ESIRK methods and variable stepsize Original Research Article

ESIRK methods are Runge-Kutta methods in which the s × s coefficient matrix A has a one point spectrum {λ} and which satisfy the generalized order conditions known as “effective order”. We assume that the stage order and the effective order are equal to s. Because ESIRK methods are generalizations of SIRK methods for which the abscissae can be chosen as any set of s distinct numbers, it might be expected that they will have serious disadvantages compared with the standard SIRK methods. In particular, changing stepsize is more complicated for the new methods, and it might be thought that difficulties arise for this aspect of the methods. However, it is found that from the point of view of truncation error, the standard SIRK choice need not be the best choice. Furthermore, there does not seem to be an overwhelming advantage in adopting the standard choice from the point of view of stability.