General error propagation in the RKrGLm method

The RKrrGLmm method is a numerical method for solving initial value problems in ordinary differential equations of the form y′=f(x,y) and is based on a combination of a Runge–Kutta method of order rr and mm-point Gauss–Legendre quadrature. In this paper we describe the propagation of local errors in this method, and we give an inductive proof of the form of the global error in RKrrGLmm. We show that, for a suitable choice of rr and mm, the global order of RKrrGLmm is expected to be r+1r+1, one better than the underlying Runge–Kutta method. We show that this gain in order is due to a reduction or “quenching” of the accumulated local error at every (m+1)(m+1)th node. We also show how a Hermite interpolating polynomial of degree 2m+12m+1 may be employed to estimate f(x,y) if the nodes to be used for the Gauss–Legendre quadrature component are not suitably placed.