Numerical Algorithms

, Volume 53, Issue 2–3, pp 397–417 | Cite as

An efficient unified approach for the numerical solution of delay differential equations

  • Hossein ZivariPiranEmail author
  • Wayne H. Enright
Original Paper


In this paper we propose a new framework for designing a delay differential equation (DDE) solver which works with any supplied initial value problem (IVP) solver that is based on a standard step-by-step approach, such as Runge-Kutta or linear multi-step methods, and can provide dense output. This is done by treating a general DDE as a special example of a discontinuous IVP. Using this interpretation we develop an efficient technique to solve the resulting discontinuous IVP. We also give a more clear process for the numerical techniques used when solving the implicit equations that arise on a time step, such as when the underlying IVP solver is implicit or the delay vanishes. The new modular design for the resulting simulator we introduce, helps to accelerate the utilization of advances in the different components of an effective numerical method. Such components include the underlying discrete formula, the interpolant for dense output, the strategy for handling discontinuities and the iteration scheme for solving any implicit equations that arise.


Delay differential equations Discontinuous ordinary differential equations Runge-Kutta methods Linear multistep methods 


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Copyright information

© Springer Science+Business Media, LLC. 2009

Authors and Affiliations

  1. 1.Department of Computer ScienceUniversity of TorontoTorontoCanada

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