A computational approach to developing two-derivative ODE-solving formulations: γβI-(2+3)P method

This paper presents the first set of two-derivative γβ formulations for time-integration of initial value (IV) ordinary differential equations (ODEs) in applied science. It belongs to the extended families of general linear methods (GLMs) and Runge-Kutta (RK) methods covering both linear and nonlinear ODEs. The present formulation is an advanced version of the basic form $\alpha I-(q+r)P$, previously published by the author [1]. The key idea behind these formulations is the body decomposition of the RK methods and GLMs into two distinctive parts including interpolation and integration. This interesting idea has many advantages. First, it increases the flexibility of the formulation process. Second, each of these parts is supported by strong theorems in numerical analysis and can be developed independently through its own theories. In addition to these advantages, a knowledge-based approach, strengthened with swarm intelligence, is employed to formulate the integrator. Accordingly, a significant level of expertise is utilized in formulating the integrators. It leads to a series of interconnectivity relations between the weights of the integrators. These are known as weighting rules (WRs), which come in different types. The interpolators are obtained from approximation theory in which a polynomial is fitted to a given set of data. Consequently, a number of high-precision interpolators are developed to collaborate with the extended integrator. They approximate solution values at intermediate stages of the integration step, while the integrator bridges between the start and end points of the step. Working with interpolators has the advantages of generating solution values at all stages. It enables us to report the solution at more points rather than merely the mesh points. All the WRs, integrator, interpolators, and the ODE are systematically arranged in a specific order to construct the new algorithms of $\gamma \beta I-(q+r)P$. Butcher tableaus are also provided for the new methods. Finally, they are carefully verified on several IVPs, including long-term and high-frequency problems. The obtained results demonstrate the practicality and efficiency of the formulations, and confirm that the collaboration of WRs, integrators, and interpolators performs exceptionally well.