Nested Second Derivative Two-Step Runge–Kutta Methods

Two-step Runge–Kutta (TSRK) methods are Runge–Kutta methods that depend on stage values at two consecutive steps. Second derivative Two-step Runge–Kutta (SD-TSRK) methods are extension of TSRK methods in which second derivatives as well as first derivatives are computed. General linear methods (GLMs) were introduced as a generalization of Runge–Kutta methods and linear multistep methods, and have also been extended to second derivative general linear methods (SD-GLMs). This paper presents SD-TSRK methods that are nested in their stages and mono-implicit in their output as SD-GLMs; these methods are referred to as nested second derivative two-step Runge–Kutta methods. L-stable members have been developed for the numerical integration of ordinary differential equations and how possible instances of order reduction can be avoided along with other theoretical order analysis are also considered. © 2021, The Author(s), under exclusive licence to Springer Nature India Private Limited.