An eight-step semi-embedded predictor–corrector method for orbital problems and related IVPs with oscillatory solutions for which the frequency is unknown

Our new linear symmetric semi-embedded predictor–corrector method (SEPCM) presented here is based on the multistep symmetric method of Quinlan and 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 related IVPs with oscillatory solutions for which the frequency is unknown. Firstly we present a SEPCM (see Panopoulos and Simos, 2013 [36,37]) in pair form. This form has the advantage that reduces the computational expense. From this form we construct a new symmetric eight-step method. The new scheme has constant coefficients and algebraic order ten. We tested the efficiency of our newly developed scheme against some well known methods from the literature. We measure the efficiency of the methods and conclude that the new scheme is the most efficient of all the compared methods and for all the problems solved.