12,100 research outputs found
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
Geometry of the ergodic quotient reveals coherent structures in flows
Dynamical systems that exhibit diverse behaviors can rarely be completely
understood using a single approach. However, by identifying coherent structures
in their state spaces, i.e., regions of uniform and simpler behavior, we could
hope to study each of the structures separately and then form the understanding
of the system as a whole. The method we present in this paper uses trajectory
averages of scalar functions on the state space to: (a) identify invariant sets
in the state space, (b) form coherent structures by aggregating invariant sets
that are similar across multiple spatial scales. First, we construct the
ergodic quotient, the object obtained by mapping trajectories to the space of
trajectory averages of a function basis on the state space. Second, we endow
the ergodic quotient with a metric structure that successfully captures how
similar the invariant sets are in the state space. Finally, we parametrize the
ergodic quotient using intrinsic diffusion modes on it. By segmenting the
ergodic quotient based on the diffusion modes, we extract coherent features in
the state space of the dynamical system. The algorithm is validated by
analyzing the Arnold-Beltrami-Childress flow, which was the test-bed for
alternative approaches: the Ulam's approximation of the transfer operator and
the computation of Lagrangian Coherent Structures. Furthermore, we explain how
the method extends the Poincar\'e map analysis for periodic flows. As a
demonstration, we apply the method to a periodically-driven three-dimensional
Hill's vortex flow, discovering unknown coherent structures in its state space.
In the end, we discuss differences between the ergodic quotient and
alternatives, propose a generalization to analysis of (quasi-)periodic
structures, and lay out future research directions.Comment: Submitted to Elsevier Physica D: Nonlinear Phenomen
High order and energy preserving discontinuous Galerkin methods for the Vlasov-Poisson system
We present a computational study for a family of discontinuous Galerkin
methods for the one dimensional Vlasov-Poisson system that has been recently
introduced. We introduce a slight modification of the methods to allow for
feasible computations while preserving the properties of the original methods.
We study numerically the verification of the theoretical and convergence
analysis, discussing also the conservation properties of the schemes. The
methods are validated through their application to some of the benchmarks in
the simulation of plasma physics.Comment: 44 pages, 28 figure
Effects of discrete energy and helicity conservation in numerical simulations of helical turbulence
Helicity is the scalar product between velocity and vorticity and, just like
energy, its integral is an in-viscid invariant of the three-dimensional
incompressible Navier-Stokes equations. However, space-and time-discretization
methods typically corrupt this property, leading to violation of the inviscid
conservation principles. This work investigates the discrete helicity
conservation properties of spectral and finite-differencing methods, in
relation to the form employed for the convective term. Effects due to
Runge-Kutta time-advancement schemes are also taken into consideration in the
analysis. The theoretical results are proved against inviscid numerical
simulations, while a scale-dependent analysis of energy, helicity and their
non-linear transfers is performed to further characterize the discretization
errors of the different forms in forced helical turbulence simulations
Effects of discrete energy and helicity conservation in numerical simulations of helical turbulence
Helicity is the scalar product between velocity and vorticity and, just like
energy, its integral is an in-viscid invariant of the three-dimensional
incompressible Navier-Stokes equations. However, space-and time-discretization
methods typically corrupt this property, leading to violation of the inviscid
conservation principles. This work investigates the discrete helicity
conservation properties of spectral and finite-differencing methods, in
relation to the form employed for the convective term. Effects due to
Runge-Kutta time-advancement schemes are also taken into consideration in the
analysis. The theoretical results are proved against inviscid numerical
simulations, while a scale-dependent analysis of energy, helicity and their
non-linear transfers is performed to further characterize the discretization
errors of the different forms in forced helical turbulence simulations
- …