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Spatial and Temporal Adaptivity in Numerical Studies of Instabilities, with Applications to Fluid Flows

Chapter
Part of the Computational Methods in Applied Sciences book series (COMPUTMETHODS, volume 50)

Abstract

In this article we discuss how to formulate numerical methods for calculating branches of steady solutions, periodic orbits and bifurcations of partial differential equations that are adaptive in both space and time. The methods are implemented within the open-source software framework oomph-lib and examples of their use in current research problems of fluid flow past cylinders and free surface flows on rotating cylinders are presented.

Notes

Acknowledgements

This work was initiated in 2012 while the author was a visiting scientist at Sandia National Laboratories, hosted by A.G. Salinger. The ideas presented within have been refined over numerous discussions with many colleagues at workshops and conferences and many of the computations have been conducted by Ph.D. students. Here, I would particularly like to thank A. von Borries Lopes who worked on the rotating cylinder problem and P. Matharu who is working on the flow past a cylinder. Oomph-lib would not exist without the tireless work of my colleague M. Heil and much of the bifurcation tracking and detection capabilities were developed from extensive discussions with R. Hewitt, T. Mullin and K.A. Cliffe. U. Thiele contributed to the work on partially wetting fluids in Sect. 5.1. Finally, I would also like to thank L. van Veen and S. Altmeyer for their comments which have helped to improve an earlier draft of this chapter.

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© Springer International Publishing AG, part of Springer Nature 2019

Authors and Affiliations

  1. 1.School of MathematicsThe University of ManchesterManchesterUK

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