Preconditioning in parallel Runge-Kutta methods for stiff initial value problems

Abstract

From a theoretical point of view, Runge-Kutta methods of collocation type belong to the most attractive step-by-step methods for integrating stiff problems. These methods combine excellent stability features with the property of superconvergence at the step points. Like the initial-value problem itself, they only need the given initial value without requiring additional starting values, and therefore, are a natural discretization 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-value problem, respectively. In contrast, linear multistep methods, the main 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 the implicit systems in such a way that the resulting methods become efficient step-by-step methods for integrating stiff initial-value problems.