AbstractThe concept of collocation, originally used by Wilson in the development of dissipative algorithms for structural dynamics, is systematically generalized and analysed. Optimal schemes within this class are developed and compared with a recently proposed family of dissipative algorithms, called a methods. The α methods are found to be superior on the basis of standard measures of dissipation and dispersion.It is pointed out that the tendency to overshoot is an important and independent factor which should be considered in an evaluation of an implicit scheme. The basis for studying overshoot is discussed and the optimal collocation and α methods are compared. It is found that pathological overshooting is an inherent property of collocation schemes, whereas the overshooting characteristics of the α methods are go