Optimal Combination of EEFs for the Model Reduction of Nonlinear Partial Differential Equations

Abstract

Proper orthogonal decomposition is a popular approach for determining the principal spatial structures from the measured data. Generally, model reduction using empirical eigenfunctions (EEFs) can generate a relatively low-dimensional model among all linear expansions. However, the neglectful modes representing only a tiny amount of energy will be crucial in the modeling for certain type of nonlinear partial differential equations (PDEs). In this paper, an optimal combination of EEFs is proposed for model reduction of nonlinear partial differential equations (PDEs), obtained by the basis function transformation from the initial EEFs. The transformation matrix is derived from straightforward optimization techniques. The present new EEFs can keep the dynamical information of neglectful modes and generate a lower-dimensional and more precise dynamical system for the PDEs. The numerical example shows its effectiveness and feasibility for model reduction of the nonlinear PDEs.