PRECONDITIONING IN PARALLEL RUNGE-KUTTA METHODS FOR STIFF INITIAL-VALUE PROBLEMS

From a theoretical point of view, Runge-Kutta methods of collocation t ype belong to the most attractive step-by-step methods for integrating stiff problems. These methods combine excellent stability features wi th the property of superconvergence at the step points. Like the initi al-value problem itself, they only need the given initial value withou t requiring additional starting values, and therefore, are a natural d iscretization of the initial-value problem. On the other hand, from a practical point of view, these methods have the drawback of requiring in each step the solution of a system of equations of dimension sd, s and d being the number of stages and the dimension of the initial-valu e problem, respectively. In contrast, linear multistep methods, the ma in competitor of Runge-Kutta methods, require the solution of systems of dimension d. However, parallel computers have changed the scene and have motivated us to design parallel iteration methods for solving th e implicit systems in such a way that the resulting methods become eff icient step-by-step methods for integrating stiff initial-value proble ms.