The Use of Phase Lag and Amplification Error Derivatives for the Construction of a Modified Runge-Kutta-Nyström Method

and Applied Analysis 3 Lemma 6. For the construction of a method with nullification of phase lag, amplification error, and their derivatives, onemust satisfy the conditions R(z) = 2 cos(z), Q(z) = 1, R(z) = −2 sin(z), Q(z) = 0. 4. Derivation of the New Modified RKN Method The new method that we are going to develop in this section, is a four-stage explicit Runge-Kutta-Nyström method with the FSAL technique (first stage as last), so themethod actually uses three stages at each step for the function evaluations. From (2) and (3), the four-stage explicit modified RKN method can be written in the following form: y n = y n−1 g 4 + hy 󸀠 n−1 + h 2 (b 1 f 1 + b 2 f 2 + b 3 f 3 + b 4 f 4 ) , y 󸀠 n = y 󸀠 n−1 + h (b 󸀠 1 f 1 + b 󸀠 2 f 2 + b 󸀠 3 f 3 + b 󸀠 4 f 4 ) , (14) where f 1 = f (t n−1 , y n−1 g 1 ) , f 2 = f (t n−1 + c 2 h, y n−1 g 2 + c 2 hy 󸀠 n−1 + h 2 a 21 f 1 ) , f 3 = f (t n−1 + c 3 h, y n−1 g 3 + c 3 hy 󸀠 n−1 + h 2 (a 31 f 1 + a 32 f 2 )) f 4 = f (t n−1 + c 4 h, y n−1 g 4 + c 4 hy 󸀠 n−1 + h 2 (a 41 f 1 + a 42 f 2 + a 43 f 3 )) . (15) By substituting the coefficients that have been used by the DEP algorithm in [15], (14) takes the following form: y n = y n−1 g 4 + hy 󸀠 n−1 + h 2 ( 1 14 f 1 + 8 27 f 2 + 25 189 f 3 ) , y 󸀠 n = y 󸀠 n−1 + h ( 1