ARK methods up to order five

Almost Runge–Kutta methods (or “ARK methods”) have many of the advantages of Runge–Kutta methods but, for many problems, are capable of greater accuracy. In this paper a complete classification of fourth order ARK methods with 4 stages is presented. The paper also analyzes fifth order methods with 5 or with 6 stages. Some limited numerical experiments show that the new methods are capable of excellent performance, comparable to that of known highly efficient Runge–Kutta methods. This revised version was published online in June 2006 with corrections to the Cover Date.