The DUNE-FEM-DG module.

  • Andreas Dedner (Author)
    University of Warwick
  • Stefan Girke (Author)
    University of Münster
  • Robert Klöfkorn (Author)
    International Research Institute of Stavanger
  • Tobias Malkmus (Author)
    University of Freiburg

Identifiers (Article)

Abstract

In this paper we discuss the new publicly released Dune-Fem-DG module. This module provides highly ecient implementations of the Discontinuous Galerkin (DG) method for solving a wide range of non linear partial dierential equations (PDE). The interfaces used are highly flexible and customizable, providing for example mechanisms for using distributed parallelization, local grid adaptivity with dynamic load balancing, and check pointing. We discuss methods for solving stationary problems as well as a matrix-free implementation for time dependent problems. Both parabolic and first order hyperbolic PDE are discussed in detail including models for compressible and incompressible flows, i.e., the compressible Navier-Stokes equations.


For the spatial discretization a wide range of DG methods are implemented ranging from the standard interior penalty method to methods like LDG and CDG2. Upwinding numerical fluxes for first order terms are also available, including limiter bases stabilization for convection dominated PDEs. For the temporal discretization Runge-Kutta methods are used including higher order explicit, diagonally implicit and IMEX schemes. We discuss asynchronous communication, shared memory parallelization, and automated code generation which combined result in a high floating point performance of the code.

Statistics

loading

References

MPI: A Message-Passing Interface Standard. Version 2.2. High Performance Computing Center Stuttgart (HLRS), 2009.

J. Ahrens, B. Geveci, C. Law, C. Hansen, and C. Johnson.

-paraview: An end-user tool for large-data visualization, 2005.

M. Alkämper, A. Dedner, R. Klöfkorn, and M. Nolte.The DUNE-ALUGrid Module. Archive of Numerical Software, 4(1):1-28, 2016.

URL http://dx.doi.org/10.11588/ans.2016.1.23252.

D. Arnold, F. Brezzi, B. Cockburn, and L. Marini.

Unified analysis of discontinuous Galerkin methods for elliptic problems.

SIAM J. Numer. Anal., 39(5):1749-1779, 2002.

S. Balay, W. D. Gropp, L. C. McInnes, and B. F. Smith.

Efficient management of parallelism in object oriented numerical software libraries.In E. Arge, A. M. Bruaset, and H. P. Langtangen, editors, Modern Software Tools in Scientific Computing, pages 163-202. Birkhäuser Press, 1997.

S. Balay, S. Abhyankar, M. F. Adams, J. Brown, P. Brune, K. Buschelman, L. Dalcin, V. Eijkhout, W. D. Gropp, D. Kaushik, M. G. Knepley, L. C. McInnes, K. Rupp, B. F. Smith, S. Zampini, and H. Zhang. PETSc users manual.Technical Report ANL-95/11 - Revision 3.6, Argonne National Laboratory, 2015.URL http://www.mcs.anl.gov/petsc.

W. Bangerth, R. Hartmann, and G. Kanschat. deal.II – a general purpose object oriented finite element library. ACM Trans. Math. Softw., 33(4):24/1–24/27, 2007. URL http://dealii.org/.

P. Bastian, M. Blatt, A. Dedner, C. Engwer, R. Klöfkorn, R. Kornhuber, M. Ohlberger, and O. Sander. A Generic Grid Interface for Parallel and Adaptive Scientific Computing. Part II: Implementation and Tests in DUNE. Computing, 82(2-3):121-138, 2008a.URL http://dx.doi.org/10.1007/s00607-008-0004-9.

P. Bastian, M. Blatt, A. Dedner, C. Engwer, R. Klöfkorn, M. Ohlberger, and O. Sander. A Generic Grid Interface for Parallel and Adaptive Scientific Computing. Part I: Abstract Framework.Computing, 82(2-3):103-119, 2008b.

URL http://dx.doi.org/10.1007/s00607-008-0003-x.

P. Bastian, F. Heimann, and S. Marnach. Generic implementation of finite element methods in the Distributed and Unified Numerics Environment (DUNE). Kybernetika, 46:294–315, 2010.

