Extendible and efficient Python framework for solving evolution equations with stabilized discontinuous Galerkin methods. (English) Zbl 07534235

Summary: This paper discusses a Python interface for the recently published Dune-Fem-DG module which provides highly efficient implementations of the discontinuous Galerkin (DG) method for solving a wide range of nonlinear partial differential equations (PDEs). Although the C++ interfaces of Dune-Fem-DG are highly flexible and customizable, a solid knowledge of C++ is necessary to make use of this powerful tool. With this work, easier user interfaces based on Python and the unified form language are provided to open Dune-Fem-DG for a broader audience. The Python interfaces are demonstrated for both parabolic and first-order hyperbolic PDEs.


65M08 Finite volume methods for initial value and initial-boundary value problems involving PDEs
65M60 Finite element, Rayleigh-Ritz and Galerkin methods for initial value and initial-boundary value problems involving PDEs
35Q31 Euler equations
35Q90 PDEs in connection with mathematical programming
68N99 Theory of software
Full Text: DOI


[1] Alnæs, M.S., Logg, A., Ølgaard, K.B., Rognes, M.E., Wells, G.N.: Unified form language: adomain-specific language for weak formulations of partial differential equations. CoRR abs/1211.4047 (2012). arxiv: 1211.4047 · Zbl 1308.65175
[2] Andersson, C.; Führer, C.; Åkesson, J., Assimulo: a unified framework for ODE solvers, Math. Comput. Simul., 116, 26-43 (2015) · Zbl 07313385
[3] Balay, S., Abhyankar, S., Adams, M.F., Brown, J., Brune, P., Buschelman, K., Dalcin, L., Dener, A., Eijkhout, V., Gropp, W.D., Karpeyev, D., Kaushik, D., Knepley, M.G., May, D.A., McInnes, L.C., Mills, R.T., Munson, T., Rupp, K., Sanan, P., Smith, B.F., Zampini, S., Zhang, H., Zhang, H.: PETSc users manual. Tech. Rep. ANL-95/11 - Revision 3.14, Argonne National Laboratory (2020). https://www.mcs.anl.gov/petsc
[4] Bangerth, W.; Hartmann, R.; Kanschat, G., deal.II - A general-purpose object-oriented finite element library, ACM Trans. Math. Softw., 33, 4, 24/1-24/27 (2007) · Zbl 1365.65248
[5] Bastian, P.; Blatt, M.; Dedner, A.; Engwer, C.; Klöfkorn, R.; Kornhuber, R.; Ohlberger, M.; Sander, O., A generic grid interface for parallel and adaptive scientific computing. Part II: implementation and tests in DUNE, Computing, 82, 2-3, 121-138 (2008) · Zbl 1151.65088
[6] Bastian, P.; Blatt, M.; Dedner, M.; Dreier, NA; Engwer, Ch; Fritze, R.; Gräser, C.; Grüninger, C.; Kempf, D.; Klöfkorn, R.; Ohlberger, M.; Sander, O., The Dune framework: basic concepts and recent developments, Comput. Math. Appl., 81, 75-112 (2008) · Zbl 07288707
[7] Birken, P.; Gassner, GJ; Versbach, LM, Subcell finite volume multigrid preconditioning for high-order discontinuous Galerkin methods, Int. J. Comput. Fluid Dyn., 33, 9, 353-361 (2019) · Zbl 07474499
[8] Brdar, S.; Baldauf, M.; Dedner, A.; Klöfkorn, R., Comparison of dynamical cores for NWP models: comparison of COSMO and DUNE, Theoretical Comput. Fluid Dyn., 27, 3-4, 453-472 (2013)
[9] Brdar, S.; Dedner, A.; Klöfkorn, R., Compact and stable discontinuous Galerkin methods for convection-diffusion problems, SIAM J. Sci. Comput., 34, 1, 263-282 (2012) · Zbl 1251.65137
[10] Chen, L.; Li, R., An integrated linear reconstruction for finite volume scheme on unstructured grids, J. Sci. Comput., 68, 1172-1197 (2016) · Zbl 1365.65203
[11] Chen, T.; Shu, C-W, Review article: review of entropy stable discontinuous Galerkin methods for systems of conservation laws on unstructured simplex meshes, CSIAM Trans. Appl. Math., 1, 1, 1-52 (2020)
[12] Cheng, Y.; Li, F.; Qiu, J.; Xu, L., Positivity-preserving DG and central DG methods for ideal MHD equations, J. Comput. Phys., 238, 255-280 (2013) · Zbl 1286.76162
[13] Cockburn, B.; Shu, C-W, Runge-Kutta discontinuous Galerkin methods for convection-dominated problems, J. Sci. Comput., 16, 3, 173-261 (2001) · Zbl 1065.76135
