On the validity range and conservation properties of diffusion analogy and variable parameter Muskingum

A wide range of real-world flood routing problems can be approached using the Parabolic Approximation, widely recognized as a convenient simplification of the Saint Venant equations. From the Parabolic Approximation, head and discharge-based linear and non-linear advection-diffusion models were also derived by merging mass and momentum conservation equations into a single second-order equation to be used for one-dimensional flood routing. Both head and discharge-based advection-diffusion models were originally derived in their linear forms and extensively studied in the literature because they admit analytical solutions. The discharge-based non-linear models have been used as the basis for the derivation of the variable-parameter Muskingum routing method by setting the truncation error at par with the diffusion parameter. Whereas the linear models are mass-conservative, but inadequately reproduce routing in a variety of cases, the non-linear ones have been recently declared to be non-conservative. In the literature it has also been stated that the variable-parameter Muskingum fails to approximate the discharge-based non-linear discharge based advection-diffusion model on mild slopes. This paper aims at providing a systematic overview of all these models, showing that whereas the non-linear head-based advection-diffusion model, an original version of which is derived here, is in fact conservative, also the discharge-based model, which in principle is not, remains de facto conservative if integrated over a space-time grid with sufficiently small spatial cells. To complete the overview, we also demonstrate that in real-world applications the variable parameter Muskingum gives satisfactory results on mild slopes because lowland rivers in nature are essentially prone to slowly rising hydrographs. The results of all models are intercompared by routing different flood waves through prismatic reference channels with slope 10-4 and smaller.