Holistic finite differences accurately model the dynamics of the Kuramoto-Sivashinsky equation

TL;DR

This paper develops a systematic, holistic finite difference method based on centre manifold theory to accurately model the complex dynamics of the nonlinear Kuramoto-Sivashinsky equation.

Contribution

It introduces a novel centre manifold approach that treats the entire dynamical system holistically, improving finite difference approximations for nonlinear, high-order PDEs.

Findings

Finite difference model accurately captures Kuramoto-Sivashinsky dynamics

Method systematically constructs finite difference schemes for complex PDEs

Holistic approach improves modeling of nonlinear spatio-temporal systems

Abstract

We analyse the nonlinear Kuramoto-Sivashinsky equation to develop an accurate finite difference approximation to its dynamics. The analysis is based upon centre manifold theory so we are assured that the finite difference model accurately models the dynamics and may be constructed systematically. The theory is applied after dividing the physical domain into small elements by introducing insulating internal boundaries which are later removed. The Kuramoto-Sivashinsky equation is used as an example to show how holistic finite differences may be applied to fourth order, nonlinear, spatio-temporal dynamical systems. This novel centre manifold approach is holistic in the sense that it treats the dynamical equations as a whole, not just as the sum of separate terms.