A new phase-fitted eight-step symmetric embedded predictor–corrector method (EPCM) for orbital problems and related IVPs with oscillating solutions

Our new phase-fitted embedded predictor–corrector method (EPCM) presented here is based on the multistep symmetric method of Quinlan–Tremaine (1990), with eight steps and eighth algebraic order and constructed to solve numerically the two-dimensional Kepler problem. It can also be used to integrate other orbital problems and related IVPs with oscillatory solutions. First we present a EPCM (Panopoulos et al. (2011) and Panopoulos et al. (2013)) pair form. From this form we construct a new eight-step method. The new scheme has algebraic order ten and infinite order of phase-lag. We tested the efficiency of our newly developed scheme against to some recently constructed optimized methods and other well known methods from the literature. We measure the efficiency of the methods and conclude that the new scheme is noticeably more efficient of all the compared methods and for all the problems solved, including the radial Schrödinger equation.