The optimal relaxation parameter for the SOR method applied to the Poisson equation on rectangular grids with different types of boundary conditions

TL;DR

This work develops explicit optimal SOR relaxation parameters for solving the Poisson equation on rectangular grids with $\Delta x \neq \Delta y$ under Dirichlet, Neumann, and Robin boundary conditions, for both point and line SOR. It analyzes central second-order and high-order compact (HOC) discretizations using separation of variables to obtain spectral-radius-based expressions, with closed-form $\omega_{opt}$ for the Dirichlet case and a quartic- or perturbation-based approach for the HOC point-SOR. A detailed treatment of Neumann and Robin boundaries is provided, including domain-extension techniques and eigenvalue conditions that determine admissible $k_x$ and hence $\omega_{opt}$; numerical experiments confirm faster convergence when using the derived optimal parameters. The results offer practical guidelines for configuring SOR solvers on rectangular grids with mixed BC, enabling more efficient simulations in engineering and physics applications that rely on Poisson solvers. The methodology emphasizes direct spectral analysis and perturbation techniques to achieve accurate, low-cost estimates of the optimal relaxation parameter.

Abstract

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 ($Δx \neq Δy$) 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.

Figures (12)

• Figure 1: Numerical grid (left) and natural row-wise ordering (Right).
• Figure 2: Update state of the neighboring points of $(i,j)$ for the second-order (left) and HOC (right) schemes.
• Figure 3: Number of iterations for the point SOR method of the second-order scheme (\ref{['e:central:sor']}) for different values of $\omega$. The dash-dotted line indicates the optimal value (\ref{['e:opt:2nd']}).
• Figure 4: Number of iterations for the line SOR method of the second-order scheme (\ref{['e:lsor:central2']}) for different values $\omega$. The dash-dotted line indicates the optimal value (\ref{['e:opt:lsor']}).
• Figure 5: Number of iterations for the line SOR method of the HOC scheme (\ref{['e:lsor:hoc']}) for different values of $\omega$. The dash-dotted line indicates the optimal value (\ref{['e:opt:lsor:hoc']}).