| This thesis focuses mainly on two problems in Numerical Conservation Laws: the problem of numerical boundary layers, and that of discrete shock profiles. First, we study the numerical solutions of relaxing scheme by matched multi-scale asymptotic expansions, formal analysis of the structure of numerical solutions caused by different boundary conditions, and using suitable entropy and anti-derivative method, we establish the stability of the numerical boundary layers rigorously, i.e. the boundary layers are shown to be localized. These analysis yield some clues to the problem: how to choose boundary conditions in the practical computations. Second, we study the asymptotic nonlinear stability of discrete shock profiles for the relaxing scheme approximating general system of nonlinear hyperbolic conservation laws. The existence of discrete shock profiles is established using a center manifold construction, and it is shown that weak discrete shock profiles for such scheme are nonlinearly stable in L2 norm, provided that the sums of the initial perturbations are equal to zero. These results are proved by weighted norm estimates and characteristic energy method based on the internal structures of the discrete shock profiles. Finally, for the application of relaxation scheme, we discuss the computational problem of phase transition, the difference between the relaxation scheme and the NT scheme is presented, and a high resolution method is proposed. |
