Continuous Runge-Kutta(-Nystrِm) methods with reduced phase-errors

An explicit Runge-Kutta (RK) or Runge-Kutta-Nyström (RKN) method, for the numerical approximation of the initial value problem, can be expanded by the addition of a “dense” formula which provides solutions at points within or outside the normal step intervals. In this paper, we are concerned with the construction of continuous extensions for RK and RKN methods, intended to approximate first- and second-order differential equations, respectively. First we derive the required equations of conditions that the coefficients of these extensions have to satisfy in order to produce reduced phase-errors, when applied to a linear homogeneous test equation. Moreover some particular continuous extensions of an explicit 6(5) RK and 8(6) RKN pair, respectively, are proposed and tested numerically.