A family of methods that extend Störmer formulae to variable stepsizes is introduced, analyzed and tested. The variable stepsize methods do not satisfy the generalized root condition that in their fixed stepsize counterparts is equivalent to stability, but are shown to be stable and convergent. However, numerical experience suggests that they cannot compete with modern embedded pairs of Runge-Kutta-Nyström methods.
Cano, B., & García-Archilla, B. (1996). A generalization to variable stepsizes of Störmer methods for second-order differential equations. Applied Numerical Mathematics, 19(4), 401–417. https://doi.org/10.1016/0168-9274(95)00107-7