4-Point FDF of Muskingum method based on the complete St Venant equations

Using the Froude number, a nonlinear convection–diffusion equation was derived from the St Venant equations of continuity and momentum. It was made applicable to discrete space using a mixing-cell method, resulting in a first-order nonlinear ordinary differential equation. A 4-point finite difference scheme was used to solve the first-order nonlinear ordinary differential equation to achieve a nonlinear algebraic form that is very similar in structure to the traditional Muskingum form thus its naming as the ‘4-point FDF of Muskingum Method (4-point FDF M-M)’. In this method, the space interval Δx is a characteristic channel length and a variable that is dependent on flow discharge, water depth, and flow velocity. An iterative search method was applied to simultaneously obtain the flow discharge and the optimal space interval Δx. The method developed was tested with both synthetic numerical examples and observed events and the results were compared with those of the Lambda scheme and the method of characteristics. The outflow hydrographs produced by this new 4-point FDF M-M were of comparable accuracy. The parameters used in the new method are based on the physical attributes of the channel and thus do not need calibration as required for the Muskingum method.