Abstract

Stepwise local error control using local extrapolation in Runge-Kutta methods is well-known. In this paper, we introduce an algorithm, designated RK$rv$Q$z,$ that is capable of controlling local and global errors in a stepwise manner. The algorithm utilizes three Runge-Kutta methods, of orders $r,v$ and $z$, with $r<v\ll z.$ Local error is controlled in the usual way using local extrapolation, whereas global error is controlled using a technique we have termed `quenching', which exploits the availability of a very high order solution and the use of a `safety factor', often present in local extrapolation methods. An example using RK34Q8 gives a clear indication of the effectiveness of the method.

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