A new trigonometrically-fitting technique to construct a symmetric linear multi-step method for the numerical solution of an orbital problem

Some previous works show that the linear multi-step methods with the trigonometrically-fitting technique suffer from the numerical instability due to the parameters, so that these parameters must be converted into Taylor series. In this paper, we present a general way to construct the symmetric linear multi-step method for the approximate solution of orbital problem by using a new trigonometrically-fitting technique. By using the new technique, we can eliminate the parameter instability so that the Taylor series expansion is avoided and show that the new obtained method is P-stable. Comparing with the previous trigonometrically-fitting technique for the implicit eight-step method, new technique extends the interval of periodicity from H02∼5 to infinity and cuts the error constant nearly off 28%. By using a new efficient algorithm, we can cut off about one-fifth CPU time, because the first-stage and the iterative calculation can be completely avoided. Numerical results from the application to the well-known periodic orbital problems show that the new improved eight-step method is better than the previous eight-step method in accuracy and efficiency.