The convergence proof of the sixth-order compact 9-point FDM for the 2D transport problem

Abstract: 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.