Derivation of Three-Derivative Two-Step Runge-Kutta Methods

In this paper, we develop explicit three-derivative two-step Runge-Kutta (ThDTSRK) schemes, and propose a simpler general form of the order accuracy conditions (p <= 7) by Albrecht's approach, compared to the order conditions in terms of rooted trees. The parameters of the general high-order ThDTSRK methods are determined by utilizing the order conditions. We establish a theory for the A-stability property of ThDTSRK methods and identify optimal stability coefficients. Moreover, ThDTSRK methods can achieve the intended order of convergence using fewer stages than other schemes, making them cost-effective for solving the ordinary differential equations.