Handling effectively the rejected stages in Runge–Kutta pairs implementation

Runge–Kutta (RK) pairs are widely used for numerically solving initial value problems (IVPs), but dealing with step rejections during integration is a common occurrence. Conventionally, when a step is rejected, all calculations made during that step are discarded, and a completely new set of computations is initiated. In our research, we propose a method to address this inefficiency by repurposing the previously computed RK stages from rejected steps. Our primary focus is on the renowned RKF45 pair, consisting of fifth‐ and fourth‐order methods. When a step rejection occurs, we leverage the stages computed in prior steps and introduce just three additional stages. These stages are then used to evaluate the results with a smaller step size. This approach effectively reduces computational costs in various challenging IVPs where RK algorithms with different step sizes encounter difficulties