A family of A-stable Runge–Kutta collocation methods of higher order for initial-value problems

We consider the construction of a special family of Runge–Kutta (RK) collocation methods based on intra-step nodal points of Chebyshev–Gauss–Lobatto type, with A-stability and stiffly accurate characteristics. This feature with its inherent implicitness makes them suitable for solving stiff initial-value problems. In fact, the two simplest cases consist in the well-known trapezoidal rule and the fourth-order Runge–Kutta–Lobatto IIIA method. We will present here the coefficients up to eighth order, but we provide the formulas to obtain methods of higher order. When the number of stages is odd, we have considered a new strategy for changing the step size based on the use of a pair of methods: the given RK method and a linear multistep one. Some numerical experiments are considered in order to check the behaviour of the methods when applied to a variety of initial-value problems.