Abstract: The RK5GL3 method is a numerical method for solving
initial value problems in ordinary differential equations, and is based
on a combination of a fifth-order Runge-Kutta method and 3-point
Gauss-Legendre quadrature. In this paper we describe the propagation
of local errors in this method, and show that the global order of
RK5GL3 is expected to be six, one better than the underlying Runge-
Kutta method.
Abstract: The RK1GL2X3 method is a numerical method for solving initial value problems in ordinary differential equations, and is based on the RK1GL2 method which, in turn, is a particular case of the general RKrGLm method. The RK1GL2X3 method is a fourth-order method, even though its underlying Runge-Kutta method RK1 is the first-order Euler method, and hence, RK1GL2X3 is considerably more efficient than RK1. This enhancement is achieved through an implementation involving triple-nested two-point Gauss- Legendre quadrature.