The spatial step size for the second-order accurate Muskingum-Cunge (M-C) method is determined by the spatial weighting factor. Both the spatial weighting factor and the time step must be selected judiciously to obtain accurate solutions. In this study, accuracy criteria for the linearised diffusion routing problem are discussed. Starting from the truncation error analysis of the general finite-difference scheme used to solve the kinematic wave equation (of which the M-C method is a special case), conditions necessary to obtain second-, third- and fourth- order accurate solutions to the linearised diffusion routing equation are derived. For given diffusion coefficient and celerity, the spatial step of the fourth- order scheme is fixed, whereas third- and second-order solutions are available for independently selected spatial steps. In order to achieve optimal solutions to the second- order accurate scheme, the truncation error criteria are combined with a condition derived from the concept of column holdup. This combination is shown to produce results as good as those from the third-and fourth-order accurate schemes. The simplest explicit method is shown to give satisfactory results for flood data from the River Wye (UK).