An analysis of the Prothero–Robinson example for constructing new DIRK and ROW methods

In this note the order reduction phenomenon of diagonally implicit Runge-Kutta methods (DIRK--methods) and Rosenbrock--Wanner methods (ROW--methods) applied on the Prothero-Robinson example is analysed. New order conditions to avoid order reduction are derived and a new second order DIRK and ROW--method is created. The new schemes are applied on the Prothero--Robinson example and on the semi-discretised incompressible Navier--Stokes equations. Numerical examples show that the new methods converge with second order for velocity and pressure.