The Discontinuous Finite Element Method (fem) To Study A Class Of Nonlinear Diffusion Equation

It has become a hot problem to obtain solutions of nonlinear diffusion equations involving the fluid dynamics, heat transfer, biology, chemistry and pattern dynamics. Because of the complexity of the nonlinear problems, in the most cases, we can’t obtain their exact solutions. Then, it is more important to solve the nonlinear diffusion equations numerically.The discontinuous Galerkin (DG) method has various advantages such as local conservation, allowance of the discontinuities, easy realization of adaptive calculation and parallation. With these advantages, the DG methods have been found widely applications in diverse areas in recent years. The present dissertation is focusing on the DG finite element method for the numerical simulation of some nonlinear diffusion problems, such as the Burgers type equation, N-carrier systems, diffusion equation with discontinuous coefficient, nonlinear Schrodinger equation, reaction diffusion equation and radiation diffusion equation.Our work is outlined in the following several aspects.In chapter1, the various DG methods are reviewed for the diffusion problems, these DG methods include the lcoal DG (LDG) method, the Baumann-Oden DG method, the direct DG (DDG) method, the DG method based on the diffusive Generalized Riemann Problem, the DG method proposed by Cheng and Shu, and interior penalty DG methods. The analysis of the correlation between these DG methods is presented, and a comparison of their features is made. The strong stability-preserving Runge-Kutta method (SSP RK), exponential time differencing method (ETD) and implicit integration factor (IIF) are addressed for time discretization. Among of these methods, the IIF method can avoid the serve time stepsize limit of the explicit method, but also preserves the "local" property of the DG method, thus it is appropriate for the semi-discrete form arising from DG method.In chapter2, the LDG method is used to solve Burgers type equations and N-carrier system. Numerical results are compared with the exact solution and other results to show the accuracy and reliability of the method.In chapter3, we study the diffusion equations with discontinuous coefficient. For the linear parabolic equation with discontinuous coefficient, we propose a new weighted numerical flux satisfying the continuous flow condition. The stability and error estimate are proved for the new weighted DG method. We also develop a new weighted interior penalty method for the elliptic equation with discontinuous diffusivity. The convergence analysis is presented based on the coercivity and continuity of bilinear form. Numerical examples demonstrate the validity of DG method for parabolic and elliptic problems with strongly discontinuous coefficients.In chapter4, the DG method is discussed for the nonlinear Schrodinger equation. The DDG method is applied to numerical simulation of the one-dimensional, two-dimensional and coupled nonlinear Schrodinger equation. We proved that numerical solution by DG method preserves the discrete mass conservation. In contrast to the LDG method, the DDG method can be used without introducing any auxiliary variables, hence it can reduce storage and computational cost. Various numerical examples show that the DDG scheme can get excellent numerical results.In chapter5, we study the reaction diffusion equation by the DG method. We use IIF method for the time discretization coupled with the DDG method for the space discretization. The IIF method can get the high order accuracy and maintain good stability while avoiding the severe time-stepsize restriction of the explicit method. And more importantly, it introduces a small nonlinear algebraic systems solved element by element and avoids solving a large system of nonlinear equations. The IIF method can preserve the DG’s local property.In chapter6, we solve the non-equilibrium radiation diffusion equation by the DG methods coupled with integration factor method. The diffusion coefficients are discontinuous on the media interface with several quantity differences. We expand the weighted numerical flux in Chapter3and develop a new weighted DG method. For the1D radiation diffusion equation, implicit method is applied for the semi-discrete form. For the2D case, we first treat the nonlinear diffusion coefficients explicitly by linearization, then apply the implicit-explicit integration factor method into the nonlinear ordinary differential equations and obtain a new implicit-explicit integration factor DG method. To the best of our knowledge, the DG method is firstly introduced to the tightly coupled, multi-material, and highly nonlinear radiation diffusion equations. The results in this paper have opened a new field for DG method application.In the last chapter, we summarize the main conclusions of the dissertation and prospect the next study of the work.