Backward Solution (in-time) of the Pollution Transport Equation in River Using Group Preserving Scheme

Document Type : Original Article


1 Department of Water Engineering and Management, Faculty of Agriculture, Tarbiat Modares University, Tehran, Iran

2 Department of Water Engineering and Management, Faculty of Agriculture, Tarbiat Modares University, Tehran, Iran.



Today, most of the surface water and groundwater resources are exposed to contamination by different materials sewage. because most of the methods have been used to restore the pollution intensity function for groundwater, representing a method to restore pollution intensity function in rivers with more complex flow conditions is considerable. this research aim is to calculate the intensity function of the contamination source, by the numerical solution of group preserving scheme, which has not been observed in other researches done so far. Group preserving scheme is an accurate method to solve ill-posed problems. By implementing this method the one-dimensional advection-dispersion equation with variable coefficients has been solved. The principle of the introduced backward solution method is that by solving the dynamic systems in negative time steps, a general equation will be obtained, which can solve ordinary differential equations. The responses of this equation will lead to convergence of the equation and prevent the divergence of the data. three examples have been presented to show the accuracy of the forward and backward group preserving scheme, Firstly, using direct solutions of the river, the intensity of pollution concentration in the river is calculated, and the implicit finite difference method is applied to verify the accuracy of the direct solutions. In the direct method, the results of the two models were compared with statistical indexes in order to demonstrate the conformity of the two models. In the next step, by solving the backward group preserving scheme in two different examples, the concentration of pollutants upstream is simulated. Following the simulation and verification of the inverse model by direct solution, statistical indexes are used to evaluate the effectiveness of this method.


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