Semi implicit hybrid methods with higher order dispersion for solving oscillatory problems (Q369713)

Summary: We construct three two-step semi-implicit hybrid methods (SIHMs) for solving oscillatory second-order ordinary differential equations (ODEs). The first two methods are three-stage fourth-order and three-stage fifth-order with dispersion order six and zero dissipation. The third is a four-stage fifth-order method with dispersion order eight and dissipation order five. Numerical results show that SIHMs are more accurate as compared to the existing hybrid methods, Runge-Kutta Nyström and Runge-Kutta methods of the same order and diagonally implicit Runge-Kutta Nyström (DIRKN) method of the same stage. The intervals of absolute stability or periodicity of SIHM for ODE are also presented.