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An optimally efficient technique for the solution of systems of nonlinear parabolic partial differential equations

Yang, Feng-Wei; Goodyer, Christopher E.; Hubbard, Matthew E.; Jimack, Peter K.

Authors

Feng-Wei Yang F.W.Yang@sussex.ac.uk

Christopher E. Goodyer

Matthew E. Hubbard Matthew.Hubbard@nottingham.ac.uk

Peter K. Jimack P.K.Jimack@leeds.ac.uk



Abstract

This paper describes a new software tool that has been developed for the efficient solution of systems of linear and nonlinear partial differential equations (PDEs) of parabolic type. Specifically, the software is designed to provide optimal computational performance for multiscale problems, which require highly stable, implicit, time-stepping schemes combined with a parallel implementation of adaptivity in both space and time. By combining these implicit, adaptive discretizations with an optimally efficient nonlinear multigrid solver it is possible to obtain computational solutions to a very high resolution with relatively modest computational resources. The first half of the paper describes the numerical methods that lie behind the software, along with details of their implementation, whilst the second half of the paper illustrates the flexibility and robustness of the tool by applying it to two very different example problems. These represent models of a thin film flow of a spreading viscous droplet and a multi-phase-field model of tumour growth. We conclude with a discussion of the challenges of obtaining highly scalable parallel performance for a software tool that combines both local mesh adaptivity, requiring efficient dynamic load-balancing, and a multigrid solver, requiring careful implementation of coarse grid operations and inter-grid transfer operations in parallel.

Journal Article Type Article
Publication Date Jan 2, 2017
Journal Advances in Engineering Software
Print ISSN 0965-9978
Electronic ISSN 0965-9978
Publisher Elsevier
Peer Reviewed Peer Reviewed
Volume 103
APA6 Citation Yang, F., Goodyer, C. E., Hubbard, M. E., & Jimack, P. K. (2017). An optimally efficient technique for the solution of systems of nonlinear parabolic partial differential equations. Advances in Engineering Software, 103, doi:10.1016/j.advengsoft.2016.06.003
DOI https://doi.org/10.1016/j.advengsoft.2016.06.003
Publisher URL http://www.sciencedirect.com/science/article/pii/S0965997816301259
Copyright Statement Copyright information regarding this work can be found at the following address: http://creativecommons.org/licenses/by-nc-nd/4.0

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Copyright Statement
Copyright information regarding this work can be found at the following address: http://creativecommons.org/licenses/by-nc-nd/4.0





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