Direct derivation of an alternative variable parameter McCarthy-Muskingum method from the diffusive wave equation

The well-known Muskingum-Cunge method and the variable parameter Muskingum-Cunge-Todini method (Todini 2007) have been developed by matching the diffusion coefficient of the numerical analogue of the kinematic wave equation with that of the physical diffusive wave equation governing the one dimensional flood wave propagation in channels. It is shown in this study that an alternative variable parameter Muskingum method can be directly derived from the diffusive wave equation without resorting to the matched diffusivity approach as followed in the above mentioned two methods. A comparative evaluation of the routing performances of this method with those of very two similar methods, known as the variable parameter Muskingum-Cunge-Todini (MCT) method as proposed by Todini (2007), and another Variable Parameter McCarthy-Muskingum (VPMM) method proposed by Perumal and Price (2013) is made in the study. For the purpose of comparative evaluation, all these three methods use the same set of benchmark solutions obtained by routing a given inflow hydrograph in twenty five trapezoidal channels each having the same bed width and the side slope, but each of them characterized by unique combinations of bed slopes and Manning’s roughness coefficients. Standard evaluation measures as available in the literature were used in assessing the capabilities of each of the three methods in reproducing the benchmark solutions obtained by routing the given hypothetical inflow hydrograph by the HEC-RAS model for a reach length of 100 km in each of these channels. All the three methods show equal performances in reproducing the benchmark solutions with NSE≳0.99, when the inflow hydrograph is characterized by the water surface gradient +(1/S₀)∂y/∂x≤0.5. But for the routing cases characterized by +(1/S₀)∂y/∂x>0.5, the MCT method fails to route the inflow hydrograph, while the other two methods yield results with diminished performance levels (NSE<0.99), though the proposed method performs better than the VPMM method for these cases.