Parameter Identification And Application Of Convection-diffusion Equation With Deterministic Algorithm

With the rapid development of social economy,the environmental pollution problems have been increasingly prominent,especially in recent years,the pollution emergency events of river and groundwater occurred frequently.As to these two kinds of water pollution incidents,it is imperative to quickly and efficiently determine the pollution sources,grasp the time and space distribution of the pollutants,and work out an effective emergency plan.However,according to the mathematical model of water environmental pollution events,it is the urgent problem to determine the initial value,the location,intensity and time history of the groundwater pollution source and the seepage velocity and the diffusion coefficient of different aquifers,to confirm the model parameters of the river pollution.Based on the pollutants transport model in groundwater and rivers,this thesis designed Landweber iteration method,PRP conjugate gradient method and gradient regularization method with variable step size to systematically identify the initial value,source item and model parameter of one-dimensional and two-dimensional,integer-order and fractional-order pollutants transport,which solved the problem of parameter identification for the groundwater and river emergency pollutant concentration prediction well.The main contents and results are summarized as follows:(1)A new iteration method,named as Landweber iteration method,was designed to reconstruct the initial value of one-dimensional groundwater pollutant transport model,the effectiveness of the proposed algorithm is tested by the diffusion example and convection-diffusion example.In addition,the effects of the regularization parameter,initial value and measurement error on the reconstruction results were also analyzed.When the regularization parameter is 1.9,the proposed algorithm can reconstruct the initial value with high precision after 50 iterations,which is in good agreement with the true value.Therefore,Landweber iteration method is an effective method to solve the inverse problem of the initial value of one-dimensional groundwater pollution.(2)This paper put forward PRP conjugate gradient algorithm to deal with the pollution source identification of one-dimensional groundwater pollutant transport,the effectiveness of the proposed algorithm was verified by the reconstruction of the groundwater sulfate concentration in Zibo area.Moreover,the impact of initial value and measurement error on the source identification results were also studied.The numerical results demonstrate that,when the initial value is reasonable,the proposed algorithm can identify quickly and efficiently the average infiltration intensity of sulfuric acid root with high precision,which prove that PRP conjugate gradient algorithm is effective to solve the pollution source of one-dimensional groundwater pollutant transport.To eliminate the dependence of PRP conjugate gradient method on the initial value,a new method,named as hybrid method,is born at the right moment,which couples genetic algorithm with PRP conjugate gradient method,and has the characteristics of strong global search ability and high inversion precision.The results of sulfuric acid root in Zibo show that,the hybrid algorithm is stable and effective to solve the problem of pollution source identification.(3)As to the joint reconstruction of multiple model parameters of one dimensional river pollutant transport,the gradient regularization was introduced to resolve the proposed problems.There are kinds of models,such as constant coefficients model,variable coefficients model with linearly dependent or linearly independent,to verify the rationality and reliability of the proposed algorithm The numerical results confirm that the gradient regularization method can effectively resolve the multiple model parameters identification problem of one dimensional convection diffusion equation.In anisotropic porous medium,there is non-Fickian phenomenon in the process of pollution diffusion inevitably,wherein the dispersion coefficient increases with the distance.Aiming at this kind of situation,this thesis established one dimensional fractional-order convection diffusion equation,and the gradient regularization method with variable step size was adopted to inverse the multiple model parameters of one dimensional fractional order convection diffusion equation.In addition,the impacts of the fractional order,the regularization parameter,the initial value and the measurement error were also taken into consideration.When the regularization parameter is taken as 1e-4 and the number of fractional differential exponent tends to 2.0,the proposed algorithm is an effective method to combine reconstruct the multiple model parameters of the fractional order convection-diffusion equation.(4)In order to expand the scope of environmental hydraulics inverse problem,this paper presented the gradient regularization algorithm with variable step size to solve the inverse problem of two-dimensional pollutants transport.For the inverse problem of the source identification,when the regularization parameter belongs to 1e-7(27)?(27)1e-4,the numerical results have higher precision and faster speed,furthermore,the initial value and measurement error have little effect on the numerical results.The proposed algorithm also shows good numerical performance for the joint reconstruction of the dispersion coefficient of two-dimensional pollutants transportBased on the inverse problem of the source term and parameter identification,this paper further applied the gradient regularization algorithm with variable step size to identify the hybrid inverse problem of two-dimensional transport(including the source term and dispersion coefficient),and an example was to test the effectiveness of the proposed algorithm.In addition,the effects of the initial value and measurement error on the identification results were also discussed.The numerical results demonstrate that the gradient regularization algorithm with variable step size can not only recognize the source item and dispersion coefficients quickly,but also show the initial value insensitivity and high stability.In short,the aforementioned three kinds of deterministic algorithm can effectively solve the inverse problems of the initial value,source item and model parameter of one-dimensional and two-dimensional,integer-order and fractional-order pollutants transport,which not only provide some datum support for identifying and controlling rivers and groundwater pollution,but enrich the solution methods of environmental hydraulics.Therefore,these three kinds of deterministic algorithms are of important value and of application prospect.