The RKGL method for the numerical solution of initial-value problems

We introduce the RKGL method for the numerical solution of initial-value problems of the form y′=f(x,y)y′=f(x,y), y(a)=αy(a)=α. The method is a straightforward modification of a classical explicit Runge–Kutta (RK) method, into which Gauss–Legendre (GL) quadrature has been incorporated. The idea is to enhance the efficiency of the method by reducing the number of times the derivative f(x,y)f(x,y) needs to be computed. The incorporation of GL quadrature serves to enhance the global order of the method by, relative to the underlying RK method. Indeed, the RKGL method has a global error of the form Ahr+1+Bh2mAhr+1+Bh2m, where r is the order of the RK method and m is the number of nodes used in the GL component. In this paper we derive this error expression and show that RKGL is consistent, convergent and strongly stable.