Note on explicit parallel multistep Runge—Kutta methods

AbstractThis paper investigates a family of explicit two-step, two-stage Runge—Kutta methods in which the two right-hand side evaluations can be computed in parallel, so that effectively only one right-hand side evaluation per step is required. This family is compared with the family of explicit linear two-step methods of Adams type and examples of methods with increased stability intervals and methods with increased order of accuracy are given. These methods are applied to test problems taken from the test sets of Hull et al. and Enright et al., and compared with conventional linear multistep methods. In addition to the family of two-step, two-stage Runge—Kutta methods, we describe a rather general class of k-step, m-stage Runge—Kutta methods in a which the m right-hand side evaluations can also be computed in parallel. For this class we indicate how the order equations and stability region can be derived