Structure preserving reduced order modeling for gradient systems
| Autor: | Tuğba Akman Yıldız, Bülent Karasözen, Murat Uzunca |
|---|---|
| Rok vydání: | 2018 |
| Předmět: |
0209 industrial biotechnology
Steady state Discretization Applied Mathematics 65P10 65M60 020206 networking & telecommunications 02 engineering and technology Numerical Analysis (math.NA) Dissipation Computational Mathematics Nonlinear system 020901 industrial engineering & automation Discontinuous Galerkin method Ordinary differential equation 0202 electrical engineering electronic engineering information engineering Dissipative system FOS: Mathematics Applied mathematics Mathematics - Numerical Analysis Galerkin method Mathematics |
| DOI: | 10.48550/arxiv.1806.05625 |
| Popis: | 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. |
| Databáze: | OpenAIRE |
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