Energy preserving model order reduction of the nonlinear Schr\"odinger equation

An energy preserving reduced order model is developed for two dimensional nonlinear Schr\"odinger equation (NLSE) with plane wave solutions and with an external potential. The NLSE is discretized in space by the symmetric interior penalty discontinuous Galerkin (SIPG) method. The resulting system of Hamiltonian ordinary differential equations are integrated in time by the energy preserving average vector field (AVF) method. The mass and energy preserving reduced order model (ROM) is constructed by proper orthogonal decomposition (POD) Galerkin projection. The nonlinearities are computed for the ROM efficiently by discrete empirical interpolation method (DEIM) and dynamic mode decomposition (DMD). Preservation of the semi-discrete energy and mass are shown for the full order model (FOM) and for the ROM which ensures the long term stability of the solutions. Numerical simulations illustrate the preservation of the energy and mass in the reduced order model for the two dimensional NLSE with and without the external potential. The POD-DMD makes a remarkable improvement in computational speed-up over the POD-DEIM. Both methods approximate accurately the FOM, whereas POD-DEIM is more accurate than the POD-DMD

Low-dimensional models for turbulent plane Couette flow in a minimal flow unit

We model turbulent plane Couette flow in the minimal flow unit (MFU) – a domain whose spanwise and streamwise extent is just sufficient to maintain turbulence – by expanding the velocity field as a sum of optimal modes calculated via proper orthogonal decomposition from numerical data. Ordinary differential equations are obtained by Galerkin projection of the Navier–Stokes equations onto these modes. We first consider a 6-mode (11-dimensional) model and study the effects of including losses to neglected modes. Ignoring these, the model reproduces turbulent statistics acceptably, but fails to reproduce dynamics; including them, we find a stable periodic orbit that captures the regeneration cycle dynamics and agrees well with direct numerical simulations. However, restriction to as few as six modes artificially constrains the relative magnitudes of streamwise vortices and streaks and so cannot reproduce stability of the laminar state or properly account for bifurcations to turbulence as Reynolds number increases. To address this issue, we develop a second class of models based on ‘uncoupled’ eigenfunctions that allow independence among streamwise and cross-stream velocity components. A 9-mode (31-dimensional) model produces bifurcation diagrams for steady and periodic states in qualitative agreement with numerical Navier–Stokes solutions, while preserving the regeneration cycle dynamics. Together, the models provide empirical evidence that the ‘backbone’ for MFU turbulence is a periodic orbit, and support the roll–streak–breakdown–roll reformation picture of shear-driven turbulence

A Stable Finite Difference Method for the Elastic Wave Equation on Complex Geometries with Free Surfaces

A stable and explicit second order accurate finite difference method for the elastic wave equation in curvilinear coordinates is presented. The discretization of the spatial operators in the method is shown to be self-adjoint for free-surface, Dirichlet and periodic boundary conditions. The fully discrete version of the method conserves a discrete energy to machine precision