398 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
The design of conservative finite element discretisations for the vectorial modified KdV equation
We design a consistent Galerkin scheme for the approximation of the vectorial
modified Korteweg-de Vries equation. We demonstrate that the scheme conserves
energy up to machine precision. In this sense the method is consistent with the
energy balance of the continuous system. This energy balance ensures there is
no numerical dissipation allowing for extremely accurate long time simulations
free from numerical artifacts. Various numerical experiments are shown
demonstrating the asymptotic convergence of the method with respect to the
discretisation parameters. Some simulations are also presented that correctly
capture the unusual interactions between solitons in the vectorial setting
Reduced Order Optimal Control of the Convective FitzHugh-Nagumo Equation
In this paper, we compare three model order reduction methods: the proper
orthogonal decomposition (POD), discrete empirical interpolation method (DEIM)
and dynamic mode decomposition (DMD) for the optimal control of the convective
FitzHugh-Nagumo (FHN) equations. The convective FHN equations consists of the
semi-linear activator and the linear inhibitor equations, modeling blood
coagulation in moving excitable media. The semilinear activator equation leads
to a non-convex optimal control problem (OCP). The most commonly used method in
reduced optimal control is POD. We use DEIM and DMD to approximate efficiently
the nonlinear terms in reduced order models. We compare the accuracy and
computational times of three reduced-order optimal control solutions with the
full order discontinuous Galerkin finite element solution of the convection
dominated FHN equations with terminal controls. Numerical results show that POD
is the most accurate whereas POD-DMD is the fastest
Metriplectic Integrators for the Landau Collision Operator
We present a novel framework for addressing the nonlinear Landau collision
integral in terms of finite element and other subspace projection methods. We
employ the underlying metriplectic structure of the Landau collision integral
and, using a Galerkin discretization for the velocity space, we transform the
infinite-dimensional system into a finite-dimensional, time-continuous
metriplectic system. Temporal discretization is accomplished using the concept
of discrete gradients. The conservation of energy, momentum, and particle
densities, as well as the production of entropy is demonstrated algebraically
for the fully discrete system. Due to the generality of our approach, the
conservation properties and the monotonic behavior of entropy are guaranteed
for finite element discretizations in general, independently of the mesh
configuration.Comment: 24 pages. Comments welcom
- …