Numerical Methods And Theoretical Analysis Of A Class Of Groundwater Pollution Problems

At present,groundwater pollution problems have received widespread atten-tion from governments,enterprises and academia.Numerical simulation technology has gradually become an indispensable way to study and analyze various problems of water resource.Many mathematical models of groundwater pollution problems are attributed to the convection-diffusion equation.The study on the numerical solution of the convection-diffusion equation has important significance on both theory and practice for the natural ecological protection of water resource.This paper mainly studies the numerical methods and theoretical analysis of a class of groundwater pollution problems.We consider the practical problem of sewage seepage caused by the two-dimensional dispersion of the tracer in the plane one-way flow field.The mathematical model is a kind of two-dimensional convection-diffusion equation but the convection term is only in the x direction:(?)?C(x,y,t)/(?)t?=Dx(?)2C(x,y,t)/(?)x2+Dy(?)2C(x,y,t)/(?)y2-v(?)C(x,y,t)/(?)x+f(x,y,t),where C(x,y,t)is the solute concentration,Dx and Dy are the horizontal and vertical diffusion coefficients,v is the average velocity,and ? is the order of the time fractional derivative,f(x,y,t)is the source term.The research content mainly includes three parts:? the case of transient injection(?=1,f(x.y,t)=0);?the model with the source item(?=1,f(x,y,t)?0);? the case of time fractional order(0<?<1,f(x,y,t)?0).Our main work is as follows.First of all,the appropriate initial value and boundary value conditions are established according to the actual problem of groundwater pollution.What's more,with the idea of dimensionality reduction,the original equation is rewritten into two equations equivalently.In order to simu-late the migration of pollutants in porous media,the finite difference method or the compact finite difference method is used to construct a suitable numerical scheme to study the numerical solution of the original problem.Firstly,for the spatial derivative,to obtain the seconded-order accuracy in part ?.the first and second-order derivative terms are replaced by the first and second-order center difference quotients;the three-points fourth-order compact fi-nite difference scheme of one-dimensional problem is used in part ? and part ?to obtain the fourth-order accuracy.Secondly,for the time derivative,to get the second-order accuracy in part ? and part ?,the Crank-Nicolson(C-N)scheme is applied;the L1 interpolation approximation of the Caputo fractional derivative is used in part ? and the(2-?)order accuracy is derived.Finally,based on the theoretical analysis of the existence,uniqueness,stability and convergence of the compact finite difference scheme,we give several numerical examples.We verify that the proposed schemes are accurate,effective and reliable by progressing on MATLAB software.The results demonstrate that our methods can simulate the concentration distribution of pollutants more accurately,which providing a quick and intuitive decision-making basis for water resources protection,especially for sudden incidents of water pollution.