M. Blatt and P. Bastian.The iterative solver template library.

In B. Kagström, E. Elmroth, J. Dongarra, and J. Wasniewski, editors, Applied Parallel Computing - State of the Art in Scientific Computing, pages 666-675, Berlin/Heidelberg, 2007. Springer.

S. Brdar. A higher order locally adaptive discontinuous Galerkin approach for atmospheric simulations.PhD thesis, Albert-Ludwigs-Universität Freiburg, 2012.https://www.freidok.uni-freiburg.de/data/8862.

S. Brdar, A. Dedner, and R. Klöfkorn. Compact and Stable Discontinuous Galerkin Methods with Application to Atmospheric Flows. In I. K. et al., editor, Computational Methods in Science and Engineering: Proceedings of the Workshop SimLabs@KIT, pages 109-116. KIT Scientific Publishing, 2011a.URL http://dx.doi.org/10.5445/KSP/1000023323.

S. Brdar, A. Dedner, R. Klöfkorn, M. Kränkel, and D. Kröner. Simulation of Geophysical Problems with DUNE-FEM. In E. K. et al., editor, Computational Science and High Performance Computing IV, volume 115, pages 93-106. Springer, 2011b.URL http://dx.doi.org/10.1007/978-3-642-17770-5_8.

S. Brdar, A. Dedner, and R. Klöfkorn.Compact and stable Discontinuous Galerkin methods for convection-diffusion problems.SIAM J. Sci. Comput., 34(1):263-282, 2012a.URL http://dx.doi.org/10.1137/100817528.

S. Brdar, A. Dedner, and R. Klöfkorn. CDG Method for Navier-Stokes Equations.In S. Jiang and T. Li, editors, Hyperbolic Problems - Theory, Numerics and Applications, pages 320-327. World Scientific Publishing Co Pte Ltd, 2012b.

S. Brdar, M. Baldauf, A. Dedner, and R. Klöfkorn.Comparison of dynamical cores for NWP models: comparison of COSMO and DUNE. Theoretical and Computational Fluid Dynamics, 27(3-4):453-472, 2013. URL http://dx.doi.org/10.1007/s00162-012-0264-z.

A. Burri, A. Dedner, D. Diehl, R. Klöfkorn, and M. Ohlberger.

A general object oriented framework for discretizing non-linear evolution equations.In Y. S. et al., editor, Advances in High Performance Computing and Computational Sciences, volume 93, pages 69-87. Springer, 2006.

URL http://dx.doi.org/10.1007/978-3-540-33844-4_7.

T. A. Davis.Algorithm 832: UMFPACK v4.3--an unsymmetric-pattern multifrontal method.ACM Trans. Math. Softw., 30(2):196-199, 2004.

URL http://dx.doi.org/10.1145/992200.992206.

A. Dedner and J. Giesselmann. A posteriori analysis of fully discrete method of lines dg schemes for systems of conservation laws. arXiv preprint 1510.05430, 2015. URL http://arxiv.org/abs/1510.05430.

A. Dedner and Klöfkorn. On Efficient Time Stepping using the Discontinuous Galerkin Method for Numerical Weather Prediction. In Advances in Parallel Computing, volume 27, pages 627 – 636. Springer, 2016. URL http://dx.doi.org/10.3233/978-1-61499-621-7-627.

A. Dedner and R. Klöfkorn.The compact discontinuous Galerkin method for elliptic problems.In Finite volumes for complex applications V, pages 761-776. ISTE, London, 2008.

A. Dedner and R. Klöfkorn.Stabilization for Discontinuous Galerkin Methods Applied to Systems of Conservation Laws.In E. T. et al., editor, Proc. of the 12th International Conference on Hyperbolic Problems, Proceedings of Symposia in Applied Mathematics 67, Part 1, 253-268, 2009.

A. Dedner and R. Klöfkorn. A Generic Stabilization Approach for Higher Order Discontinuous Galerkin Methods for Convection Dominated Problems.

