Optimizing the Numerical Integration of Initial Value Problems in Shooting Methods for Linear Boundary Value Problems

Abstract

In this paper two new embedded Runge-Kutta (RK) methods RK5(4) specially designed for linear ODEs [L. F. Shampine, J. Comput. Appl. Math., 15 (1986), pp. 293–300] are derived with (a) “small” principal truncation terms in the fifth-order formula, and (b) minimum number of Jacobian matrix evaluations per step. Numerical tests comparing their efficiency to the classical RKF5(4)#2 [E. Fehlberg, Tech. Rep. R-315, NASA, 1969], with the SUPORT code [M. R. Scott and H. A. Watts, SIAM J. Nmmer. Anal., 1 (1977), pp. 40–70], are presented.