Crank-Nicholson Difference Scheme On Nonuniform Meshes For A Kind Of Nonlinear Parabolic Equation System

At first, the uniform meshes were adopted in numerical solutions of partial differential equations when dividing definition domains. Uniform meshes made difference schemes and numerical computations relatively simple. Such a scheme must be refined as far as possible for improving the numerical precision. In order to reduce computation, non-uniform meshes were adopted in the establishment of difference schemes. However, the difference schemes in non-uniform meshes both in form and in error estimating is much more complicated than the difference schemes in uniform meshes. For various initial/boundary-value problems of nonlinear parabolic equations, it is also complicated to establish high-precision difference schemes by non-uniform meshes.In this thesis, the numerical solution of the initial/boundary-value problem for the nonlinear parabolic equations (System) A( x , t , u , u_x )u_t = u_xx + f ( x , t , u , u_x) Was mainly discussed. The main content is as follows:In the second chapter, the Crank-Nicholson scheme was introduced. Based on the idea of the C-N scheme, an improved C-N difference scheme for the Dirichlet problem of the nonlinear parabolic equation by using non-uniform meshes was established. The convergence of the improved C-N scheme was proved and the error estimation was given. Finally, numerical examples on three different meshes were given to verify the validity of the difference scheme, and an analysis on how to make reasonable non-uniform meshes was given.In the third chapter, according to the method of the second chapter, the Neumann initial/boundary-value problems of the nonlinear parabolic equations for the numerical solution were discussed. A three-layer difference scheme on non-uniform meshes was also established. The convergence of the improved C-N scheme for the Neumann initial/boundary-value problems was also proved and the error estimation was given. Numerical examples were given to verify the validity of the results.