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

This note analyses the order reduction phenomenon of diagonally implicit Runge-Kutta methods (DIRK methods) and Rosenbrock-Wanner methods (ROW methods) applied on the Prothero-Robinson example. New order conditions to reduce order reduction are derived, and a new third-order DIRK and ROW method is created. The new schemes are applied to the Prothero-Robinson example and on the semi-discretised incompressible Navier-Stokes equations. Numerical examples show that the new methods have better convergence properties than comparable methods.