Explicit Runge–Kutta methods for starting integration of Lane–Emden problem

Abstract Traditionally, when constructing explicit Runge–Kutta methods we demand the satisfaction of the trivial simplifying assumption. Thus, f 1 = f ( x 0 , y 0 ) is always used as the first stage of these methods when applied to the Initial Value Problem (IVP): y ′ ( x ) = f ( x , y ) , y ( x 0 ) = y 0 . Here we examine the case with f 1 = f ( x 0 + c 1 h , y 0 ) , ( h : the step) and c 1  ≠ 0. We derive the order conditions for arbitrary order and construct a 5th order method at the standard cost of six stages per step. This method is found to outperform other classical Runge–Kutta pairs with orders 5(4) when applied to problems with singularity at the beginning (e.g. Lane–Emden problem).