An analysis of the Prothero-Robinson example for constructing new adaptive ESDIRK methods of order 3 and 4

Diagonally implicit Runge-Kutta methods with an explicit first stage (ESDIRK--methods) have usually order reduction if they are applied on stiff ODEs such as the example of Prothero and Robinson. It can be observed that the numerical order of convergence drops down to the stage order, which is limited by two. In this paper we analyse the Prothero-Robinson example and derive new order conditions to avoid order reduction. New third and fourth order ESDIRK--methods are created. The new schemes are applied on the Prothero--Robinson example and on an index-2 DAE. Numerical examples show that the new methods have better convergence properties than usual ESDIRK methods.