ARK methods for stiff problems

“Almost Runge-Kutta”, or ARK, methods form a sub-class of general linear methods, with properties very close to those of traditional Runge-Kutta methods. They have previously been applied only to non-stiff problems. In this paper we explore their possible generalisation to diagonally implicit methods, for the solution of stiff problems. A characteristic feature of these new methods is that, although some of the data computed in a step has order limited to 2, this inaccuracy does not effect the order of the subsequent steps. Consequences of these assumptions are explored and this leads on to the derivation of some new methods of orders three and four. Preliminary numerical experiments are promising for some of these methods.