Parallelization of an unstructured Navier–Stokes solver using a multi-color ordering method for OpenMP

•A multi-color Gauss–Seidel (MCGS) method is developed for OpenMP parallelization.•Using MCGS, an unstructured-grid Navier–Stokes solver is parallelized.•An algorithm for painting multi-colors is developed for MCGS.•The uniqueness of the solution of MCGS is confirmed with double precision accuracy....

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Vydáno v:	Computers & Fluids Ročník 88; s. 496 - 509
Hlavní autoři:	Sato, Yohei, Hino, Takanori, Ohashi, Kunihide
Médium:	Journal Article
Jazyk:	angličtina
Vydáno:	Elsevier Ltd 01.12.2013 Elsevier BV
Témata:	Central processing units Computer simulation Mathematical analysis Mathematical models Multi color ordering method Multi grid method Navier-Stokes equations OpenMP Order disorder Parallel processing Parallelization Solvers Unstructured grid Unstructured grid OpenMP Multi grid method Multi color ordering method Parallelization
ISSN:	0045-7930, 1879-0747
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Abstract	•A multi-color Gauss–Seidel (MCGS) method is developed for OpenMP parallelization.•Using MCGS, an unstructured-grid Navier–Stokes solver is parallelized.•An algorithm for painting multi-colors is developed for MCGS.•The uniqueness of the solution of MCGS is confirmed with double precision accuracy. A multi-color ordering method has been developed for the Gauss–Seidel (GS) method in the framework of the unstructured-grid based Navier–Stokes equations solver using OpenMP. The multi-color ordering method is required to avoid the data race condition in do-loop parallelization and to achieve the uniqueness of a solution of GS. A coloring algorithm of painting neighbor cells with different colors is proposed for the multi-color ordering method. The method is tested for four sample simulation cases: one case of two-dimensional simulation and three cases of three-dimensional simulation. Through the sample simulations, the uniqueness of the solution of the Multi-Color ordering Gauss Seidel (MCGS) method is verified, and the convergence ratio of MCGS is found to be in the similar level to that of GS and better than the Jacobi method. The parallel efficiency is examined for workstations with two hexa-core CPUs or two octa-core CPUs. Although the parallel efficiency is dependent on computer systems and simulation cases, the speed up ratio of MCGS reaches 14 using two octa-core CPUs in the maximum case using 14 million cells.
AbstractList	•A multi-color Gauss–Seidel (MCGS) method is developed for OpenMP parallelization.•Using MCGS, an unstructured-grid Navier–Stokes solver is parallelized.•An algorithm for painting multi-colors is developed for MCGS.•The uniqueness of the solution of MCGS is confirmed with double precision accuracy. A multi-color ordering method has been developed for the Gauss–Seidel (GS) method in the framework of the unstructured-grid based Navier–Stokes equations solver using OpenMP. The multi-color ordering method is required to avoid the data race condition in do-loop parallelization and to achieve the uniqueness of a solution of GS. A coloring algorithm of painting neighbor cells with different colors is proposed for the multi-color ordering method. The method is tested for four sample simulation cases: one case of two-dimensional simulation and three cases of three-dimensional simulation. Through the sample simulations, the uniqueness of the solution of the Multi-Color ordering Gauss Seidel (MCGS) method is verified, and the convergence ratio of MCGS is found to be in the similar level to that of GS and better than the Jacobi method. The parallel efficiency is examined for workstations with two hexa-core CPUs or two octa-core CPUs. Although the parallel efficiency is dependent on computer systems and simulation cases, the speed up ratio of MCGS reaches 14 using two octa-core CPUs in the maximum case using 14 million cells. A multi-color ordering method has been developed for the Gauss-Seidel (GS) method in the framework of the unstructured-grid based Navier-Stokes equations solver using OpenMP. The multi-color ordering method is required to avoid the data race condition in do-loop parallelization and to achieve the uniqueness of a solution of GS. A coloring algorithm of painting neighbor cells with different colors is proposed for the multi-color ordering method. The method is tested for four sample simulation cases: one case of two-dimensional simulation and three cases of three-dimensional simulation. Through the sample simulations, the uniqueness of the solution of the Multi-Color ordering Gauss Seidel (MCGS) method is verified, and the convergence ratio of MCGS is found to be in the similar level to that of GS and better than the Jacobi method. The parallel efficiency is examined for workstations with two hexa-core CPUs or two octa-core CPUs. Although the parallel efficiency is dependent on computer systems and simulation cases, the speed up ratio of MCGS reaches 14 using two octa-core CPUs in the maximum case using 14 million cells.
Author	Ohashi, Kunihide Hino, Takanori Sato, Yohei
Author_xml	– sequence: 1 givenname: Yohei orcidid: 0000-0001-5313-5767 surname: Sato fullname: Sato, Yohei email: yohei.sato@psi.ch organization: Paul Scherrer Institute, Villigen PSI 5232, Switzerland – sequence: 2 givenname: Takanori orcidid: 0000-0002-5811-7688 surname: Hino fullname: Hino, Takanori organization: Yokohama National University, Japan – sequence: 3 givenname: Kunihide surname: Ohashi fullname: Ohashi, Kunihide organization: National Maritime Research Institute, Japan
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Cites_doi	10.2514/6.1987-1137 10.1006/jcph.1994.1155 10.1016/0021-9991(77)90095-X 10.2534/jjasnaoe1968.1997.182_9 10.1016/j.compfluid.2010.12.024 10.1016/j.parco.2011.02.002 10.2514/6.1995-345 10.1016/0377-0427(91)90165-G 10.1016/j.jpdc.2004.11.010 10.1016/S0167-8191(84)90380-6 10.2514/2.392 10.1016/j.jpdc.2005.06.019 10.1016/S0021-9991(03)00194-3 10.1016/0021-9991(67)90037-X 10.1016/j.compfluid.2012.07.010 10.1146/annurev.fl.18.010186.002005 10.2534/jjasnaoe1968.1999.186_177 10.1016/j.compstruc.2004.04.014 10.1016/j.parco.2011.05.004 10.1137/0911045
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Snippet	•A multi-color Gauss–Seidel (MCGS) method is developed for OpenMP parallelization.•Using MCGS, an unstructured-grid Navier–Stokes solver is parallelized.•An... A multi-color ordering method has been developed for the Gauss-Seidel (GS) method in the framework of the unstructured-grid based Navier-Stokes equations...
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SubjectTerms	Central processing units Computer simulation Mathematical analysis Mathematical models Multi color ordering method Multi grid method Navier-Stokes equations OpenMP Order disorder Parallel processing Parallelization Solvers Unstructured grid
Title	Parallelization of an unstructured Navier–Stokes solver using a multi-color ordering method for OpenMP
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