Parallel implicit solvers for 2D numerical models on structured meshes

Authors: ZHANG, YAOXIN - University Of Mississippi; AL-HAMDAN, MOHAMMAD - University Of Mississippi; CHAO, XIAOBO - University Of Mississippi

Submitted to: Mathematics
Publication Type: Peer Reviewed Journal
Publication Acceptance Date: 7/9/2024
Publication Date: 7/12/2024
Citation: Zhang, Y., Al-Hamdan, M., Chao, X. 2024. Parallel implicit solvers for 2D numerical models on structured meshes. Mathematics. 12(14):2184. https://doi.org/10.3390/math12142184.
DOI: https://doi.org/10.3390/math12142184

Interpretive Summary: The movement of sediments with water is called sediment transport. It is a common phenomenon in rivers, reservoirs, and lakes. Sediment transport can be described by mathematic equations, so we can develop mathematic models to solve those equations. However, it usually takes long time to run sediment transport models on a computer's CPU (Central Process Unit) for events that occur over long periods of time. To shorten the run time, a device other than CPU can be used, defined as a general-purpose GPU (Graphic Process Unit). The GPU has the capability to break down larger problems into smaller parts and then solves those smaller parts at the same time often defined as parallel computing. This study proposes two parallel methods to solve mathematic equations, so models can process information much faster on GPU. The proposed parallel methods can speed up the process as much as 2 to 3 times. This approach can provide information to decision-makers quickly to guide the implementation of practices to address sediment deposition and erosion in rivers, reservoirs and gullies.

Technical Abstract: 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 CUDA on GPU, the P-ADI solver adopted two mapping methods: one block-thread with-iterations (OBM-it) and multi-block-threads (MBM), 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 new 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, CCHE2D (Center of Computational Hydroscinece and Engineering), model, developed by National Center for Computational Hydro-science and Engineering at University of Mississippi. Comparisons of these two parallel solvers and their efficiency were demonstrated by examples and application. Both parallel solvers demonstrated higher efficiency than their serial counterparts on CPU (Central Processing Unit). In general, the P-ADI solver is faster than the P-SIP solver; and for the P-SIP solver, the new mapping method WF-all significantly improved the conventional mapping method WF-e.