Performance analysis of effective methods for solving band matrix SLAEs after parabolic nonlinear PDEs

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 in Effective methods for solving band SLAEs after parabolic nonlinear PDEs (2018) [3]. 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.