Optimally zero stable explicit peer methods with variable nodes

Abstract

In this paper, explicit peer methods are studied in which some of the stage values are copies of stage values from previous steps. This allows to reduce the number of function calls per step and can be interpreted as being a generalization of the first-same-as-last principle known from Runge–Kutta methods. The variable step size implementation is more complex as the nodes depend on the history of previous step size changes. Optimally zero stable explicit peer methods up to order \(p=8\) are constructed using constraint numerical optimization. In addition the constructed methods are superconvergent of order \(s+1\) for constant step sizes. The new methods show their efficiency in comparison with the Matlab codes ode23, ode45 and ode113 in numerical experiments.