Singular Perturbation Finite Difference Method For Convection-diffusion Problems

Differential equations with a small parameter e multiplying the highest order derivative terms are said to be singularly perturbed and normally boundary layers occur in their solutions. Singularly perturbed differential equations are ubiquitous in mathematical problems in the sciences and engineering. For example,the Navier-Stokes equations of fluid flow at high Reynolds number, the equations governing flow in porous media, the drift diffusion equations of semiconductor device physics,and mathematical models of liquid crystal materials and of chemical reactions.Classical numerical methods usually give unsatisfactory numerical results when the singular perturbation parameter e is small.In particular, the pointwise errors in numerical methods based on centered or upwinded differences on uniform meshes depend inversely on a power of e. In this paper we consruct finite differnce method on layer-adapted meshes for solving singularly perturbed convection-diffusion equations whose convergence behavior in the global maximum norm is ε-uniform. The numerical results are clear illustrations of the convergence estimate. They indicate that the theoretical results are fairly sharp.The paper is organized as follows. In chapter 1, a singularly perturbed convection-diffusion problems in one dimension is studied. First we derive a general classification for layer-adapted meshes. Second we study the stability of the continuous operator and the properties of the exact solution. Finally, based on different stability properties of the first-order upwind difference scheme, we give three convergence analysises on layer-adapted meshes.In chapter 2, for one dimensional problems we analyse second-order accurate difference schemes. Our hybrid difference scheme uses central difference on the fine part of the mesh and the midpoint upwind scheme on the coarse part. We show that the hybrid scheme is almost second-order convergent in the maximum norm which is independent of singular perturbation parameter. We also present two convergence acceleration techniques: Richardson extrapolation and defect correction to get higher-order schemes. Finally, a singularly perturbed convection diffusion problem with a discontinuous convection coefficient is considered. Due to the discontinuity an interior layer appears in the solution. The problem is solved using a hybrid difference scheme on a Shishkin mesh.In chapter 3, a singularly perturbed quasilinear two-point boundary value problem is considered.The problem is discretized using two upwind difference schemes...