Stepwise Global Error Control in an Explicit Runge-Kutta Method Using Local Extrapolation with High-Order Selective Quenching

Received: December 9, 2010 Accepted: January 6, 2011 doi:10.5539/jmr.v3n2p126 Abstract Stepwise local error control using local extrapolation in Runge-Kutta methods is well-known. In this paper, we introduce an algorithm, designated RKrvQz, 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 ! 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.