On the equivalence between the Scheduled Relaxation Jacobi method and Richardson's non-stationary method

TL;DR

This paper establishes the equivalence between the Scheduled Relaxation Jacobi method and a non-stationary Richardson's method, providing explicit optimal weights and demonstrating improved convergence for elliptic problems and high-order discretizations.

Contribution

It derives explicit optimal relaxation factors for SRJ, showing their equivalence to a non-stationary Richardson method, and offers practical benefits for high-order discretizations.

Findings

Optimal weights replace eigenvalue calculations with frequency analysis.

The method achieves fastest convergence for given grid structures.

Effective for high-order Laplacian discretizations without multi-coloring.

Abstract

The Scheduled Relaxation Jacobi (SRJ) method is an extension of the classical Jacobi iterative method to solve linear systems of equations ($Au=b$) associated with elliptic problems. It inherits its robustness and accelerates its convergence rate computing a set of $P$ relaxation factors that result from a minimization problem. In a typical SRJ scheme, the former set of factors is employed in cycles of $M$ consecutive iterations until a prescribed tolerance is reached. We present the analytic form for the optimal set of relaxation factors for the case in which all of them are different, and find that the resulting algorithm is equivalent to a non-stationary generalized Richardson's method. Our method to estimate the weights has the advantage that the explicit computation of the maximum and minimum eigenvalues of the matrix $A$ is replaced by the (much easier) calculation of the maximum and minimum frequencies derived from a von Neumann analysis. This set of weights is also optimal for the general problem, resulting in the fastest convergence of all possible SRJ schemes for a given grid structure. We also show that with the set of weights computed for the optimal SRJ scheme for a fixed cycle size it is possible to estimate numerically the optimal value of the parameter $\omega$ in the Successive Overtaxation (SOR) method in some cases. Finally, we demonstrate with practical examples that our method also works very well for Poisson-like problems in which a high-order discretization of the Laplacian operator is employed. This is of interest since the former discretizations do not yield consistently ordered $A$ matrices. Furthermore, the optimal SRJ schemes here deduced, are advantageous over existing SOR implementations for high-order discretizations of the Laplacian operator in as much as they do not need to resort to multi-coloring schemes for their parallel implementation. (abridged)