Solving ODEs and DDEs with residual control

https://doi.org/10.1016/j.apnum.2004.07.003Get rights and content

Abstract

We first consider the numerical integration of ordinary differential equations (ODEs) with Runge–Kutta methods that have continuous extensions. For some methods of this kind we develop robust and inexpensive estimates of both the local error and the size of the residual. We then develop an effective program, ddesd, to solve delay differential equations (DDEs) with time- and state-dependent delays. To get reliable results for these difficult problems, the code estimates and controls the size of the residual. The user interface of ddesd makes it easy to formulate and solve DDEs, even those with complications like event location and restarts.

References (16)

  • L.F. Shampine et al.

    Solving DDEs in Matlab

    Appl. Numer. Math.

    (2001)
  • S.P. Corwin et al.

    DKLAG6: A code based on continuously imbedded sixth order Runge–Kutta methods for the solution of state dependent functional differential equations

    Appl. Numer. Math.

    (1997)
  • W.H. Enright

    A new error-control for initial value solvers

    Appl. Math. Comput.

    (1989)
  • W.H. Enright et al.

    A delay differential equation solver based on a continuous Runge–Kutta method with defect control

    Numer. Algorithms

    (1997)
  • W.H. Enright et al.

    The evaluation of numerical software for delay differential equations

  • W.H. Enright et al.

    Runge–Kutta software with defect control for boundary value ODEs

    SIAM J. Sci. Comput.

    (1996)
  • I. Gladwell et al.

    Practical aspects of interpolation in Runge–Kutta codes

    SIAM J. Sci. Statist. Comput.

    (1987)
  • D.J. Higham

    Robust defect control with Runge–Kutta schemes

    SIAM J. Numer. Anal.

    (1989)
There are more references available in the full text version of this article.

Cited by (83)

View all citing articles on Scopus
View full text