Park 123,)
  • Rosenberger, Henrik (Centrum Wiskunde & Informatica, Science Park 123)
  • Klein, Robin (Centrum Wiskunde & Informatica, Science Park 123)

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An issue of increasing interest in projection-based reduced order modeling of conservation laws is the preservation of the conservative structure underlying such equations at the reduced level [2,3]. A non-linearly stable POD-Galerkin reduced order model (ROM) of the incompressible Navier-Stokes equations that is pressure-free and globally conserves kinetic energy (in the inviscid limit), momentum and mass on periodic domains was constructed in [2]. However, this formulation used a rather expensive tensor decomposition of the convective terms and did not include the treatment of time-dependent boundary conditions. In this work we propose a novel hyper-reduction approach that keeps the non-linear stability and energy-conserving property of the original (tensor-decomposition) ROM by adapting the discrete empirical interpolation method (DEIM) [1]. We relax the condition of exact correspondence between the full order model (FOM) nonlinearity and the DEIM approximation in the measurement space and instead solve a minimization problem which is constrained such that the DEIM approximation mimics the energy-conservation property of the FOM convection operator. Furthermore, we decouple the DEIM dimension from the measurement space dimension. The resulting Decoupled Least-Squares DEIM (DLSDEIM) allows us to oversample and is suitable for problems with a large range of spatio-temporal features. Furthermore, in order to keep the pressure-free (`velocity-only') property of the ROM in the presence of time-dependent boundary conditions, we propose a time-dependent lifting function based on a POD approximation of the boundary conditions. The lifting function, inspired by the Helmholtz-Hodge decomposition, is orthogonal to the basis of the velocity field, and thus eliminates the pressure term from the equations. We have implemented the structure-preserving formulation DLSDEIM and time-dependent boundary conditions in the ROM framework proposed in [2]. We study several convection-dominated flow configurations with slow Kolmogorov N-width decay, for example shear-layer instabilities and the Taylor-Green vortex, and assess the computational cost and accuracy of our proposed ROM.