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

Explicit singly-diagonally-implicit (ESDIRK) Runge-Kutta 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 decreases to the stage order, which is limited to 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, which are applied to the Prothero-Robinson example and to an index-2 DAE. Numerical examples show that the new methods have better convergence properties than usual ESDIRK methods.