A fast parallel algorithm for solving block-tridiagonal systems of linear equations including the domain decomposition method

TL;DR

This paper introduces a new parallel algorithm for efficiently solving block-tridiagonal systems of linear equations, enhancing the domain decomposition method and achieving near-linear speedup on supercomputers.

Contribution

It generalizes the parallel dichotomy algorithm for block-tridiagonal systems and implements a parallel domain decomposition method with improved efficiency.

Findings

Achieves near-linear speedup with increasing processors.

Effective for solving engineering problems on supercomputers.

Improves computational efficiency of acoustic wave field calculations.

Abstract

In this study, we develop a new parallel algorithm for solving systems of linear algebraic equations with the same block-tridiagonal matrix but with different right-hand sides. The method is a generalization of the parallel dichotomy algorithm for solving systems of linear equations with tridiagonal matrices \cite{terekhov:Dichotomy}. Using this approach, we propose a parallel realization of the domain decomposition method (\mbox{the Schur} complement method). The calculation of acoustic wave fields using the spectral-difference technique improves the efficiency of the parallel algorithms. A near-linear dependence of the speedup with the number of processors is attained using both several and several thousands of processors. This study is innovative because the parallel algorithm developed for solving block-tridiagonal systems of equations is an effective and simple set of procedures for solving engineering tasks on a supercomputer.