Scheduled Relaxation Jacobi schemes for non-elliptic partial differential equations

TL;DR

This paper extends the Scheduled Relaxation Jacobi method to non-elliptic PDEs by developing schemes that ensure convergence for nonsymmetric linear systems, demonstrated on advection-diffusion equations.

Contribution

The paper introduces a methodology for constructing SRJ schemes suitable for non-elliptic PDEs, broadening the method's applicability beyond elliptic problems.

Findings

Schemes accelerate convergence for 1D and 2D advection-diffusion equations.

Error attenuation is guaranteed for eigenvalues in specific complex plane regions.

Method extends SRJ applicability to nonsymmetric linear systems.

Abstract

The Scheduled Relaxation Jacobi (SRJ) method is a linear solver algorithm which greatly improves the convergence of the Jacobi iteration through the use of judiciously chosen relaxation factors (an SRJ scheme) which attenuate the solution error. Until now, the method has primarily been used to accelerate the solution of elliptic PDEs (e.g. Laplace, Poisson's equation) as the currently available schemes are restricted to solving this class of problems. The goal of this paper is to present a methodology for constructing SRJ schemes which are suitable for solving non-elliptic PDEs (or equivalent, nonsymmetric linear systems arising from the discretization of these PDEs), thereby extending the applicability of this method to a broader class of problems. These schemes are obtained by numerically solving a constrained minimization problem which guarantees the solution error will not grow as long as the linear system has eigenvalues which lie in certain regions of the complex plane. We demonstrate that these schemes are able to accelerate the convergence of standard Jacobi iteration for the nonsymmetric linear systems arising from discretization of the 1D and 2D steady advection-diffusion equations.