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

Abstract

In this work, we explore a methodology to compute the empirical eigenfunctions for the order-reduction of nonlinear parabolic partial differential equations (PDEs) system with time-varying domain. The idea behind this method is to obtain the mapping functional, which relates the time-evolution scalar physical property solution ensemble of the nonlinear 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-Lo'eve 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 time-varying domain by an appropriate mapping resulting in the basis for the construction of the reduced-order model of the parabolic PDEs with time-varying domain. Finally, this methodology is applied in the representative cases of calculation of empirical eigenfunctions in the case of one and two dimensional model of nonlinear reaction-diffusion parabolic PDE systems with analytically defined domain evolutions. In particular, the design of both mappings which relate the raw data and function spaces transformations from the time-varying to time-invariant domain are designed to preserve dynamic features of the scalar physical property and we provide comparisons among reduced and high order fidelity models.