2025/06/14 by Qiwei Feng, Feng, Qiwei
Engineering · Mathematics · #Differential Equations and Numerical Methods #FOS: Mathematics #Numerical Analysis (math.NA) #Optimization and Packing Problems #Traffic control and management
paper · pdf · doi:10.48550/arxiv.2506.12549
openalex publication_date 2025/06/14 · openalex created_date 2025/10/13 · openalex updated_date 2026/07/28
It is widely acknowledged that the convergence proof of the error in the l∞ norm of the high-order finite difference method (FDM) and finite element method (FEM) in 2D is challenging. In this paper, we derive the sixth-order compact 9-point FDM with the explicit stencil for the 2D transport problem with the constant coefficient and the Dirichlet boundary condition in a unit square. The proposed sixth-order FDM forms an M-matrix for the any mesh size h employing the uniform Cartesian mesh. The explicit formula of our FDM also enables us to construct the comparison function with the explicit expression to rigorously prove the sixth-order convergence rate of the maximum pointwise error by the discrete maximum principle. Most importantly, we demonstrate that the sixth-order convergence proof is valid for any mesh size h. The numerical results are consistent with sixth-order accuracy in the l∞ norm. Our theoretical convergence proof is clear and the proposed sixth-order FDM is straightforward to be implemented, facilitating the reproduction of our numerical results.