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BIT Numerical Mathematics

, Volume 54, Issue 3, pp 673–689 | Cite as

Implicit Euler and Lie splitting discretizations of nonlinear parabolic equations with delay

  • Eskil Hansen
  • Tony Stillfjord
Article

Abstract

A convergence analysis is presented for the implicit Euler and Lie splitting schemes when applied to nonlinear parabolic equations with delay. More precisely, we consider a vector field which is the sum of an unbounded dissipative operator and a delay term, where both point delays and distributed delays fit into the framework. Such equations are frequently encountered, e.g. in population dynamics. The main theoretical result is that both schemes converge with an order (of at least) \(q=1/2\), without any artificial regularity assumptions. We discuss implementation details for the methods, and the convergence results are verified by numerical experiments demonstrating both the correct order, as well as the efficiency gain of Lie splitting as compared to the implicit Euler scheme.

Keywords

Nonlinear parabolic equations Delay differential equations  Convergence orders Implicit Euler Lie splitting 

Mathematics Subject Classification (2000)

65L03 65J08 65M15 

Notes

Acknowledgments

We would like to thank Alexandru Aleman and Erik Henningsson for helpful comments during the preparation of the manuscript, and Stefano Maset for providing us with the report [12].

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Copyright information

© Springer Science+Business Media Dordrecht 2014

Authors and Affiliations

  1. 1.Centre for Mathematical SciencesLund UniversityLundSweden

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