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New explicit hybrid Numerov type methods are presented in this paper. These efficient methods are constructed using a new approach, where we do not need the use of the intermediate high accuracy interpolatory nodes, since only the Taylor expansion of the internal points is needed. The methods share sixth algebraic order at a cost of five stages per step while their phase-lag order is 14 and partly satisfy the dissipation order conditions. It has be seen that the property of phase-lag is more important than the nonempty interval in constructing numerical methods for the solution of Schrödinger equation and related problems. 1-3 Numerical results over some well known problems in physics and mechanics indicate the superiority of the new methods.

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Runge-Kutta-Nyström methods possess a separate theory from the classical RungeKutta schemes for the derivation of their order conditions and principal truncation error terms. A new code using the Mathematica programming language properties and tensor products is proved very efficient in this task.