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New stability results for Runge–Kutta methods adapted to delay differential equations

✍ Scribed by Guido Van den Heuvel

Publisher
Elsevier Science
Year
2000
Tongue
English
Weight
145 KB
Volume
34
Category
Article
ISSN
0168-9274

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✦ Synopsis

By using the Kreiss resolvent condition we establish upper bounds for the growth of errors in Runge-Kutta schemes for the numerical solution of delay differential equations. We consider application of such a scheme to the well-known linear test equation Z (t) = λZ(t) + µZ(tτ ), and prove that at any given point in the so-called stability region of the scheme errors grow at most linearly with the number of timesteps and with the dimension involved. Moreover, we present a sufficient condition on the Runge-Kutta method for which this kind of error growth is valid uniformly within the stability region. For the important case where the Runge-Kutta method is A-stable, we show that this condition is also necessary.

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