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. This revised version was published online in June 2006 with corrections to the Cover Date.