[14] Dedner, A.; Girke, S.; Klöfkorn, R.; Malkmus, T., The DUNE-FEM-DG module, ANS (2019)
[15] Dedner, A.; Kane, B.; Klöfkorn, R.; Nolte, M., Python framework for hp-adaptive discontinuous Galerkin methods for two-phase flow in porous media, AMM, 67, 179-200 (2019) · Zbl 1481.65180
[16] Dedner, A., Kloefkorn, R., Nolte, M.: Python bindings for the DUNE-FEM module (2020). doi:10.5281/zenodo.3706994
[17] Dedner, A.; Klöfkorn, R., A generic stabilization approach for higher order discontinuous Galerkin methods for convection dominated problems, J. Sci. Comput., 47, 3, 365-388 (2011) · Zbl 1229.65175
[18] Dedner, A., Klöfkorn, R.: The DUNE-FEM-DG Module. https://gitlab.dune-project.org/dune-fem/dune-fem-dg (2019)
[19] Dedner, A.; Klöfkorn, R.; Klöfkorn, R.; Keilegavlen, E.; Radu, FA; Fuhrmann, J., A Python framework for solving advection-diffusion problems, Finite Volumes for Complex Applications IX - Methods, Theoretical Aspects, Examples, 695-703 (2020), Cham: Springer International Publishing, Cham · Zbl 1454.65084
[20] Dedner, A.; Klöfkorn, R.; Nolte, M.; Ohlberger, M., A generic interface for parallel and adaptive scientific computing: abstraction principles and the DUNE-FEM module, Computing, 90, 3-4, 165-196 (2010) · Zbl 1201.65178
[21] Dedner, A.; Makridakis, C.; Ohlberger, M., Error control for a class of Runge-Kutta discontinuous Galerkin methods for nonlinear conservation laws, SIAM J. Numer. Anal., 45, 2, 514-538 (2007) · Zbl 1145.65072
[22] Dedner, A.; Nolte, M.; Dedner, A.; Flemisch, B.; Klöfkorn, R., Construction of local finite element spaces using the generic reference elements, Advances in DUNE, 3-16 (2012), Berlin, Heidelberg: Springer, Berlin, Heidelberg
[23] Dedner, A., Nolte, M.: The Dune-Python Module. CoRR abs/1807.05252 (2018). arxiv: 1807.05252
[24] Discacciati, N.; Hesthaven, JS; Ray, D., Controlling oscillations in high-order discontinuous Galerkin schemes using artificial viscosity tuned by neural networks, J. Comput. Phys., 409, 109304 (2020) · Zbl 1435.65156
[25] Dolejší, V.; Feistauer, M.; Schwab, C., On some aspects of the discontinuous Galerkin finite element method for conservation laws, Math. Comput. Simul., 61, 3, 333-346 (2003) · Zbl 1013.65108
[26] Dumbser, M.; Balsara, DS; Toro, EF; Munz, CD, A unified framework for the construction of one-step finite volume and discontinuous Galerkin schemes on unstructured meshes, J. Comput. Phys., 227, 18, 8209-8253 (2008) · Zbl 1147.65075
[27] The Feel++ Consortium: The Feel++ Book (2015). https://www.gitbook.com/book/feelpp/feelpp-book
[28] Feistauer, M.; Kučera, V.; Benzoni-Gavage, S.; Serre, D., A new technique for the numerical solution of the compressible Euler equations with arbitrary Mach numbers, Hyperbolic Problems: Theory, Numerics and Applications, 523-531 (2008), Berlin, Heidelberg: Springer, Berlin, Heidelberg · Zbl 1138.65090
[29] Gottlieb, S.; Shu, C-W; Tadmor, E., Strong stability-preserving high-order time discretization methods, SIAM Rev., 43, 1, 89-112 (2001) · Zbl 0967.65098
[30] Guermond, JL; Pasquetti, R.; Popov, B., Entropy viscosity method for nonlinear conservation laws, J. Comput. Phys., 230, 11, 4248-4267 (2011) · Zbl 1220.65134
[31] Hindenlang, F.; Gassner, GJ; Altmann, C.; Beck, A.; Staudenmaier, M.; Munz, CD, Explicit discontinuous Galerkin methods for unsteady problems, Comput. Fluids, 61, 86-93 (2012) · Zbl 1365.76117
[32] Hönig, J.; Koch, M.; Rüde, U.; Engwer, C.; Köstler, H.; Foster, I.; Joubert, GR; Kucera, L.; Nagel, WE; Peters, F., Unified generation of DG-kernels for different HPC frameworks, Advances in Parallel Computing, 376-386 (2020), IOS Press BV
[33] Houston, P.; Sime, N., Automatic symbolic computation for discontinuous Galerkin finite element methods, SIAM J. Sci. Comput., 40, 3, C327-C357 (2018) · Zbl 1397.65268
