On explicit ninth-order, two-step methods addressing y″=f(x,y)

We present a new family of ninth-order hybrid explicit Numerov-type methods, effectively utilizing only eight stages, for solving the special second-order initial value problem. After applying a number of simplifying assumptions, we arrive to a reduced set of order conditions. Then, we derive an optimal method with constant coefficients that requires one less stage than standard methods found in the literature that use nine stages at this moment. Numerical tests are conducted using quadruple precision arithmetic on several well-known problems and the superiority of the new method is clear. Finally, in Section 6, a Mathematica package is presented that implements the corresponding algorithm.