An efficient family of strongly AA-stable Runge–Kutta collocation methods for stiff systems and DAEs. Part I: Stability and order results

For each integer s≥3s≥3, a new uniparametric family of stiffly accurate, strongly AA-stable, ss-stage Runge–Kutta methods is obtained. These are collocation methods with a first internal stage of explicit type. The methods are based on interpolatory quadrature rules, with precision degree equal to 2s−42s−4, and all of them have two prefixed nodes, c1=0c1=0 and cs=1cs=1. The amount of implicitness of our ss-stage method is similar to that involved with the ss-stage LobattoIIIA method or with the (s−1)(s−1)-stage RadauIIA method. The new family of Runge–Kutta methods proves to be of interest for the numerical integration of stiff systems and Differential Algebraic Equations. In fact, on several stiff test problems taken from the current literature, two methods selected in our 4-stage family, seem to be slightly more efficient than the 33-stage RadauIIA method and also more robust than the 44-stage LobattoIIIA method.