An Inversion Method Based On B-Spline Wavelet And Its Application Research On Groundwater Contaminant Identification

In recent years, the pollution of groundwater in most cities and regions of our country is more and more serious. The water shortage which is caused by the contaminant groundwater can badly influence our daily life since one-third of the water is supplied by the groundwater. The governance problem of groundwater pollution needs urgent attention for contemporary society. And the primary problem is to find out pollution sources. The transport process of pollutant in groundwater can be described with the diffusion equation or the advection-diffusion-reaction equation. Therefore, some scholars use applied mathematical method to inverse pollution sources based on the mathematical models.At present, there are some methods for groundwater contaminant source identification, and these methods are generally divided into two classes, iterative method and direct method. Direct method is based on the meshless method, it is of high calculation efficiency and accuracy. Among the direct method, one of the relatively mature methods is radial basis collocation method. However, for inverse problem of groundwater contaminant source identification, the numerical solution obtained by the radial basis collocation method is not agree well with the analytical solution. Wavelet theory has many good properties, such as local compactly support, orthogonality, symmetry and so on. Inspired by the radial basis collocation method used for inverse problem, in this paper, an inversion method based on the scaling function of the B-spline wavelet is proposed to solve the partial differential equations inversion problems. Furthermore, the B-spline wavelet method is applied to the source inversion of groundwater contaminant mathematical models and obtains some preferable results.In this paper, the properties and characteristics of B-spline wavelet are firstly introduced. The multi-resolution analysis method, construction of B-spline wavelet scaling function and discretization process of differential equation are introduced as well. Then, the B-spline wavelet method is applied to inverse the unknown boundary condition of Laplace equation. The inverse problem is transformed into the solution of linear algebraic equations system, and then the least squares method is used to solve it. Next, based on the assumption that time domain and space domain are independent, the B-spline wavelet method is used to solve the one-dimensional diffusion equation in both time and space domain. Lastly, the resolution of one-dimensional advection-diffusionreaction equation inverse problem with convective term is considered. Based on the comparison and analysis of the numerical results, it is concluded that when the inverse function is smooth, B-spline wavelet method can solve previous three kinds of inverse problems effectively. Compared with radial basis collocation method and modified radial basis collocation method, the numerical results obtained with B-spline wavelet method have higher accuracy and agree with the analytical solution better.