General error propagation in the RKrGLm method

The RK r GL m 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 r and m -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 RK r GL m . We show that, for a suitable choice of r and m , the global order of RK r GL m is expected to be r + 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 ) th node. We also show how a Hermite interpolating polynomial of degree 2 m + 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.