DESIRE: diagonally extended singly implicit Runge-Kutta effective order methods

To overcome known disadvantages in the use of singly-implicit methods in the solution of stiff differential equations, two generalizations have been recently introduced. These are the use of additional diagonally-implicit stages and the use of effective order as a weakening of the conditions on the method. In this paper, these two generalizations are combined into a unified scheme. The resulting DESIRE methods are very promising alternatives to standard methods for solving stiff problems.