Research On Nonstandard Finite Difference Schemes For Several Partial Differential Equations

Partial differential equations (PDE) arose from many natural sciences and engineer-ing applications two hundred years ago. PDE was originally a small topic in the physical and geometric problems and has become into a separate branch of mathematics. Now, the issues of partial differential equations include not only problems in classical physics and geometry, but also problems in disciplines, mechanics, biology, chemistry and other dis-ciplines. In recent decades, PDE, especially nonlinear PDE, develops very fast, and there are many amazing results. Among them, difference method is one of the best methods.In this thesis, we firstly introduced the background and history of the numerical algo-rithms for PDE. For difference methods, we presented the definition, principle, classical methods and the important concept called "dynamic consistency". We also elaborat-ed the development of basic ideas and recent advances of exact finite difference scheme, nonstandard finite difference scheme and θ method. And also pointed out that the exact finite difference is a special class of non-standard finite difference method. Moreover, the development of exact difference scheme and nonstandard finite difference scheme and the situation of Fisher equation, Burgers equation, Burgers-Fisher equation, coupled Burgers equation and Convection-diffusion equation are also described.Secondly, we obtained the stepsize function based on the travelling waves solutions. This work developed exact finite difference scheme for the Fisher equation. The form of the exact difference scheme is not simple enough, hence we extended the exact finite difference scheme to two nonstandard finite difference schemes using the method of tak-ing the limit for the mash grid step to0for solving the Fisher equation. And we also gave the local truncation errors of our nonstandard finite difference schemes. Numerical experiments are presented and the results show the nonstandard finite difference schemes are effective.Thirdly, a new version of exact finite difference scheme for Burgers equation and Burgers-Fisher equation are proposed using the solitary wave solution. Then nonstan-dard finite difference schemes are constructed to solve two equations based on the exact difference schemes. We used the method in Chapter2to construct another nonstandard finite difference schemes. And we analyzed the local truncation errors of the two kinds of nonstandard finite difference schemes. At the end of Chapter3, numerical experi-ments are presented to verify accuracy and efficiency of our nonstandard finite difference schemes. We also compared our method with domain decomposition method to show the advantages of our methods.Then, we developed exact finite difference schemes for the two dimensional nonlin-ear coupled viscous Burgers’equation using the solitary wave solution. We extended the explicit nonstandard finite difference schemes on the basis of the exact finite difference schemes to solve the coupled Burgers’equation. We carried out two numerical exam-ples of different initial-boundary conditions to verify the efficiency and accuracy of the methods. We also compared our numerical solutions with Srivastava, Bahadir and Jain’ s numerical results in the second numerical simulation. We can see that our methods are effective from the numerical experiments.Finally, we constructed a nonstandard θ-method for Convection-diffusion equation in this thesis. A matrix form of the0-method with Neumann boundary and Dirichlet boundary conditions are given. By analysis we know that the coefficient matrix is irre-ducible matrix. In addition, numerical simulations indicate that the numerical error of our method is smaller than implicit finite difference method.