Structure preserving reduced order modeling for gradient systems

Download

index.pdf

Date

2019-04-15

Author

Akman Yildiz, Tugba
UZUNCA, MURAT
Karasözen, Bülent

Metadata

Show full item record

This work is licensed under a Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International License.

Item Usage Stats

169
views

98
downloads

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.

Subject Keywords

Gradient systems, Pattern formation, Discontinuous Galerkin method, Average vector field method, Proper orthogonal decomposition, Discrete empirical interpolation

URI

https://hdl.handle.net/11511/30109

Journal

APPLIED MATHEMATICS AND COMPUTATION

DOI

https://doi.org/10.1016/j.amc.2018.11.008

Collections

Graduate School of Applied Mathematics, Article

Suggestions

OpenMETU
Core

Energy preserving model order reduction of the nonlinear Schrodinger equation Karasözen, Bülent (2018-12-01) An energy preserving reduced order model is developed for two dimensional nonlinear Schrodinger 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 orth...
Time-Space Adaptive Method of Time Layers for the Advective Allen-Cahn Equation UZUNCA, MURAT; Karasözen, Bülent; Sariaydin-Filibelioglu, Ayse (2015-09-18) We develop an adaptive method of time layers with a linearly implicit Rosenbrock method as time integrator and symmetric interior penalty Galerkin method for space discretization for the advective Allen-Cahn equation with nondivergence-free velocity fields. Numerical simulations for convection dominated problems demonstrate the accuracy and efficiency of the adaptive algorithm for resolving the sharp layers occurring in interface problems with small surface tension.
Energy Stable Discontinuous Galerkin Finite Element Method for the Allen-Cahn Equation Karasözen, Bülent; Sariaydin-Filibelioglu, Ayse; Yücel, Hamdullah (2018-05-01) In this paper, we investigate numerical solution of Allen-Cahn equation with constant and degenerate mobility, and with polynomial and logarithmic energy functionals. We discretize the model equation by symmetric interior penalty Galerkin (SIPG) method in space, and by average vector field (AVF) method in time. We show that the energy stable AVF method as the time integrator for gradient systems like the Allen-Cahn equation satisfies the energy decreasing property for fully discrete scheme. Numerical result...
Reduced order optimal control of the convective FitzHugh-Nagumo equations Karasözen, Bülent; KÜÇÜKSEYHAN, TUĞBA (2020-02-15) 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 consist 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 ...
Structure preserving model order reduction of shallow water equations Karasözen, Bülent; UZUNCA, MURAT (2020-07-01) In this paper, we present two different approaches for constructing reduced-order models (ROMs) for the two-dimensional shallow water equation (SWE). The first one is based on the noncanonical Hamiltonian/Poisson form of the SWE. After integration in time by the fully implicit average vector field method, ROMs are constructed with proper orthogonal decomposition(POD)/discrete empirical interpolation method that preserves the Hamiltonian structure. In the second approach, the SWE as a partial differential eq...

Citation Formats

T. Akman Yildiz, M. UZUNCA, and B. Karasözen, “Structure preserving reduced order modeling for gradient systems,” APPLIED MATHEMATICS AND COMPUTATION, pp. 194–209, 2019, Accessed: 00, 2020. [Online]. Available: https://hdl.handle.net/11511/30109.