A class of hybrid methods for direct integration of fourth-order ordinary differential equations (Q1746726)

For a special fourth-order ordinary differential equations (ODE), a class of four-step hybrid Runge-Kutta (RK) methods are proposed. Unlike the other RK methods, these methods do not depend on the derivatives of the solution. Using the B-series with fourth-order ODEs trees, algebraic order conditions are derived. Using these order conditions, one-stage and two-stage methods are constructed. Numerical results reveal that the new methods are more accurate and efficient when compared with the hybrid methods for solving special second-order ODEs and RK methods for solving special fourth-order ODEs directly and some linear multistep methods.