The Difference Method Of A Type Of Nonlinear Partial Differential Equations And Its Application On The Simulation Of Sewage Disposal

Many mathematical models in the fields of physics, chemistry, biology and geology can be boiled down to the fixed solution problem of linear or nonlinear partial differential equations (PDE.). It's so hard for people to get the accurate solution of nonlinear PDE as people have to switch to the numerical solution. Despite the long history of the numerical solution method of PDE, it had advance little due to the restraints of people's computing ability. It is not until the 70s of the 20th, with the rapid development of computer, people has much strengthened his computing ability and focused the research on the nonlinear science which includes the important aspect of numerical solution of PDE.In practice, people has developed many kinds of numerical solution methods which covers the Galerkin method and all kinds of mixed or boundary or infinite element methods as well as the wavelet method for the problem of fixed solution of PDE., but of all the methods , finite difference method and finite element method are used most often.The basic principle of finite difference method is replacing the differential quotient of each order in the equations with difference quotient of respective order. With respect to the specific problem and the difference scheme along with the proper initial and boundary conditions, we can get a linear or nonlinear algebraic equations. The chief task of difference method is constructing a reasonable difference scheme which keeps the main property of the initial problem as well is accurate enough. A good and feasible difference scheme should satisfy the consistency, stability and convergence. In this paper, we mainly consider the following typical initial and boundary value problem of nonlinear coupling PDE.,　　　　 , of which is an unknown vector function of dimensions. is the positive definite coefficient matrix with the order, is a known vector function of dimensions , () belongs to the rectangle domain. is known vector function of dimensions.Aiming at the fixed solution problem of above PDE, we construct the difference scheme as followings:,,and give the proof of stability under certain conditions. With regard to the above fixed solution problem, if we adopt the implicit scheme, then we'll get a nonlinear difference equations which is related to the iterative method of solving nonlinear equations. In Chapter 3 , the author give a brief introduction of the iterative method of solving nonlinear equations and with regard to the damp coefficient scheme method, , the author gives the proof the necessary and sufficient condition of its stability.At the last, using the former schemes, the author give a numerical simulation of 3N concentration during the sewage disposal process. The results show that the numerical solution bears a good convergence and simulation, and consequently indicate the high practical significance of the adoptive difference scheme and the damp coefficient method in solving nonlinear equations.