Exponential high-order compact finite difference method for convection-dominated diffusion problems on nonuniform grids

In this article, an exponential high-order compact (EHOC) difference method is introduced for solving the convection-dominated convection diffusion problems on nonuniform grid without any coordinate transformation from the physical space to the computational space. The derived EHOC schemes on nonuniform grid can not only preserve nonoscillation property and yield high accuracy approximation solutions, but also efficiently handle the convection diffusion problems with boundary layers by employing a flexible discretized grid that can be adapted to the singularity in the domain. To demonstrate the performances in computational accuracy, efficiency, and stability of the proposed EHOC schemes, several problems with boundary or interior layers where sharp gradients may appear are solved numerically. The results obtained by the present EHOC schemes on nonuniform grid are compared with analytical solutions and those using the EHOC on uniform grid and other methods. The present EHOC schemes on nonuniform grid provide excellent results for all test problems. It is shown that the proposed EHOC schemes on nonuniform grid are accurate, efficient, and stable, as well as have the advantage of better-scale resolution for convection-dominated diffusion problems.