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 paper derives the optimal relaxation parameter for the SOR method applied to discretized Poisson equations with various boundary conditions on rectangular grids with unequal mesh sizes, enhancing convergence efficiency.

Contribution

It provides the first explicit formulas for the optimal relaxation parameter for mixed boundary conditions and different discretization schemes on non-uniform grids.

Findings

Optimal relaxation parameters are derived for various boundary conditions.

Numerical verification confirms the effectiveness of the optimal parameters.

Results improve convergence speed of the SOR method in practical applications.

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 ($\Delta x \neq \Delta y$) 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.