Study Of Scheme For Solving Convection-Dominated Diffusion Equations

The characteristics method is combined with the finite volume method and the mixed finite element method to treat the convection dominated diffusion equations. The characteristic finite volume method and the characteristic mixed finite element method are constructed. As the traditional methods of linearization and iteration have the deficiencies of accuracy and convergence speed for the nonlinear convection-dominated diffusion equations. The two-grid method of characteristic finite volume and the two-grid method of characteristic mixed finite element are constructed. The nonlinearities are expanded about the coarse grid solution, and the resulting linear system is solved on a fine grid. We discuss the convergences and error estimates of these methods theoretically and numerically. The theoretical and numerical results show these new methods have no loss in order of accuracy and can reduce the numerical oscillations of convection-dominated diffusion equations and improve the computational efficiency.In chapter 1, the background and introduction of convection-diffusion equations and several existing numerical methods on solving convection-diffusion equations are introduced. In chapter 2, a two-grid method of characteristic finite volume is constructed for 1D and 2D nonlinear convection-dominated diffusion equation, convergence analysis and error estimates are also established. Numerical example is presented at last. In chapter 3, characteristic mixed finite element method is given for 2D linear convection dominated diffusion equation. Convergence analysis and error estimates are established in detail. In chapter 4, two-grid method of characteristic mixed finite element is constructed for 2D nonlinear convection-dominated diffusion equation and error estimate is also established. From the conclusion, we can see the method maintains numerical accuracy, the convergence speed is accelerated and the approximation of the solution is obtain. The numerical results show that these methods are effective and afford a remarkably efficient linearization. Chapter 5 is conclusion of the paper, main results are given. And the work needs to complete later are enumerated.