Direct integrators of Runge–Kutta type for special third-order ordinary differential equations

• Formulation of schemes of direct integrators of Runge–Kutta type for special third-order ordinary differential equations (RKT).
• New tri-colored tree theory and corresponding B-series theory for RKT methods.
• Derivation of order conditions for RKT methods and construction of practical (explicit and implicit) RKT integrators.
• Collocation approach for a class of implicit RKT methods.
• Numerical experiments showing the effectiveness and competence of RKT methods compared with the traditional RK methods.

This paper is devoted to the investigation of direct integrators of Runge–Kutta type for third-order ordinary differential equations (RKT). A new tri-colored tree theory and the corresponding B-series theory are built systematically, based on which the order conditions for RKT methods are derived. A two-stage explicit RKT method of order four and a three-stage explicit RKT method of order five are constructed. Implicit RKT methods of collocation type are considered. The results of numerical experiments show that our explicit RKT methods are more efficient than the traditional RK methods of the same algebraic order.