J. Sci. Comput., 47(3):365-388, 2011. URL http://dx.doi.org/10.1007/s10915-010-9448-0.

A. Dedner, R. Klöfkorn, and D. Kröner. Higher Order Adaptive and Parallel Simulations Including Dynamic Load Balancing with the Software Package DUNE. In W. N. et al., editor, High Performance Computing in Science and Engineering '09, pages 229-239. Springer, 2010a. URL http://dx.doi.org/10.1007/978-3-642-04665-0_16.

A. Dedner, R. Klöfkorn, M. Nolte, and M. Ohlberger. A generic interface for parallel and adaptive scientific computing: Abstraction principles and the DUNE-FEM.Computing, 90(3–4):165–196, 2010b. URL http://dx.doi.org/10.1007/s00607-010-0110-3.

A. Dedner, M. Fein, R. Klöfkorn, D. Kröner, D. Lebiedz, J. Siehr, and J. Unger. On the computation of slow manifolds in chemical kinetics via optimization and their use as reduced models in reactive flow systems.

In Proceedings of the 13th International Conference on Numerical Combustion, 2011a.

A. Dedner, D. Kröner, and N. Shokina. Computational Science and High Performance Computing IV: The 4th Russian-German Advanced Research Workshop, Freiburg, Germany, 2009, chapter Adaptive Modelling of Two-Dimensional Shallow Water Flows with Wetting and Drying, pages 1–15. Springer, 2011b. URL http://dx.doi.org/10.1007/978-3-642-17770-5_1.

A. Dedner, M. Fein, R. Klöfkorn, and D. Lebiedz.On the use of chemistry-based slow invariant manifolds in discontinuous Galerkin methods for reactive flows. Technical report, Institute für Numerische Mathematik, Universität Ulm, 2013. URL https://www.uni-ulm.de/fileadmin/website_uni_ulm/mawi2/forschung/preprint-server/2013/1302_dednerfeinkloefkorn.pdf.

A. Dedner, R. Klöfkorn, and M. Kränkel.Continuous Finite-Elements on Non-Conforming Grids Using Discontinuous Galerkin Stabilization. In J. F. et al., editor, Finite Volumes for Complex Applications VII, volume 77 of Springer Proceedings in Mathematics & Statistics, pages 207-215. Springer, 2014. URL http://dx.doi.org/10.1007/978-3-319-05684-5_19.

R. Eymard, G. Henry, R. Herbin, F. Hubert, R. Klöfkorn, and G. Manzini.

D Benchmark on Discretization Schemes for Anisotropic Diffusion Problems on General Grids. In J. Fort, J. Fürst, J. Halama, R. Herbin, and F. Hubert, editors, Finite Volumes for Complex Applications VI Problems & Perspectives, volume 4 of Springer Proceedings in Mathematics, pages 895-930. Springer Berlin Heidelberg, 2011. URL http://dx.doi.org/10.1007/978-3-642-20671-9_89.

The Feel++ Consortium. The Feel++ Book. 2015. URL https://www.gitbook.com/book/feelpp/feelpp-book.

G. J. Gassner. Discontinuous Galerkin Methods for the Unsteady Compressible Navier-Stokes Equations.PhD thesis, Universität Stuttgart, 2009. URL http://elib.uni-stuttgart.de/opus/volltexte/2009/3948/.

S. Girke. Parallel and Efficient Simulation of Atherosclerotic Plaque Formation Using Higher Order Discontinuous Galerkin Schemes.

PhD thesis (work in progress), University of Münster, 2017.

S. Girke, R. Klöfkorn, and M. Ohlberger. Efficient Parallel Simulation of Atherosclerotic Plaque Formation Using Higher Order Discontinuous Galerkin Schemes. In J. F. et al., editor, Finite Volumes for Complex Applications VII, volume 78 of Springer Proceedings in Mathematics & Statistics, pages 617-625. Springer, 2014. URL http://dx.doi.org/10.1007/978-3-319-05591-6_61.

G. Guennebaud, B. Jacob, et al. Eigen v3, 2010.

URL http://eigen.tuxfamily.org.

