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Order-reduction of parabolic PDEs with time-varying domain using empirical eigenfunctions



A novel methodology for the order-reduction of parabolic partial differential equation (PDE) systems with time-varying domain is explored. In this method, a mapping functional is obtained, which relates the time-evolution of the solution of a parabolic PDE with time-varying domain to a fixed reference domain, while preserving space invariant properties of the initial solution ensemble. Subsequently, the Karhunen–Loève decomposition is applied to the solution ensemble on fixed spatial domain resulting in a set of optimal eigenfunctions. Further, the low dimensional set of empirical eigenfunctions is mapped on the original time-varying domain by an appropriate mapping, resulting in the basis for the construction of the reduced-order model of the parabolic PDE system with time-varying domain. This methodology is used in three representative cases, one- and two-dimensional (1-D and 2-D) models of nonlinear reaction-diffusion systems with analytically defined domain evolutions, and the 2-D model of the Czochralski crystal growth process with nontrivial geometry. © 2013 American Institute of Chemical Engineers AIChE J, 59: 4142–4150, 2013