×

zbMATH — the first resource for mathematics

Task-based parallelization of an implicit kinetic scheme. (English) Zbl 1408.35143
Summary: In this paper, we present and implement the palindromic discontinuous Galerkin (PDG) method in dimensions higher than one. The method has already been exposed and tested in [the third author et al., in: Finite volumes for complex applications VIII – hyperbolic, elliptic and parabolic problems. Cham: Springer. 171–178 (2017; Zbl 1365.76251)] in the one-dimensional context. The PDG method is a general implicit high order method for approximating systems of conservation laws. It relies on a kinetic interpretation of the conservation laws containing stiff relaxation terms. The kinetic system is approximated with an asymptotic-preserving high order DG method. We describe the parallel implementation of the method, based on the StarPU runtime library. Then, we apply it on preliminary test cases.
MSC:
35Q35 PDEs in connection with fluid mechanics
65M60 Finite element, Rayleigh-Ritz and Galerkin methods for initial value and initial-boundary value problems involving PDEs
65Y10 Numerical algorithms for specific classes of architectures
Software:
Gmsh; StarPU; KLU
PDF BibTeX Cite
Full Text: DOI
References:
[1] Denise Aregba-Driollet and Roberto Natalini. Discrete kinetic schemes for multidimensional systems of conservation laws. SIAM Journal on Numerical Analysis, 37(6):1973–2004, 2000. · Zbl 0964.65096
[2] Cédric Augonnet, Olivier Aumage, Nathalie Furmento, Raymond Namyst, and Samuel Thibault. StarPU-MPI: Task Programming over Clusters of Machines Enhanced with Accelerators. In Siegfried Benkner Jesper Larsson Träff and Jack Dongarra, editors, EuroMPI 2012, volume 7490 of LNCS. Springer, September 2012. Poster Session.
[3] François Bouchut, François Golse, and Mario Pulvirenti. Kinetic equations and asymptotic theory. Elsevier, 2000.
[4] David Coulette, Emmanuel Franck, Philippe Helluy, Michel Mehrenberger, and Laurent Navoret. Palindromic Discontinuous Galerkin Method, pages 171–178. Springer International Publishing, Cham, 2017. · Zbl 1365.76251
[5] Timothy A Davis and Ekanathan Palamadai Natarajan. Algorithm 907: KLU, a direct sparse solver for circuit simulation problems. ACM Transactions on Mathematical Software (TOMS), 37(3):36, 2010. · Zbl 1364.65066
[6] Christophe Geuzaine and Jean-François Remacle. Gmsh: A 3-D finite element mesh generator with built-in pre-and postprocessing facilities. International Journal for Numerical Methods in Engineering, 79(11):1309–1331, 2009. · Zbl 1176.74181
[7] Benjamin Graille. Approximation of mono-dimensional hyperbolic systems: A lattice Boltzmann scheme as a relaxation method. Journal of Computational Physics, 266:74–88, 2014. · Zbl 1310.76145
[8] Ernst Hairer, Christian Lubich, and Gerhard Wanner. Geometric numerical integration: structure-preserving algorithms for ordinary differential equations, volume 31. Springer Science & Business Media, 2006. · Zbl 1094.65125
[9] Claes Johnson, Uno Nävert, and Juhani Pitkäranta. Finite element methods for linear hyperbolic problems. Computer methods in applied mechanics and engineering, 45(1):285–312, 1984. · Zbl 0526.76087
[10] Robert I McLachlan and G Reinout W Quispel. Splitting methods. Acta Numerica, 11:341–434, 2002. · Zbl 1105.65341
[11] Salli Moustafa, Mathieu Faverge, Laurent Plagne, and Pierre Ramet.
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.