Parallel Implicit Solvers for 2D Numerical Models on Structured Meshes

• Published in: Mathematics (Basel) Vol. 12; no. 14; p. 2184
• Main Authors: Zhang, Yaoxin, Al-Hamdan, Mohammad Z., Chao, Xiaobo
• Format: Journal Article
• Language: English
• Published: Basel MDPI AG 01.07.2024
• Subjects: ADI solver; Algorithms; Analysis; Central processing units; Computational grids; CPUs; Differential equations; Efficiency; Finite volume method; Flow simulation; Fluid dynamics; Graphics processing units; implicit; Linear systems; Load; Mapping; Mathematical analysis; Mathematical models; Microprocessors; Numerical models; parallel; Partial differential equations; Sediment transport; Semiconductor industry; Simulation; SIP solver; Solvers; structured mesh; Turbulence models; Velocity; Viscosity; Wave fronts; United States
• ISSN: 2227-7390; 2227-7390
• DOI: 10.3390/math12142184

This paper presents the parallelization of two widely used implicit numerical solvers for the solution of partial differential equations on structured meshes, namely, the ADI (Alternating-Direction Implicit) solver for tridiagonal linear systems and the SIP (Strongly Implicit Procedure) solver for the penta-diagonal systems. Both solvers were parallelized using CUDA (Computer Unified Device Architecture) Fortran on GPGPUs (General-Purpose Graphics Processing Units). The parallel ADI solver (P-ADI) is based on the Parallel Cyclic Reduction (PCR) algorithm, while the parallel SIP solver (P-SIP) uses the wave front method (WF) following a diagonal line calculation strategy. To map the solution schemes onto the hierarchical block-threads framework of the CUDA on the GPU, the P-ADI solver adopted two mapping methods, one block thread with iterations (OBM-it) and multi-block threads (MBMs), while the P-SIP solver also used two mappings, one conventional mapping using effective WF lines (WF-e) with matrix coefficients and solution variables defined on original computational mesh, and a newly proposed mapping using all WF mesh (WF-all), on which matrix coefficients and solution variables are defined. Both the P-ADI and the P-SIP have been integrated into a two-dimensional (2D) hydrodynamic model, the CCHE2D (Center of Computational Hydroscience and Engineering) model, developed by the National Center for Computational Hydroscience and Engineering at the University of Mississippi. This study for the first time compared these two parallel solvers and their efficiency using examples and applications in complex geometries, which can provide valuable guidance for future uses of these two parallel implicit solvers in computational fluids dynamics (CFD). Both parallel solvers demonstrated higher efficiency than their serial counterparts on the CPU (Central Processing Unit): 3.73~4.98 speedup ratio for flow simulations, and 2.166~3.648 speedup ratio for sediment transport simulations. In general, the P-ADI solver is faster than but not as stable as the P-SIP solver; and for the P-SIP solver, the newly developed mapping method WF-all significantly improved the conventional mapping method WF-e.