High-Effectiveness and -Accuracy Difference Scheme Based on Nonuniform Grids for Solving Convection–Diffusion Equations with Boundary Layers

Abstract

In this paper, some rational high-accuracy compact finite difference schemes on nonuniform grids (NRHOC) are introduced for solving convection–diffusion equations. The derived NRHOC schemes not only can suppress the oscillatory property of numerical solutions but can also obtain a high-accuracy approximate solution, and they can effectively solve the convection–diffusion problem with boundary layers by flexibly adjusting the discrete grid, which can be obtained with the singularity in the computational region. Three numerical experiments with boundary layers are conducted to verify the accuracy of the proposed NRHOC schemes. We compare the computed results with the analytical solutions, the results of the rational high-accuracy compact finite difference schemes on uniform grids (RHOC), and the other schemes in the literature. For all test problems, good computed results are obtained with the presented NRHOC schemes. It is shown that the presented NRHOC schemes have a better resolution for the solution of convection-dominated problems.