The convergence proof of the sixth-order compact 9-point FDM for the 2D transport problem
Fuente:
arXiv
Saved in:
| Main Author: | |
|---|---|
| Format: | Preprint |
| Published: |
2025
|
| Subjects: | |
| Online Access: | |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866908407990059008 |
|---|---|
| author | Feng, Qiwei |
| author_facet | Feng, Qiwei |
| contents | It is widely acknowledged that the convergence proof of the error in the $l_{\infty}$ 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 $l_{\infty}$ 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2506_12549 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | The convergence proof of the sixth-order compact 9-point FDM for the 2D transport problem Feng, Qiwei Numerical Analysis It is widely acknowledged that the convergence proof of the error in the $l_{\infty}$ 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 $l_{\infty}$ 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. |
| title | The convergence proof of the sixth-order compact 9-point FDM for the 2D transport problem |
| topic | Numerical Analysis |
| url | https://arxiv.org/abs/2506.12549 |