Adaptive Numerical Method For Solving Singularly Perturbed Convection-diffusion Problems

Singularly perturbed convection-diffusion problems arise in many branches of science and engineering such as modelling of water quality problems in river networks, fluid flow at high Reynolds numbers, convective heat transport field problem in moving media, financial modelling and turbulence model. Such problems eventually can be summarized for various boundary conditions under a variety of different forms of singular perturbation boundary value problem of existence, uniqueness and analytic studies. The characteristic of such problems is that there is a perturbation parameter in differential equation, this parameter can be naturally occurring to reflect certain physical properties, and also be artificially introduced. The solution of such problems changes sharp in thin region of the solution area. Singularly perturbed convection-diffusion problems arise in a lot of important fields, and these problems have attracted many scholars to study.In this paper, there are two parts. In the first part, we discuss the midpoint upwind scheme on the Bakhvalov-Shishkin mesh for solving singularly perturbed convection-diffusion two-point boundary value problems. Its elaborate ε-uniform pointwise convergence is proved by using the comparison principle and the barrier functions. The numerical experiments support the theoretical result and indicate that the estimate is sharp. Additionally, the comparison of numerical results on the Bakhvalov-Shishkin mesh and the Shishkin mesh is provided. In the second part, we propose a class of hybrid difference schemes on the Bakhvalov-Shishkin mesh for quasilinear singularly perturbed convection-diffusion boundary value problems. The ε-uniform second-order convergence in the maximum norm is proved. The numerical experiments support the schemes and the second-order estimate.