Nonlinear Reduced Order Modelling for Korteweg-de Vries Equation

Abstract

Efficient computational techniques that maintain the accuracy and invariant preservation property of the Korteweg-de Vries (KdV) equations have been studied by a wide range of researchers. In this paper, we introduce a reduced order model technique utilizing kernel principle component analysis (KPCA), a nonlinear version of the classical principle component analysis, in a non-intrusive way. The KPCA is applied to the data matrix, which is formed by the discrete solution vectors of KdV equation. In order to obtain the discrete solutions, the finite differences are used for spatial discretization, and linearly implicit Kahan's method for the temporal one. The back-mapping from the reduced dimensional space, is handled by a non-iterative formula based on the idea of multidimensional scaling (MDS) method. Through KPCA, we illustrate that the reduced order approximations conserve the invariants, i.e., Hamiltonian, momentum and mass structure of the KdV equation. The accuracy of reduced solutions, conservation of invariants, and computational speed enhancements facilitated by classical (linear) PCA and KPCA are exemplified through one-dimensional KdV equation.