The Application Of Algebraic Multigrid Methods On Grids With Local Refinement AND The Modified Method Of Upwind Scheme For Seawater Intrusion

The multigrid methods are important methods for solving large linear systems of equations. Nowadays,they are used widely used for practical computations and are generally considered as the fastest numerical method for differential equations.There are two kinds of multigird methods:the geometric multigrid method(GMG)and the algebraic multigrid method(AMG). The geometric multigrid method is based on the geometric characteristics of the problem to be solved,which makes the method too problem-dependent and inconvenient to apply.The algebraic multigrid approach is a kind of multilevel method developed to solve matrix equations using exclusively the information of the equations,which avoids the defect of the GMG method, and has advantage of little storage,high precision of convergence,and little computational time, etc.The application of algebraic multigrid method on grids with local refinement for elliptical boundary value problem is considered in this paper,and numerical computational example is given.In the end of this paper,we extend this approach to a convection-diffusion problem.In modern times,along with economy development of the littoral province,people pay more and more attention to seawater intrusion problem.In littoral province,because of the excessive exploitation of groundwater or the durative droughty climate or the raise of sea level and so on, seawater intrudes the groundwater store of the littoral area,and seawater intrusion has become a important geological disaster.The area where seawater intrudes is mostly littoral alluvial plain with good soil,rich groundwater and developed agriculture,after seawater intrusion,the basification of groundwater causes great loss to national economy.It is significant to do research on the relation of the exploitation of groundwater and seawater intrusion,and to tackle seawater intrusion.The numerical model of seawater intrusion is a coupled system of nonlinear partial differential equations consisting of water head equation and salt concentration equation.A finite difference scheme is constructed in this paper and error analysis is given.This paper has two chapters.In Chapter 1 the application of algebraic multigrid method on grids with local refinemerit is considered,first the fundamental iceas and algorithm of algebraic multigrid method are introduced,and an explanation is given for the construction of the AMG components:coarse grids,interpolation operators,restriction operators,coarse grid operators and smooth operators. Then the solution of algebraic multigrid method on grids with local refinement is proposed,and numerical computational example is given in the end.In Chapter 2 the modified upwind finite difference scheme for seawater intrusion model is developed.First a finite difference scheme is constructed for seawater intrusion model.The implicit finite difference scheme is used for approximating the water head equation,and the modified upwind finite difference scheme is referred when approximating the salt concentration equation.In the end error analysis is given.In the end of this paper the program used in Chapter 1 for numerical computation is given.