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 $B_{PR}$-consistency. We show that for strongly $A$-stable methods $B_{PR}$-consistency implies $B_{PR}$-convergence. Finally we analyse methods from literature, derive new $B_{PR}$-consistent methods and present numerical examples. This analysis shows that Runge--Kutta methods and Rosenbrock--Wanner methods which are not stiffly accurate and are only consistent converge with order 2 in the stiff case, but the error constant may be large. As an improvement stiffly accurate methods can be considered, since the numerical error is now smaller, but the method converges only with order 1. The numerical results and the order of convergence can be improved if the derived order conditions are satisfied.