Performance Analysis of Effective Methods for Solving Band Matrix SLAEs after Parabolic Nonlinear PDEs

TL;DR

This paper compares the performance of direct, symbolic, and iterative algorithms, including the SIP method, for solving band matrix SLAEs from parabolic nonlinear PDEs using HPC clusters, highlighting their efficiencies and limitations.

Contribution

It introduces an experimental performance analysis of multiple algorithms, including a novel application of the Hotelling-Bodewig iterative method for solving band matrix SLAEs.

Findings

Iterative SIP method with ILU(0) shows competitive performance.

Hotelling-Bodewig algorithm offers an alternative to standard substitutions.

Performance varies significantly across different HPC systems.

Abstract

This paper presents an experimental performance study of implementations of three different types of algorithms for solving band matrix systems of linear algebraic equations (SLAEs) after parabolic nonlinear partial differential equations -- direct, symbolic, and iterative, the former two of which were introduced in Veneva and Ayriyan (arXiv:1710.00428v2). An iterative algorithm is presented -- the strongly implicit procedure (SIP), also known as the Stone method. This method uses the incomplete LU (ILU(0)) decomposition. An application of the Hotelling-Bodewig iterative algorithm is suggested as a replacement of the standard forward-backward substitutions. The upsides and the downsides of the SIP method are discussed. The complexity of all the investigated methods is presented. Performance analysis of the implementations is done using the high-performance computing (HPC) clusters "HybriLIT" and "Avitohol". To that purpose, the experimental setup and the results from the conducted computations on the individual computer systems are presented and discussed.