Open AccessMultilinear POD-DEIM model reduction for 2D and 3D semilinear systems of differential equations
Author(s) -
Gerhard Kirsten
Publication year - 2022
Publication title -
journal of computational dynamics
Language(s) - English
Resource type - Journals
eISSN - 2158-2505
pISSN - 2158-2491
DOI - 10.3934/jcd.2021025
Subject(s) - mathematics , multilinear map , discretization , ode , tensor (intrinsic definition) , kronecker delta , partial differential equation , interpolation (computer graphics) , matrix (chemical analysis) , reduction (mathematics) , mathematical analysis , pure mathematics , computer science , geometry , animation , physics , computer graphics (images) , materials science , quantum mechanics , composite material
We are interested in the numerical solution of coupled semilinear partial differential equations (PDEs) in two and three dimensions. Under certain assumptions on the domain, we take advantage of the Kronecker structure arising in standard space discretizations of the differential operators and illustrate how the resulting system of ordinary differential equations (ODEs) can be treated directly in matrix or tensor form. Moreover, in the framework of the proper orthogonal decomposition (POD) and the discrete empirical interpolation method (DEIM) we derive a two- and three-sided model order reduction strategy that is applied directly to the ODE system in matrix and tensor form respectively. We discuss how to integrate the reduced order model and, in particular, how to solve the tensor-valued linear system arising at each timestep of a semi-implicit time discretization scheme. We illustrate the efficiency of the proposed method through a comparison to existing techniques on classical benchmark problems such as the two- and three-dimensional Burgers equation.