General error propagation in the RK\(r\)GL\(m\) method (Q1019806)

This paper is concerned with the error propagation of the so called RK\(r\)GL\(m\) methods introduced by the author in a previous paper [J. Comput. Appl. Math. 213, No.~2, 477--487 (2008; Zbl 1135.65348)] for the numerical solution of initial value problems (IVPs) for ordinary differential equations (ODEs). In these methods, for a given explicit Runge-Kutta(RK) method with order \(r\) and a Gauss-Legendre quadrature formula with \(m\) nodes, the numerical solution of the IVP: \( y'(x)= f(x, y(x)), y(x_0)= y_0\) at \( x_1=x_0+h\) is given by the \(m\) nodes quadrature formula \( y_1 = y_0 + h \sum_{j=1}^m b_j \; f(x_0+ c_j h, y(x_0+ c_j h) )\) where \(b_j\) are the coefficients of the formula and \( y(x_0+c_j h), j=1, \ldots ,m\) are approximated successively by the explicit RK method with appropriate step size. The author studies the local error propagation of these methods proving that the order of the global error is \(r+1\), one order higher than the underlying RK method. Further some numerical experiments are presented to show that the Gauss-Legendre quadrature slows the accumulation of the local errors introduced by the explicit RK method, but no theoretical explanation of this fact (referred to as error quenching) is given.