Development of a new Runge‐Kutta method and its economical implementation.

Abstract

[EN]A two-step hybrid block method is developed for numerically solving first-order initial-value problems. The formulas will be obtained from a continuous approximation derived via interpolation and collocation at different points where the two intermediate points are chosen through the optimization of the local truncation errors. The main characteristics of the method are discussed after considering its formulation in vector form. As with the Runge-Kutta (RK) methods, there is no need to provide starting values by using other approaches; thus, it is a self-starting method. The proposed method may be reformulated as a RK one, but in this way, its implementation requires more computational cost. Nevertheless, a straightforward reformulation of the method reducing the number of occurrences of the source term f results in the most efficient formulation. This strategy might be applied to the numerous existing block methods. Some numerical tests considering different problems that appeared in the literature show the performance of the presented schemes, confirming the best behavior of the proposed economical reformulation.