Computation of empirical eigenfunctions of parabolic PDEs with non-trivial time-varying domain

In this article, a methodology to compute the empirical eigenfunctions 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 parabolic PDE with the time-varying domain to a fixed reference domain, while preserving space invariant properties of the raw solution ensemble. Subsequently, the Karhunen-Lòeve decomposition is applied to the solution ensemble with fixed spatial domain resulting in a set of optimal eigenfunctions that capture the most energy of data. Further, the low dimensional set of empirical eigenfunctions is mapped (“pushed-back”) 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. Finally, this methodology is used for the order-reduction of the Czochralski crystal growth process model which is a two dimensional parabolic PDE system on a time-varying domain with non-trivial geometry. The transformations which relate the raw data on the time-varying and time-invariant domains are designed to preserve dynamic features of the scalar physical property and comparisons among reduced and high order fidelity models are provided.