Journal article
On ill-posedness of truncated series models for water waves
Proceedings of the Royal Society. A, Mathematical, physical, and engineering sciences, v 470(2166), 20130849
08 Jun 2014
Featured in Collection : UN Sustainable Development Goals @ Drexel
Abstract
The evolution of surface gravity waves on a large body of water, such as an ocean, is reasonably well approximated by the Euler system for ideal, free-surface flow under the influence of gravity. The well-posedness theory for initial-value problems for these equations, which has a long and distinguished history, reveals that solutions exist, are unique, and depend continuously upon initial data in various function-space contexts. This theory is subtle, and the design of stable, accurate, numerical schemes is likewise challenging. Depending upon the wave regime in question, there are many different approximate models that can be formally derived from the Euler equations. As the Euler system is known to be well-posed, it seems appropriate that associated approximate models should also have this property. This study is directed to this issue. Evidence is presented calling into question the well-posedness of a well-known class of model equations which are widely used in simulations. A simplified version of these models is shown explicitly to be ill-posed and numerical simulations of quadratic- and cubic-order water-wave models, initiated with initial data predicted by the explicit analysis of the simplified model, lends credence to the general contention that these models are ill-posed.
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Details
- Title
- On ill-posedness of truncated series models for water waves
- Creators
- David M. Ambrose (Corresponding Author) - Drexel University, MathematicsJerry L. Bona - University of Illinois ChicagoDavid P. Nicholls - University of Illinois Chicago
- Publication Details
- Proceedings of the Royal Society. A, Mathematical, physical, and engineering sciences, v 470(2166), 20130849
- Publisher
- The Royal Society
- Resource Type
- Journal article
- Language
- English
- Academic Unit
- Mathematics
- Web of Science ID
- WOS:000335326400011
- Scopus ID
- 2-s2.0-84899827334
- Other Identifier
- 991019168357004721
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- Collaboration types
- Domestic collaboration
- Web of Science research areas
- Mathematics