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 less than or equal to 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.