The Prothero and Robinson example: Convergence studies for Runge–Kutta and Rosenbrock–Wanner methods

It is well-known that one-step methods have order reduction if they are applied on stiff ODEs such as the example of Prothero–Robinson. In this paper we analyse the local error of Runge–Kutta and Rosenbrock–Wanner methods. We derive new order conditions and define with them BPRBPR-consistency. We show that for strongly A  -stable methods BPRBPR-consistency implies BPRBPR-convergence. Finally we analyse methods from literature, derive new BPRBPR-consistent methods and present numerical examples. The numerical and analytical results show the influence of different properties of the methods and of different order conditions on the numerical error and on the numerical convergence order.