A high-order nonuniform compact difference scheme coupled with adaptive mesh method for solving singularly perturbed convection-diffusion equations

A high-order nonuniform compact difference scheme coupled with an adaptive mesh method for solving singularly perturbed convection-diffusion equations is proposed. The core idea of the proposed method lies in constructing a nonuniform grid scheme with the same fourth-order accuracy at both inner and boundary points, and this construction is based on an adaptive mesh method. This approach effectively eliminates numerical oscillation near the boundaries, a common challenge in singularly perturbed problems. Specifically, we first formulate the fourth-order compact scheme on nonuniform grids, then integrate it with the adaptive mesh method and elaborate on the corresponding numerical implementation procedure. Finally, numerical experiments are conducted against exact solutions, with the proposed scheme further compared against three benchmark methods: the fourth-order compact scheme on uniform grids, the identical scheme on adaptive nonuniform grids, and the other established methods reported in existing literature. Results of all test cases demonstrate that the proposed scheme generates accurate and stable numerical solutions and exhibits enhanced resolution for singularly perturbed convection-diffusion problems.