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

2000/01/14 by Tony MacKenzie, MacKenzie, T., A. J. Roberts +1 · 1 citation
Biochemistry, Genetics and Molecular Biology · Computer Science · Engineering · #Evolution and Genetic Dynamics #FOS: Mathematics #Nonlinear Dynamics and Pattern Formation #Numerical Analysis (math.NA) #Slime Mold and Myxomycetes Research

paper · pdf · doi:10.48550/arxiv.math/0001079

openalex publication_date 2000/01/14 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

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.

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• Fully discrete finite element methods for the stochastic Kuramoto-Sivashinsky equation with multiplicative noise

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