Implementing Radau IIA methods for stiff delay differential equations. This article discusses the numerical solution of a general class of delay differential equations, including stiff problems, differential-algebraic delay equations, and neutral problems. The delays can be state dependent, and they are allowed to become small and vanish during the integration. Difficulties encounted in the implementation of implicit Runge-Kutta methods are explained, and it is shown how they can be overcome. The performance of the resulting code -- RADAR5 -- is illustrated on several examples, and it is compared to existing programs.

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  1. Linh, Vu Hoang; Truong, Nguyen Duy: On convergence of continuous half-explicit Runge-Kutta methods for a class of delay differential-algebraic equations (2020)
  2. Randall, E. Benjamin; Randolph, Nicholas Z.; Olufsen, Mette S.: Persistent instability in a nonhomogeneous delay differential equation system of the Valsalva maneuver (2020)
  3. Xu, Xiuxiu; Huang, Qiumei: Superconvergence of discontinuous Galerkin methods for nonlinear delay differential equations with vanishing delay (2019)
  4. Sieber, Jan: Local bifurcations in differential equations with state-dependent delay (2017)
  5. Khasi, M.; Ghoreishi, F.; Hadizadeh, M.: Numerical analysis of a high order method for state-dependent delay integral equations (2014)
  6. Augustin, Florian; Gilg, Albert; Paffrath, Meinhard; Rentrop, Peter; Villegas, Manuel; Wever, Utz: An accuracy comparison of polynomial chaos type methods for the propagation of uncertainties (2013)
  7. Guglielmi, Nicola; Hairer, Ernst: Regularization of neutral delay differential equations with several delays (2013)
  8. Hartung, Ferenc: On differentiability of solutions with respect to parameters in neutral differential equations with state-dependent delays (2013)
  9. Olufsen, Mette S.; Ottesen, Johnny T.: A practical approach to parameter estimation applied to model predicting heart rate regulation (2013)
  10. Khasawneh, Firas A.; Barton, David A. W.; Mann, Brian P.: Periodic solutions of nonlinear delay differential equations using spectral element method (2012)
  11. Rihan, Fathalla A.; Anwar, M. Naim: Qualitative analysis of delayed SIR epidemic model with a saturated incidence rate (2012)
  12. Boer, H. M. T.; Stötzel, C.; Röblitz, S.; Deuflhard, P.; Veerkamp, R. F.; Woelders, H.: A simple mathematical model of the bovine estrous cycle: follicle development and endocrine interactions (2011)
  13. Singh, Paramjeet; Sharma, Kapil K.: Numerical approximations to the transport equation arising in neuronal variability (2011)
  14. Barton, David A. W.; Krauskopf, Bernd; Wilson, R. Eddie: Nonlinear dynamics of torsional waves in a drill-string model with spatial extent (2010)
  15. Rihan, Fathalla A.: Computational methods for delay parabolic and time-fractional partial differential equations (2010)
  16. Ali, Ishtiaq; Brunner, Hermann; Tang, Tao: Spectral methods for pantograph-type differential and integral equations with multiple delays (2009)
  17. Bellen, Alfredo; Guglielmi, Nicola: Solving neutral delay differential equations with state-dependent delays (2009)
  18. Huang, Chengming: Delay-dependent stability of high order Runge-Kutta methods (2009)
  19. Adhikari, Mohit H.; Coutsias, Evangelos A.; McIver, John K.: Periodic solutions of a singularly perturbed delay differential equation (2008)
  20. Guglielmi, Nicola; Hairer, Ernst: Computing breaking points in implicit delay differential equations (2008)

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