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.