On the construction of nested explicit Runge–Kutta methods via null rules

In this short note, we present a simple and general procedure for constructing embedded explicit Runge–Kutta families by exploiting the null space of the order conditions. Starting from a given method, the approach generates families of embedded formulas that preserve a prescribed order of accuracy while reusing the same stages and computational cost. We analyze these families through their stability regions and show that suitable choices of the free parameters can enlarge the real stability interval on the negative axis. We also illustrate, on a stiff scalar test problem with adaptive time stepping, how different embedded formulas may affect both the adaptive step count and the final-time accuracy.