The Muskingum method is one of hydrological approaches that has been used for flood routing for many years thanks to its simplicity and reasonable accuracy over other methods. In engineering works, the calculation of the Peak section of a flood hydrograph is crucially important. In the present study, using the particle swarm optimization (PSO) algorithm, instead of using a single basic flood, the parameters of the linear Muskingum method (X, K, Δt) are calculated by computed arithmetic and geometric means relevant to two basic floods in the form of eight different models for calculating the downstream hydrograph. The results indicate that if the numerical values of the calculated flood inflow are placed in the interval of the inflow and the basic flood which the parameters X, K, Δt are from, the computation accuracy in approximating the outflow flood related to the peak section of the inflow hydrograph increases for all the mentioned models. In other words, if the arithmetic mean of X, K and the geometric mean of Δt, relevant to the two basic floods, are used instead of using values of X, K, Δt of a single basic flood, the computational accuracy in estimating the flood peak section of the hydrograph in downstream has the highest increase among all the eight models. Thus, the Mean Relative Error (MRE) relevant to the peak section of the inflow hydrograph of the third flood (observational flood) obtained by the first and second basic floods was equal to 4.89% and 2.91%, respectively, while in case of using the arithmetic mean of X and K and the geometric mean of Δt, related to the first and second basic floods (the best models presented in this study), this value is equal to 1.66%.

River water pollution is one of the most important environmental issues. Advection–dispersion equations are used to study the temporal changes in pollutant concentration along the study river reach. The use of advection–dispersion equations in investigating how the concentration of pollution is transformed requires a lot of data including river cross-section characteristics, dispersion coefficient, and upstream and downstream boundary conditions, etc. therefore, the corresponding calculations are very costly, difficult and time-consuming. In the present study, instead of using the mentioned equations, the linear Muskingum method (used in previous studies for flood routing) and the particle swarm optimization (PSO) algorithm was used for the first time to calculate the temporal changes in pollution concentration at different stream locations. The presented solution in the presented study is very accurate and only requires the temporal changes in concentration in the upstream and downstream of the study river reach and for this reason, it is very low-cost and easy to use and requires less time to collect data and perform calculations. In the proposed method, the parameters (X, K, ∆t) of the linear Muskingum method were optimized using the PSO algorithm, and by dividing the temporal changes in the input concentration into three areas of the beginning (the input concentration is greater than the output concentration), the peak (the maximum input and output concentrations) and the end (the output concentration is greater than the input concentration) areas, the accuracy of the calculations increased. The mentioned method was studied for different lengths (first case of x = 50 m (up) and x = 75 m (down), second case of x = 50 m (up) and x = 100 m (down), third case of x = 75 m (up) and x = 100 m (down)) and the mean relative error (MRE) of the total, peak area and the relative error of the maximum concentration using constant parameters for the first case were calculated as 7.08, 1.02, and 2.34 percent, for the second case as 7.41, 11.06 and 6.69 percent, and for the third case as 6.75, 3.59 and 5.42 percent, respectively. If three parameters of (X, K, ∆t) are used, the mentioned values improved by 31.3, 63.7 and 65.5 percent, respectively compared to the case of using constant parameters.

The Muskingum method is one the simplest and most applicable methods of flood routing. Optimizing the coefficients of linear Muskingum is of great importance to enhance accuracy of computations on an outflow hydrograph. In this study, considering the uncertainty of flood in the rivers and by application of the particle swarm optimization (PSO) algorithm, we used the data obtained from three floods simultaneously as basic flood to optimize parameters of linear Muskingum (X, K and ), rather than using inflow and outflow hydrographs of a single basic flood (observational flood), and optimized the outflow discharge at the beginning of flood (O1) as a percentage of inflow discharge at the beginning of flood (I1). The results suggest that the closer inflow discharge variation of basic flood to the inflow discharge variation of observational flood, the greater the accuracy of outflow hydrograph computations. Moreover, when the proposed approach is used to optimize parameters of X, K and , the accuracy of outflow hydrograph computations will increase too. In other words, if rather than using a single basic flood, the proposed approach is applied, the average values of mean relative error (MRE) of total flood for the first, second, third and fourth flood will be improved as 31, 13, 39 and 33%, respectively.