G. Karniadakis and S. Sherwin. Spectral/hp element methods for computational fluid dynamics. Oxford

University Press, 2005. URL http://www.nektar.info/.

R. Klöfkorn. Numerics for Evolution Equations -- A General Interface Based Design Concept. PhD thesis, Albert-Ludwigs-Universität Freiburg, 2009.

URL https://www.freidok.uni-freiburg.de/data/7175.

R. Klöfkorn. Benchmark 3D: The Compact Discontinuous Galerkin 2 Scheme. In J. Fort, J. Fürst, J. Halama, R. Herbin, and F. Hubert, editors, Finite Volumes for Complex Applications VI Problems & Perspectives, volume 4 of Springer Proceedings in Mathematics, pages 1023-1033. Springer Berlin Heidelberg, 2011. URL http://dx.doi.org/10.1007/978-3-642-20671-9_100.

R. Klöfkorn. Efficient Matrix-Free Implementation of Discontinuous Galerkin Methods for Compressible Flow Problems. In A. H. et al., editor, Proceedings of the ALGORITMY 2012, pages 11-21, 2012.

R. Klöfkorn and M. Nolte. Performance pitfalls in the dune grid interface.

In A. Dedner, B. Flemisch, and R. Klöfkorn, editors, Advances in DUNE, pages 45-58. Springer Berlin Heidelberg, 2012. URL http://dx.doi.org/10.1007/978-3-642-28589-9_4.

R. Klöfkorn and M. Nolte. Solving the Reactive Compressible Navier-Stokes Equations in a Moving Domain. In K. Binder, G. Münster, and M. Kremer, editors, NIC Symposium 2014 - Proceedings, volume 47. John von Neumann Institute for Computing Jülich, 2014.

D. A. Knoll and D. E. Keyes. Jacobian-free Newton-Krylov methods: a survey of approaches and applications. J. Comput. Phys., 193(2):357-397, 2004. URL http://dx.doi.org/10.1016/j.jcp.2003.08.010.

D. Kröner. Numerical Schemes for Conservation Laws. Wiley & Teubner, Stuttgart, 1997.

T. Malkmus. Fluid-Structure-Interaction -- Simulation of Non-Newtonian Fluid Interacting with Thin Ellastic Shells. PhD thesis, Albert-Ludwigs-Universität Freiburg, 2016.

K. Schloegel, G. Karypis, and V. Kumar. Wavefront Diffusion and LMSR: Algorithms for Dynamic Repartitioning of Adaptive Meshes. IEEE Transactions on Parallel and Distributed Systems, 12(5):451-466, 2001.

D. Schuster, S. Brdar, M. Baldauf, A. Dedner, R. Klöfkorn, and D. Kröner. On discontinuous Galerkin approach for atmospheric flow in the mesoscale with and without moisture. Meteorologische Zeitschrift, 23(4):449-464, 2014.

URL http://dx.doi.org/10.1127/0941-2948/2014/0565.

D. Terpstra, H. Jagode, H. You, and J. Dongarra. Collecting Performance Data with PAPI-C. In Tools for High Performance Computing 2009, pages 157–173, Dresden, Germany, 2009.

J. Treibig, G. Hager, and G. Wellein. Likwid: A lightweight performance-oriented tool suite for x86 multicore environments. In Proceedings of PSTI2010, the First International Workshop on Parallel Software Tools and Tool Infrastructures, San Diego CA, 2010.

V. Weaver and J. Dongarra. Can Hardware Performance Counters Produce Expected, Deterministic Results? In 3rd Workshop on Functionality of Hardware Performance Monitoring, Atlanta, GA, December 4, 2010.

Supplementary Content

Published
2017-04-03
Language
en
Academic discipline and sub-disciplines
Scientific Computing, Mathematics,
Keywords
Numerical Software, Dune, Discontinuous Galerkin Schemes, Hyperbolic problems, Elliptic problems, Parabolic problems, Euler, Navier-Stokes, Advection-Diffusion, Stokes, Poisson