Geodesic
Development of a parallel algorithm based on an implicit scheme for the discontinuous Galerkin method for solving diffusion type equations
Žurnal Srednevolžskogo matematičeskogo obŝestva, Tome 22 (2020) no. 1, pp. 94-106.

Voir la notice de l'article provenant de la source Math-Net.Ru

The paper presents a numerical parallel algorithm based on an implicit scheme for the Galerkin method with discontinuous basis functions for solving diffusion-type equations on triangular grids. To apply the Galerkin method with discontinuous basis functions, the initial equation of parabolic type is transformed to a system of partial differential equations of the first order. To do this, auxiliary variables are introduced, which are the components of the gradient of the desired function. To store sparse matrices and vectors, the CSR format is used in this study. The resulting system is solved numerically using a parallel algorithm based on the Nvidia AmgX library. A numerical study is carried out on the example of solving two-dimensional test parabolic initial-boundary value problems. The presented numerical results show the effectiveness of the proposed algorithm for solving parabolic problems.
Mots-clés : parabolic equations, implicit scheme
Keywords: discontinuous Galerkin method, Nvidia AmgX. 
@article{SVMO_2020_22_1_a6,
     author = {R. V. Zhalnin and N. A. Kuzmin and V. F. Masyagin},
     title = {Development of a parallel algorithm based on an implicit scheme for the discontinuous {Galerkin} method for solving diffusion type equations},
     journal = {\v{Z}urnal Srednevol\v{z}skogo matemati\v{c}eskogo ob\^{s}estva},
     pages = {94--106},
     publisher = {mathdoc},
     volume = {22},
     number = {1},
     year = {2020},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/SVMO_2020_22_1_a6/}
}
TY  - JOUR
AU  - R. V. Zhalnin
AU  - N. A. Kuzmin
AU  - V. F. Masyagin
TI  - Development of a parallel algorithm based on an implicit scheme for the discontinuous Galerkin method for solving diffusion type equations
JO  - Žurnal Srednevolžskogo matematičeskogo obŝestva
PY  - 2020
SP  - 94
EP  - 106
VL  - 22
IS  - 1
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/SVMO_2020_22_1_a6/
LA  - ru
ID  - SVMO_2020_22_1_a6
ER  - 
%0 Journal Article
%A R. V. Zhalnin
%A N. A. Kuzmin
%A V. F. Masyagin
%T Development of a parallel algorithm based on an implicit scheme for the discontinuous Galerkin method for solving diffusion type equations
%J Žurnal Srednevolžskogo matematičeskogo obŝestva
%D 2020
%P 94-106
%V 22
%N 1
%I mathdoc
%U http://geodesic.mathdoc.fr/item/SVMO_2020_22_1_a6/
%G ru
%F SVMO_2020_22_1_a6
R. V. Zhalnin; N. A. Kuzmin; V. F. Masyagin. Development of a parallel algorithm based on an implicit scheme for the discontinuous Galerkin method for solving diffusion type equations. Žurnal Srednevolžskogo matematičeskogo obŝestva, Tome 22 (2020) no. 1, pp. 94-106. http://geodesic.mathdoc.fr/item/SVMO_2020_22_1_a6/

[1] M. E. Ladonkina, O. Yu. Milyukova, V. F. Tishkin, “Numerical algorithm for solving diffusion-type equations on the basis of multigrid methods”, Comput. Math. Math. Phys., 50:8 (2010), 1367–-1390 (In Russ.) | MR | Zbl

[2] M. M. Krasnov, P. A. Kuchugov, M. E. Ladonkina, V. F. Tishkin, “Discontinuous Galerkin method on three-dimensional tetrahedral grids: Using the operator programming method”, Math. Models Comput. Simul., 9:5 (2017), 529–543 | DOI | MR | Zbl

[3] P. B. Bogdanov, A. V. Gorobets, S. A. Sukov, “Adaptation and optimization of basic operations for an unstructured mesh CFD algorithm for computation on massively parallel accelerators”, Comput. Math. Math. Phys., 53:8 (2013), 1183–1194 (In Russ.) | DOI | DOI | MR | Zbl

[4] J. Brannick, Y. Chen, X. Hu, L. Zikatanov, “Parallel unsmoothed aggregation algebraic multigrid algorithms on GPUs, in Numerical Solution of Partial Differential Equations: Theory, Algorithms, and Their Applications”, Springer Proc. Math. Stat., 45 (2013), 81–102 (In English) | MR | Zbl

[5] H. De Sterck, U. M. Yang, J. Heys, “Reducing complexity in parallel algebraic multigrid preconditioners”, SIAM Journal on Matrix Analysis and Applications, 27:4 (2006), 1019–1039 (In English) | DOI | MR | Zbl

[6] R. Barett, M. W. Berry, T. F. Chan, J. Demmel, J. Donato, J. Dongarra, V. Eijkhout, R. Pozo, C. Romine, H. van der Vorst, Templates for the Solution of Linear Systems: Building Blocks for Iterative Methods, SIAM., Philadelphia, PA., 1993, 117 pp. (In English)

[7] Y. Saad, “A flexible inner-outer preconditioned GMRES algorithm”, SIAM J. Sci. Comput, 14:2 (1993), 461–469 (In English) | DOI | MR | Zbl

[8] V. Simoncini, D. B. Szyld, “Flexible inner-outer Krylov subspace methods”, SIAM J. Numer. Anal., 40:6 (2003), 2219–2239 (In English) | DOI | MR | Zbl

[9] M. Naumov, M. Arsaev, J. Cohen, P. Castonguay, J. Eaton, J. Demouth, S. Layton, N. Markovskiy, I. Z. Reguly, N. Sakharnykh, V. Sellappan, R. Strzodka, “AmgX: A Library for GPU Accelerated Algebraic Multigrid and Preconditioned Iterative Methods”, SIAM J. Sci. Comput., 37:5 (2015), 602–626 (In English) | DOI | MR

[10] J. Gao, K. Wu, Y. Wang, P. Qi, G. He, “GPU-accelerated preconditioned GMRES method for two-dimensional Maxwell's equations”, International Journal of Computer Mathematics, 94:10 (2017), 1–24 | DOI | MR

[11] R. V. Zhalnin, M. E. Ladonkina, V. F. Masyagin, V. F. Tishkin, “Discontinuous finite-element Galerkin method for numerical solution of parabolic problems in anisotropic media on triangle grids”, Bulletin of the South Ural State University. Ser. Mathematical Modelling, Programming Computer Software, 9:3 (2016), 144–151 (In Russ.) | Zbl

[12] C. Fletcher, Computational Galerkin methods, Mir, Moscow, 1988, 352 pp. (In Russ.)

[13] F. A. Bassi, S. Rebay, “High-Order Accurate Discontinuous Finite Element Method for the Numerical Solution of the Compressible Navier – Stokes Equations”, Journal of Computational Physics, 131:2 (1997), 267–279 (In English) | DOI | MR | Zbl

[14] B. Cockburn, C.-W. Shu, “The local discontinuous Galerkin method for timedependent convection-diffusion systems”, SIAM J. Numer. Anal., 35:6 (1998), 2440–2463 (In English) | DOI | MR | Zbl

[15] H. Wang, C.-W. Shu, Q. Zhang, “Local discontinuous Galerkin methods with implicit-explicit time-marching for multi-dimensional convection-diffusion problems”, ESAIM Mathematical Modelling and Numerical Analysis, 50:4 (2016), 1083–1105 (In English) | DOI | MR | Zbl