Abstract
Numerical simulation of multi-physical processes requires a lot of processor time, especially when solving ill-conditional linear systems arising in fluid dynamics problems. This paper is devoted to the development of efficient parallel methods for such systems for FireStar3D wildfire modeling code. Two alternative approaches are discussed and analyzed, based on the MILU-preconditioned conjugate gradient method and on the algebraic multigrid, respectively. The main difficulties of parallelizing these methods are considered and solutions are presented: in the first case, nested twisted factorization with a staircase pipelining, and in the second, a multicolor technique for a new smoother for strongly anisotropic grids. A novel quasi-geometric interpolation technique is presented for solving the problem of positive off-diagonal matrix entries in the multigrid. The limits of applicability of the methods are determined depending on their flexibility, robustness and parallelization capabilities. The performance comparison demonstrates the superiority of the new methods over the widely used variants of the traditional conjugate gradient method.
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References
Morvan D, Accary G, Meradji S, Frangieh N, Bessonov O (2018) A 3D physical model to study the behavior of vegetation fires at laboratory scale. Fire Safety J 101:39–53. https://doi.org/10.1016/j.firesaf.2018.08.011
Frangieh N, Morvan D, Meradji S, Accary G, Bessonov O (2018) Numerical simulation of grassland fires behavior using an implicit physical multiphase model. Fire Safety J 102:37–47. https://doi.org/10.1016/j.firesaf.2018.06.004
Saad Y (2000) Iterative methods for sparse linear systems. PWS Publishing, Boston
Shewchuk JR (1994) An introduction to the Conjugate gradient method without the agonizing pain. Carnegie Mellon University, Pittsburgh, School of Computer Science
Bessonov O (2013) Parallelization properties of preconditioners for the Conjugate gradient methods. In: Malyshkin, V (ed.) PaCT 2013. LNCS, vol. 7979, pp. 26–36. Springer, Heidelberg. https://doi.org/10.1007/978-3-642-39958-9_3
Stüben K (1999) Algebraic multigrid (AMG): an introduction with applications. CMD Report 70, GMD – Forschungszentrum Informationstechnik GmbH, Sankt Augustin
Stüben K (2001) A review of algebraic multigrid. J Comput Appl Math 128:281–309. https://doi.org/10.1016/S0377-0427(00)00516-1
Brezina M, Falgout R, MacLachlan S, Manteuffel T, McCormick S, Ruge J (2006) Adaptive algebraic multigrid. SIAM J Sci Comput 27(4):1261–1286. https://doi.org/10.1137/040614402
Axelsson O (1986) Analysis of incomplete matrix factorizations as multigrid smoothers for vector and parallel computers. Appl Math Comput 19:3–22. https://doi.org/10.1016/0096-3003(86)90094-9
Llorente IM, Melson ND (1998) Robust multigrid smoothers for three dimensional elliptic equations with strong anisotropies. Technical Report 98-37, ICASE
Baker A, Falgout R, Gamblin T, Kolev T, Schulz M, Yang U (2015) Scaling algebraic multigrid solvers: on the road to exascale. In: Bischof C, Hegering HG, Nagel W, Wittum G (eds) Competence in High Performance Computing 2010, pp.215–226. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-24025-6_18
Rupp K, Weinbub J, Rudolf F, Morhammer A, Grasser T, Jüngel A (2015) A performance comparison of algebraic multigrid preconditioners on CPUs, GPUs, and Xeon Phis. Institute for Microelectronics, TU Wien
Xu J, Zikatanov L (2017) Algebraic Multigrid Methods. Acta Numerica 26:591–721. https://doi.org/10.1017/S0962492917000083
Gustafsson I (1978) A class of first order factorization methods. BIT 18:142–156. https://doi.org/10.1007/BF01931691
Accary G, Bessonov O, Fougère D, Gavrilov K, Meradji S, Morvan D (2009) Efficient parallelization of the preconditioned Conjugate gradient method. In: Malyshkin, V (ed.) PaCT 2009. LNCS, vol. 5698, pp. 60–72. Springer, Heidelberg. https://doi.org/10.1007/978-3-642-03275-2_7
Bessonov O (2015) Highly parallel multigrid solvers for multicore and manycore processors. In: Malyshkin, V (ed.) PaCT 2015. LNCS, vol. 9251, pp. 10–20. Springer, Heidelberg. https://doi.org/10.1007/978-3-319-21909-7_2
Bessonov O, Meradji S (2019) Efficient parallel solvers for the FireStar3D wildfire numerical simulation model. In: Malyshkin, V (ed.) PaCT 2019. LNCS, vol. 11657, pp. 140–150. Springer, Heidelberg. https://doi.org/10.1007/978-3-030-25636-4_11
Patankar SV (1980) Numerical heat transfer and fluid flow. Hemisphere Publishing, New York
Versteeg H, Malalasekera W (2007) An introduction to Computational Fluid Dynamics: the Finite method. Prentice Hall, Harlow
Moukalled F, Darwish M (2000) A unified formulation of the segregated class of algorithms for fluid flow at all speed. Numer Heat Transfer, Part B 37:103–139. https://doi.org/10.1080/104077900275576
van der Vorst HA (1987) Large tridiagonal and block tridiagonal linear systems on vector and parallel computers. Par Comp 5:45–54. https://doi.org/10.1016/0167-8191(87)90005-6
Elizarova T, Chetverushkin B (1992) Implementation of multiprocessor transputer system for computer simulation of computational physics problems. Math Model 4(11):75–100 (in Russian)
Frangieh N, Accary G, Morvan D, Meradji S, Bessonov O (2020) Wildfires front dynamics: 3D structures and intensity at small and large scales. Combust Flame 211:54–67. https://doi.org/10.1016/j.firesaf.2018.08.0110
Acknowledgements
This work was supported by the Russian State Assignment under Contract No. AAAA-A20-120011690131-7. Centre de Calcul Intensif d’Aix-Marseille (France) is acknowledged for granting access to its high-performance computing resources.
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Bessonov, O., Meradji, S. Parallel modeling of wildfires using efficient solvers for ill-conditioned linear systems. J Supercomput 77, 9365–9379 (2021). https://doi.org/10.1007/s11227-021-03632-8
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DOI: https://doi.org/10.1007/s11227-021-03632-8