83 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
Convergent Analysis of Energy Conservative Algorithm for the Nonlinear Schrödinger Equation
Using average vector field method in time and Fourier pseudospectral method in space, we obtain an energy-preserving scheme for the nonlinear Schrödinger equation. We prove that the proposed method conserves the discrete global energy exactly. A deduction argument is used to prove that the numerical solution is convergent to the exact solution in discrete L2 norm. Some numerical results are reported to illustrate the efficiency of the numerical scheme in preserving the energy conservation law
Structure preserving reduced order modeling for gradient systems
Minimization of energy in gradient systems leads to formation of oscillatory
and Turing patterns in reaction-diffusion systems. These patterns should be
accurately computed using fine space and time meshes over long time horizons to
reach the spatially inhomogeneous steady state. In this paper, a reduced order
model (ROM) is developed which preserves the gradient dissipative structure.
The coupled system of reaction-diffusion equations are discretized in space by
the symmetric interior penalty discontinuous Galerkin (SIPG) method. The
resulting system of ordinary differential equations (ODEs) are integrated in
time by the average vector field (AVF) method, which preserves the energy
dissipation of the gradient systems. The ROMs are constructed by the proper
orthogonal decomposition (POD) with Galerkin projection. The nonlinear reaction
terms are computed efficiently by discrete empirical interpolation method
(DEIM). Preservation of the discrete energy of the FOMs and ROMs with POD-DEIM
ensures the long term stability of the steady state solutions. Numerical
simulations are performed for the gradient dissipative systems with two
specific equations; real Ginzburg-Landau equation and Swift-Hohenberg equation.
Numerical results demonstrate that the POD-DEIM reduced order solutions
preserve well the energy dissipation over time and at the steady state
Adaptive Energy Preserving Methods for Partial Differential Equations
A method for constructing first integral preserving numerical schemes for
time-dependent partial differential equations on non-uniform grids is
presented. The method can be used with both finite difference and partition of
unity approaches, thereby also including finite element approaches. The schemes
are then extended to accommodate -, - and -adaptivity. The method is
applied to the Korteweg-de Vries equation and the Sine-Gordon equation and
results from numerical experiments are presented.Comment: 27 pages; some changes to notation and figure
Linearly implicit exponential integrators for damped Hamiltonian PDEs
Structure-preserving linearly implicit exponential integrators are
constructed for Hamiltonian partial differential equations with linear constant
damping. Linearly implicit integrators are derived by polarizing the polynomial
terms of the Hamiltonian function and portioning out the nonlinearly of
consecutive time steps. They require only a solution of one linear system at
each time step. Therefore they are computationally more advantageous than
implicit integrators. We also construct an exponential version of the
well-known one-step Kahan's method by polarizing the quadratic vector field.
These integrators are applied to one-dimensional damped Burger's,
Korteweg-de-Vries, and nonlinear Schr\"odinger equations. Preservation of the
dissipation rate of linear and quadratic conformal invariants and the
Hamiltonian is illustrated by numerical experiments
Energy Stable Interior Penalty Discontinuous Galerkin Finite Element Method for Cahn-Hilliard Equation
An energy stable conservative method is developed for the Cahn-Hilliard (CH) equation with the degenerate mobility. The CH equation is discretized in space with the mass conserving symmetric interior penalty discontinuous Galerkin (SIPG) method. The resulting semi-discrete nonlinear system of ordinary differential equations are solved in time by the unconditionally energy stable average vector field (AVF) method. We prove that the AVF method preserves the energy decreasing property of the fully discretized CH equation. Numerical results for the quartic double-well and the logarithmic potential functions with constant and degenerate mobility confirm the theoretical convergence rates, accuracy and the performance of the proposed approach
- …