[34] Karniadakis, G.; Sherwin, S., Spectral/hp Element Methods for Computational Fluid Dynamics (2005), New York: Oxford University Press, New York · Zbl 1116.76002
[35] Ketcheson, DI, Highly efficient strong stability-preserving Runge-Kutta methods with low-storage implementations, SIAM J. Sci. Comput., 30, 4, 2113-2136 (2008) · Zbl 1168.65382
[36] Klieber, W.; Rivière, B., Adaptive simulations of two-phase flow by discontinuous Galerkin methods, Comput. Methods Appl. Mech. Eng., 196, 1-2, 404-419 (2006) · Zbl 1120.76327
[37] Klöckner, A.; Warburton, T.; Hesthaven, JS, Viscous shock capturing in a time-explicit discontinuous Galerkin method, Math. Model. Nat. Phenom., 6, 3, 57-83 (2011) · Zbl 1220.65165
[38] Klöfkorn, R.; Handlovicova, A., Efficient matrix-free implementation of discontinuous Galerkin methods for compressible flow problems, Proceedings of the ALGORITMY 2012, 11-21 (2012), SlovakiaSlovakia: Slovak University of Technology in Bratislava, Publishing House of STU, SlovakiaSlovakia · Zbl 1278.35157
[39] Klöfkorn, R.; Kvashchuk, A.; Nolte, M., Comparison of linear reconstructions for second-order finite volume schemes on polyhedral grids, Comput. Geosci., 21, 5, 909-919 (2017) · Zbl 1396.76057
[40] Knoll, DA; Keyes, DE, Jacobian-free Newton-Krylov methods: a survey of approaches and applications, J. Comput. Phys., 193, 2, 357-397 (2004) · Zbl 1036.65045
[41] Kopriva, DA; Gassner, G., On the quadrature and weak form choices in collocation type discontinuous Galerkin spectral element methods, J. Sci. Comput., 44, 136-155 (2010) · Zbl 1203.65199
[42] Kopriva, DA; Woodruff, SL; Hussaini, MY, Computation of electromagnetic scattering with a non-conforming discontinuous spectral element method, Int. J. Numer. Methods Eng., 53, 1, 105-122 (2002) · Zbl 0994.78020
[43] Krivodonova, L.; Xin, J.; Remacle, JF; Chevaugeon, N.; Flaherty, JE, Shock detection and limiting with discontinuous Galerkin methods for hyperbolic conservation laws, Appl. Numer. Math., 48, 3-4, 323-338 (2004) · Zbl 1038.65096
[44] Logg, A.; Mardal, KA; Wells, G., Automated Solution of Differential Equations by the Finite Element Method: the FEniCS Book (2012), Berlin: Springer Publishing Company Incorporated, Berlin · Zbl 1247.65105
[45] Mandli, KT; Ahmadia, AJ; Berger, M.; Calhoun, D.; George, DL; Hadjimichael, Y.; Ketcheson, DI, Lemoine, G.I., LeVeque, R.J.: Clawpack: building an open source ecosystem for solving hyperbolic PDEs, Peer J. Comput. Sci., 2, e68 (2013)
[46] May, S.; Berger, M., Two-dimensional slope limiters for finite volume schemes on non-coordinate-aligned meshes, SIAM J. Sci. Comput., 35, 5, A2163-A2187 (2013) · Zbl 1281.65116
[47] Persson, P.O., Peraire, J.: Sub-cell shock capturing for discontinuous Galerkin methods. In: 44th AIAA Aerospace Sciences Meeting and Exhibit, AIAA-2006-0112, Reno, Nevada (2006). doi:10.2514/6.2006-112
[48] Rathgeber, F.; Ham, DA; Mitchell, L.; Lange, M.; Luporini, F.; McRae, ATT; Bercea, GT; Markall, GR, Firedrake: automating the finite element method by composing abstractions, ACM Trans. Math. Softw., 43, 3, 24/1-24/7 (2016) · Zbl 1396.65144
[49] Schuster, D.; Brdar, S.; Baldauf, M.; Dedner, A.; Klöfkorn, R.; Kröner, D., On discontinuous Galerkin approach for atmospheric flow in the mesoscale with and without moisture, Meteorologische Zeitschrift, 23, 4, 449-464 (2014)
[50] Shu, C-W, High order WENO and DG methods for time-dependent convection-dominated PDEs: a brief survey of several recent developments, J. Comput. Phys., 316, 598-613 (2016) · Zbl 1349.65486
[51] Wallwork, JG; Barral, N.; Kramer, SC; Ham, DA; Piggott, MD, Goal-oriented error estimation and mesh adaptation for shallow water modelling, SN Appl. Sci., 2, 1053 (2020)
[52] Zhang, X.; Shu, C-W, On positivity-preserving high order discontinuous Galerkin schemes for compressible Euler equations on rectangular meshes, J. Comput. Phys., 229, 23, 8918-8934 (2010) · Zbl 1282.76128
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