Structure preserving reduced order modeling for gradient systems
For researchers simulating pattern formation in gradient systems, this work provides a structure-preserving ROM that ensures long-term stability, though it is an incremental extension of existing POD-DEIM methods.
This paper develops a reduced order model (ROM) that preserves the gradient dissipative structure for reaction-diffusion systems, enabling accurate long-time simulation of pattern formation. Numerical results on the Ginzburg-Landau and Swift-Hohenberg equations show that the POD-DEIM ROM preserves energy dissipation and steady-state solutions.
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.