We consider a class of multistage single-step methods introduced by Zienkicwicz ef al. ' and developed by Thomas and Gladwell.' From this class, we derive methods which are particularly suitable for the numerical solution of oscillatory problems. We propose a particular pair of methods which permit local error estimation while retaining a 'one-step' mode of operation. We discuss the implementation of this pair for both linear and nonlinear problems, and illustrate the discussion with numerical results.
1 . INTRODUCTION
Thomas and Gladwell' developed a class of multistage algorithms for the linear second-order system (1) These algorithms are extensions of the methods of Zienkiewicz et d., ' and extend straightforwardly to the nonlinear system
(Here, M and R are nonsingular.) The methods of Thomas and Gladwell' are designed to provide a set of efficient methods which can be used with local error estimators based on embedding techniques. Codes implementing their algorithms are given by Gladwell and Thomas.
Here, we consider the special second-order system
such as arises commonly in nonlinear oscillation problems. We derive an efficient formula pair which is particularly suitable for such problems. The methods derived permit local error estimation using embedding techniques, while retaining a 'one-step' mode of operation. We discuss the implementation of these methods for both linear and nonlinear problems. t and formerly of Department of Mathematics, UMlST