Accurately model the Kuramoto-Sivashinsky dynamics with holistic discretisation

MacKenzie, T. and Roberts, A. J. (2006) Accurately model the Kuramoto-Sivashinsky dynamics with holistic discretisation. SIAM Journal on Applied Dynamical Systems, 5 (3). pp. 365-402.

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We analyze the nonlinear Kuramoto-­Sivashinsky equation to develop accurate discretizations modeling its dynamics on coarse grids. The analysis is based upon center manifold theory, so we are assured that the discretization accurately models the dynamics and may be constructed systematically. The theory is applied after dividing the physical domain into small elements by introducing isolating
internal boundaries which are later removed. Comprehensive numerical solutions and simulations show that the holistic discretizations excellently reproduce the steady states and the dynamics of the Kuramoto­Sivashinsky equation. The Kuramoto­Sivashinsky equation is used as an example to show how holistic discretization may be successfully applied to fourth-order, nonlinear, spatio-temporal dynamical systems. This novel center 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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Item Type: Article (Commonwealth Reporting Category C)
Refereed: Yes
Item Status: Live Archive
Additional Information: Deposited according to Publisher's requirements.
Faculty / Department / School: Historic - Faculty of Sciences - Department of Maths and Computing
Date Deposited: 11 Oct 2007 00:37
Last Modified: 26 Sep 2016 02:56
Uncontrolled Keywords: Kuramoto-­Sivashinsky equation; low-dimensional modeling; computational discretizations; multiscale modeling
Fields of Research : 01 Mathematical Sciences > 0103 Numerical and Computational Mathematics > 010301 Numerical Analysis
01 Mathematical Sciences > 0101 Pure Mathematics > 010109 Ordinary Differential Equations, Difference Equations and Dynamical Systems
01 Mathematical Sciences > 0101 Pure Mathematics > 010104 Combinatorics and Discrete Mathematics (excl. Physical Combinatorics)
Socio-Economic Objective: E Expanding Knowledge > 97 Expanding Knowledge > 970101 Expanding Knowledge in the Mathematical Sciences
Identification Number or DOI: 10.1137/050627733

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