On explicit two-derivative two-step Runge–Kutta methods

Abstract

We introduce a class of methods for the numerical solution of ordinary differential equations. These methods called as two-derivative two-step Runge–Kutta methods are extension of the two-step Runge–Kutta methods in which the second derivative of the solution is included. These methods are a special class of second-derivative general linear methods studied by many authors Butcher et al. (Numer Algorithms 40:415–429, 2005), Abdi et al. (Numer Algorithms 57:149–167, 2011), Okuonghae and Ikhile (Numer Algorithms 67(3):637–654, 2014). The order conditions are derived based on the algebraic theory of Butcher (Mathe Comput 26:79–106, 1972) and the $$\mathcal {B}-$$ series theory Hairer and Wanner (Computing 13:1–15, 1974), in a similar way to Chan and Chan (2006). In this study, special explicit two-derivative two-step Runge–Kutta methods that possess one evaluation of the first derivative and many evaluations of the second derivative per step are introduced. Methods with stages up to five and of order up to eight are presented. The numerical calculations have been performed on some non-stiff and mildly stiff problems and comparisons have been made with the accessible methods in the literature.