Explicit pseudo two-step RKN methods with stepsize control

Abstract

This paper is devoted to variable stepsize strategy implementations of a class of explicit pseudo two-step Runge–Kutta–Nyström methods of arbitrarily high order for solving nonstiff problems for systems of special second-order differential equations. The constant stepsize explicit pseudo two-step Runge–Kutta–Nyström methods are developed into variable stepsize ones and equipped with embedded formulas giving a cheap error estimate for stepsize control. By two examples of widely-used test problems, a pseudo two-step Runge–Kutta–Nyström method of order 8 implemented with variable stepsize strategy is shown to be much more efficient than parallel and sequential codes available in the literature. With stringent error tolerances, this new explicit pseudo two-step Runge–Kutta–Nyström method is even superior to sequential codes in a sequential computer.