An introduction to “Almost Runge-Kutta” methods Original Research Article

In this paper we consider numerical methods which differ from Runge-Kutta methods in that they are multivalue and have a greater stage order than is possible with explicit Runge-Kutta methods. On the other hand they share many other properties with one-step methods. In particular, they have stability properties as for Runge-Kutta methods and it is easy to start the methods and to change stepsize. Because of the high stage order, continuous output is easier to achieve than for Runge-Kutta methods and this gives them a potential for the solution of delay problems as well as for ordinary differential equations. These new methods will be known as “ARK” or “Almost Runge-Kutta” methods. Even though they exist for orders at least as high as 6, we will concentrate here on methods of order 4, for which we present four sample methods. For one of these methods, we will present some numerical results for some initial value ordinary differential equation problems and some delay differential equations, illustrating various features of the new methods.