A Unified Framework for Double Sweep Methods for the Helmholtz Equation

TL;DR

This paper presents a unified theoretical framework for double sweep methods used as preconditioners in solving the Helmholtz equation, enabling comparison and development of new algorithms.

Contribution

It unifies the derivation and convergence analysis of double sweep methods, allowing for theoretical comparison and the introduction of a new sweeping algorithm.

Findings

Framework enables comparison of existing methods

Numerical tests validate theoretical predictions

New sweeping algorithm proposed and analyzed

Abstract

We consider sweeping domain decomposition preconditioners to solve the Helmholtz equation in the case of stripwise domain decomposition with or without overlaps. We unify their derivation and convergence studies by expressing them as Jacobi, Gauss-Seidel, and Symmetric Gauss-Seidel methods for different numbering of the unknowns. The proposed framework enables theoretical comparisons between the double sweep methods in [Nataf and Nier (1997), Vion and Geuzaine (2018)] and those in [Stolk (2013, 2017), Vion and Geuzaine (2014)]. Additionally, it facilitates the introduction of a new sweeping algorithm. We provide numerical test cases to assess the validity of the theoretical studies.