Some New Methods Of Hamilton-Jacobi Equations And Convection-diffusion Equations

In this thesis, we present some new methods for numerically solving Hamilton-Jacobi equations and convection-diffusion equations. We investigate the stabilities and convergence properties and numerical performances of these methods.In Chapter 2, based on the monotone numerical flux and piecewise linear reconstruction method for derivative, we construct a finite difference scheme-MUSCL scheme for time depen-dent Hamilton-Jacobi equations. It is proved that the MUSCL scheme possesses the property of TVB(Total Variational Bounded) stability in one dimensional case. We also do extensive nu-merical experiments. The results show that it is a second order scheme and free from spurious oscillations. In addition, it has good resolution at the corner-like discontinuity.In Chapter 3, we present a relaxation Lax-Friedrichs sweeping method (RLxFSM) for static Hamilton-Jacobi equations with arbitrary spatial dimensions. The method generalizes the LxFSM scheme and includes the LxFSM scheme as a special case. In the RLxFSM scheme we use SOR relaxation iteration instead of Gauss-Seidel iteration in LxFSM. When the relaxation factorω=1, the RLxFSM scheme reduces to the LxFSM scheme. It is proved that the RLxFSM scheme enjoys some nice properties of the well-known Fast Sweeping Method [83], such as the non-increasing, monotonicity and order preserving properties. In addition, it inherits the main advantage of the LxFSM scheme and can deal with both convex and nonconvex Hamiltonians, no matter differentiable or not they are. The presented extensive numerical results show that the number of iterations is reduced evidently whenωis slightly larger than one.In Chapter 4, by incooperating the local discontinuous Galerkin (LDG) method and the conforming finite element method (CFEM) we develop a so-called LDG/CFEM coupled method, for solving convection-diffusion problems. The basic idea of the LDG/CFEM coupled method is to split the domain into two disjoint subdomains. Taking into accout the advantages of the LDG and CFEM methods, we adopt the LDG in the subdomain where the solution varies rapidly while the CFEM in the other subdomain due to its lower computational cost. The LDG/CFEM coupled method reserves the good stability property of the LDG method. Meanwhile, it has the lower computation cost advantage of the CFEM. The stability and a priori error estimate are established on the quasi-uniform mesh. The convergence rate of the LDG/CFEM coupled method is proved to be O(((?)1/2+h1/2)hk) in an associated DG-norm, where (?) is the diffusion coefficient, h is the mesh size and k is the degree of polynomial. The reported numerical results verify the sharpness of our theorectical results. Although in the present work, we have only analyzed the LDG case in the DG-CFEM coupled approach, our method can be generalized to all other DG methods belong to the unify framework of Arnold et al [106]. In Chapter 5, we further investigate the stability and the uniform convergence of the LDG/CFEM coupled method proposed in Chaper 4 for one dimensional singularly perturbed problem with convection-diffusion type on Shishkin mesh. In the case of linear element, we give a simple way to prove the uniform convergence. Specifically, we need not to use the solution decomposition technique, which is generally necessary in the proof of the uniform convergence on layer-adapted meshes. Instead, by the use of the interpolation operator introduced by To-biska in [91], we first prove the uniform convergence of the LDG/CFEM coupled method for higher order element case. We show that the uniform convergence rate is O(N-k Ink N) in an associated DG-norm, where k≥1 is the degree of polynomial, N is the degrees of freedom of the Shishkin mesh of the solution domain. Our numerical results are in accordance with the theoretical results.In Chapter 6, we further analyze the stability and the uniform convergence of the LDG/CFEM coupled method proposed in Chaper 4 for two dimensional singularly perturbed problem with convection-diffusion type on Shishkin mesh. Due to the difficulty of the gener-alization of the interpolation operator of Tobiska to two dimensional problems, we focus our attention to the uniform convergence for the bilinear element. We show that the uniform conver-gence rate is O(N-1 ln N) in an associated DG-norm. Numerical results support the theorectial result. Moreover, they seem to indicate an optimal uniform convergence under the L2 norm. In other words, it is likely that the uniform convergence rate is O(N-2) under the L2 norm.