BUTCHER-KUNTZMANN METHODS FOR NONSTIFF PROBLEMS ON PARALLEL COMPUTERS

From a theoretical point of view, the Butcher-Kuntzmann Runge-Kutta me thods belong to the best step-by-step methods for nonstiff problems. T hese methods integrate first-order initial-value problems by means of formulas based on Gauss-Legendre quadrature, and combine excellent sta bility features with the property of superconvergence at the step poin ts. Like the IVP itself, they only need the given initial value withou t requiring additional starting values, and therefore are a natural di scretization of the initial value problem. On the other hand, from a p ractical point of view, these methods have the drawback of requiring i n each step an approximation to the solution of a system of equations of dimension sd, s and d being the number of stages and the dimension of the initial-value problem, respectively. However, parallel computer s have changed the scene and enable us to design parallel iteration me thods for approximating the solution of the implicit systems such that the Butcher-Kuntzmann methods became efficient step-by-step methods f or integrating initial-value problems. In this contribution, we addres s nonstiff initial-value problems and we investigate the possibility o f introducing preconditioners into the iteration method. In particular , the iteration error will be analysed. By a number of numerical exper iments it will be shown that the Butcher-Kuntzmann method, in combinat ion with the preconditioned, parallel iteration scheme, performs much more efficiently than the best sequential methods.