A new family of 7 stages, eighth‐order explicit Numerov‐type methods

In this paper, we consider the integration of the special second‐order initial value problem. Hybrid Numerov methods are used, which are constructed in the sense of Runge‐Kutta ones. Thus, the Taylor expansions at the internal points are matched properly in the final expression. A new family of such methods attaining eighth algebraic order is given at a cost of only 7 function evaluations per step. The new family provides us with an extra parameter, which is used to increase phase‐lag order to 18. We proceed with numerical tests over a standard set of problems for these cases. Appendices implementing the symbolic construction of the methods and derivation of their coefficients are also given.