a b s t r a c tOur 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 twodimensional 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.

Abstract:In this work a new optimized symmetric eight-step embedded predictor-corrector method (EPCM) with minimal phase-lag and algebraic order ten is presented. The method is based on the symmetric multistep method of Quinlan-Tremaine [1], with eight steps and eighth algebraic order and is constructed to solve numerically IVPs with oscillatory solutions. We compare the new method to some recently constructed optimized methods and other methods from the literature. We measure the efficiency of the methods and conclude that the new optimized method with minimal phase-lag is noticeably most efficient of all the compared methods and for all the problems solved including the two-dimensional Kepler problem and the radial Schrödinger equation.

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A new general multistep predictor-corrector (PC) pair form is introduced for the numerical integration of second-order initial-value problems. Using this form, a new symmetric eight-step predictor-corrector method with minimal phase-lag and algebraic order ten is also constructed. The new method is based on the multistep symmetric method of Quinlan–Tremaine,1 with eight steps and 8th algebraic order and is constructed to solve numerically the radial time-independent Schrödinger equation. It can also be used to integrate related IVPs with oscillating solutions such as orbital problems. We compare the new method to some recently constructed optimized methods from the literature. We measure the efficiency of the methods and conclude that the new method with minimal phase-lag is the most efficient of all the compared methods and for all the problems solved.

Abstract. In this paper we present two optimized eight-step symmetric implicit methods with phase-lag order ten and infinite (phase-fitted). The methods are constructed to solve numerically the radial time-independent Schrödinger equation with the use of the Woods-Saxon potential. They can also be used to integrate related IVPs with oscillating solutions such as orbital problems. We compare the two new methods to some recently constructed optimized methods from the literature. We measure the efficiency of the methods and conclude that the new method with infinite order of phase-lag is the most efficient of all the compared methods and for all the problems solved.