Butcher–Kuntzmann methods for nonstiff problems on parallel computers

Abstract

From a theoretical point of view, the Butcher–Kuntzmann Runge–Kutta methods belong to the best step- by-step methods for nonstiff problems. These methods integrate first-order initial-value problems by means of formulas based on Gauss–Legendre quadrature, and combine excellent stability features with the property of superconvergence at the step points. Like the IVP 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 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 computers have changed the scene and enable us to design parallel iteration methods for approximating the solution of the implicit systems such that the Butcher–Kuntzmann methods become efficient step-by-step methods for integrating initial-value problems. In this contribution, we address nonstiff initial-value problems and we investigate the possibility of introducing preconditioners into the iteration method. In particular, the iteration error will be analysed. By a number of numerical experiments it will be shown that the Butcher–Kuntzmann method, in combination with the preconditioned, parallel iteration scheme, performs much more efficiently than the best sequential methods.