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

TL;DR

The paper develops a holistic finite difference framework for the Kuramoto–Sivashinsky equation by partitioning the domain into elements with insulating boundaries and applying centre manifold theory to ensure the coarse-grid discretisation accurately captures the full nonlinear dynamics. By scaling to grid-wide variables and coupling elements through a parameter, the authors construct a centre manifold that yields evolution equations for element amplitudes, providing a principled justification for the finite-difference scheme. Numerical comparisons using computer algebra demonstrate that incorporating higher-order nonlinear corrections markedly improves accuracy on relatively coarse grids compared to conventional discretisations. This approach offers a systematic pathway to precise, low-dimensional models of stiff, fourth-order nonlinear spatio-temporal PDEs with potential applicability beyond the KS equation.

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.

Figures (1)

• Figure 1: Contours of an accurate solution $u(x,t)$, -----, to compare with numerical approximations (\ref{['Ecmg24']}): $\cdots$, the conventional approximation, errors ${\cal O}\left(h^2\right)$; $-~-~-$, the first correction, errors ${\cal O}\left(\|u\|^3\right)$; $-\cdot-\cdot-$ , the second correction,errors ${\cal O}\left(\|u\|^4\right)$. Ku-ra-mo-to-Siva-shin-sky equation (\ref{['Epde']}) with parameter $R=2$ is discretised on just $m=8$ elements in $[0,2\pi)$ and drawn with contour interval $\Delta u=3$.