Variable stepsize continuous two-step Runge-Kutta methods for ordinary differential equations

A general class of variable stepsize continuous two-step Runge-Kutta methods is investigated. These methods depend on stage values at two consecutive steps. The general convergence and order criteria are derived and examples of methods of order p and stage order q = p or q = p - 1 are given for p ≤ 5. Numerical examples are presented which demonstrate that high order and high stage order are preserved on nonuniform meshes with large variations in ratios between consecutive stepsizes.