Explicit two-step Runge-Kutta methods

The explicit two-step Runge-Kutta (TSRK) formulas for the numerical solution of ordinary differential equations are analyzed. The order conditions are derived and the construction of such methods based on some simplifying assumptions is described. Order barriers are also presented. It turns out that for order $p\le 5$ the minimal number of stages for explicit TSRK method of order $p$ is equal to the minimal number of stages for explicit Runge-Kutta method of order $p-1$. Numerical results are presented which demonstrate that constant step size TSRK can be both effectively and efficiently used in an Euler equation solver. Furthermore, a comparison with a variable step size formulation shows that in these solvers the variable step size formulation offers no advantages compared to the constant step size implementation.