An in-depth numerical study of the two-dimensional Kuramoto–Sivashinsky equation

Kalogirou, Anna; Keaveny, Eric E.; Papageorgiou, Demetrios T.

Abstract

The Kuramoto–Sivashinsky equation in one spatial dimension (1D KSE) is one of the most well-known and well-studied partial differential equations. It exhibits spatio-temporal chaos that emerges through various bifurcations as the domain length increases. There have been several notable analytical studies aimed at understanding how this property extends to the case of two spatial dimensions. In this study, we perform an extensive numerical study of the Kuramoto–Sivashinsky equation (2D KSE) to complement this analytical work. We explore in detail the statistics of chaotic solutions and classify the solutions that arise for domain sizes where the trivial solution is unstable and the long-time dynamics are completely two-dimensional. While we find that many of the features of the 1D KSE, including how the energy scales with system size, carry over to the 2D case, we also note several differences including the various paths to chaos that are not through period doubling.

Citation

Kalogirou, A., Keaveny, E. E., & Papageorgiou, D. T. (2015). An in-depth numerical study of the two-dimensional Kuramoto–Sivashinsky equation. Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences, 471(2179), Article 20140932. https://doi.org/10.1098/rspa.2014.0932

• Journal Article Type: Article
• Acceptance Date: Jun 1, 2015
• Online Publication Date: Jul 1, 2015
• Publication Date: Jul 8, 2015
• Deposit Date: Jun 4, 2019
• Journal: Proceedings of the Royal Society A: Mathematical, Physical and Engineering Science
• Print ISSN: 1364-5021
• Electronic ISSN: 1471-2946
• Publisher: The Royal Society
• Peer Reviewed: Peer Reviewed
• Volume: 471
• Issue: 2179
• Article Number: 20140932
• DOI: https://doi.org/10.1098/rspa.2014.0932
• Public URL: https://nottingham-repository.worktribe.com/output/2138180
• Publisher URL: https://royalsocietypublishing.org/doi/10.1098/rspa.2014.0932