The Successive Over-Relaxation (SOR) method is a useful method for solving the sparse system of linear equations which arises from finite-difference discretization of the Poisson equation. Knowing the optimal value of the relaxation parameter is crucial for fast convergence. In this manuscript, we present the optimal relaxation parameter for the discretized Poisson equation with mixed and different types of boundary conditions on a rectangular grid with unequal mesh sizes in $x$- and $y$-directions ($\Delta x \neq \Delta y$) which does not addressed in the literature. The central second-order and high-order compact (HOC) schemes are considered for the discretization and the optimal relaxation parameter is obtained for both the point and line implementation of the SOR method. Furthermore, the obtained optimal parameters are verified by numerical results.

翻译：逐次超松弛（SOR）方法是求解泊松方程有限差分离散化所产生的稀疏线性方程组的一种有效方法。确定松弛参数的最优值对于实现快速收敛至关重要。本文针对矩形网格上$x$和$y$方向不等网格尺寸（$\Delta x \neq \Delta y$）且具有混合及不同类型边界条件的离散化泊松方程，提出了文献中尚未涉及的最优松弛参数。研究考虑了中心二阶格式和高阶紧致（HOC）格式进行离散化，并针对SOR方法的点迭代和线迭代两种实现方式分别推导了最优松弛参数。此外，通过数值结果验证了所得最优参数的有效性。