The RKFHB4 method for delay — Differential equations

Oppelstrup, Jesper

doi:10.1007/bfb0067469

Cited by 27 publications

(14 citation statements)

References 8 publications

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“…In the general case fn is a complicated expression in 8t,tn' yn,yj' .•• ,yj-l" Alternative methods for solving (I.I) are the LMMs of Tavernini [JO] and the Runge-Kutta type methods proposed by Arndt [! ] and Oppelstrup [8].…”

Section: Linear Multistep Methodsmentioning

confidence: 99%

Improved Absolute Stability of Predictor-Corrector Methods for Retarded Differential Equations

Houwen¹,

Sommeijer²

1983

Differential-Difference Equations/Differential-Differenzengleichungen

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137The absolute stability of predictor-corrector type methods is investigated for retarded differential equations. The stability test equation is of the form dy(t)/dt = WJY(t) + w2y(t-w) where w1,w2 and ware constants (w>O). By generalizing the conventional predictor-corrector methods it is possible to improve the stability region in the (w16t,w26t) -plane considerably. In particular, methods based on extrapolation-predictors and backward differentiation-correctors are studied.

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Section: Linear Multistep Methodsmentioning

confidence: 99%

Improved Absolute Stability of Predictor-Corrector Methods for Retarded Differential Equations

Houwen¹,

Sommeijer²

1983

Differential-Difference Equations/Differential-Differenzengleichungen

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“…Error estimates for (2.1) have been given by many authors, e.g. Oberle and Pesch [16], Oppelstrup [17,18], Tavernini [24]. These are of the form…”

Section: Global Errormentioning

confidence: 98%

“…On the other hand, Oppelstrup [17,18] has shown that with regard to the point 2 = x 0 there is an asymptotic expansion of the form…”

Section: Z'(x) =F(x Z(x)) Z(:2) = U(s H)mentioning

confidence: 99%

Numerical solution of retarded initial value problems: Local and global error and stepsize control

Arndt

1984

Numer. Math.

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Summary.Retarded initial value problems are routinely replaced by an initial value problem of ordinary differential equations along with an appropriate interpolation scheme. Hence one can control the global error of the modified problem but not directly the actual global error of the original problem. In this paper we give an estimate for the actual global error in terms of controllable quantities. Further we show that the notion of local error as inherited from the theory of ordinary differential equations must be generalized for retarded problems. Along with the new definition we are led to developing a reliable basis for a step selection scheme.

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“…The other approach to the retarded argument is interpolation (see e.g. [11]), which we will use. Since for the numerical solution of the differential equation, the time is split into discrete time steps, labeled with index n, we have the approximated values of the solution at the discrete times t j (0 ≤ j < n), available when calculating y n = y(t n ) for n ≥ 0.…”

Section: Treatment Of the Retarded Valuementioning

confidence: 99%

QMC Methods for the solution of delay differential equations

Kainhofer

Tichy

2003

Proc Appl Math and Mech

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In this paper the Quasi-Monte Carlo methods for Runge Kutta solution techniques of differential equations, which were developed by Stengle, Lécot, Coulibaly and Koudiraty, are extended to delay differential equations of the form f (t, y(t), y(t − τ (t))). The retarded arguments are approximated by interpolation, after which the conventional (Quasi-)Monte Carlo Runge Kutta methods can be applied. We give a proof of the convergence of this method and its order in a general form, which does not depend on a specific Quasi-Monte Carlo Runge Kutta method. Finally, a numerical investigation shows that -as with ordinary differential equations -for heavily oscillating delay differential equations, this quasi-randomized method leads to an improvement compared even to high order Runge Kutta schemes.

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The RKFHB4 method for delay — Differential equations

Cited by 27 publications

References 8 publications

Improved Absolute Stability of Predictor-Corrector Methods for Retarded Differential Equations

Improved Absolute Stability of Predictor-Corrector Methods for Retarded Differential Equations

Numerical solution of retarded initial value problems: Local and global error and stepsize control

QMC Methods for the solution of delay differential equations

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