Extendible and Efcient Python Framework for Solving Evolution Equations with Stabilized Discontinuous Galerkin Methods

doi:10.1007/s42967-021-00134-5

摘要/Abstract

摘要： This paper discusses a Python interface for the recently published Dune-Fem-DG module which provides highly efcient implementations of the discontinuous Galerkin (DG) method for solving a wide range of nonlinear partial diferential equations (PDEs). Although the C++ interfaces of Dune-Fem-DG are highly fexible 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 unifed form language are provided to open Dune-Fem-DG for a broader audience. The Python interfaces are demonstrated for both parabolic and frst-order hyperbolic PDEs.

关键词: Dune, Dune-Fem, Discontinuous Galerkin, Finite volume, Python, Advection-difusion, Euler, Navier-Stokes

Abstract: This paper discusses a Python interface for the recently published Dune-Fem-DG module which provides highly efcient implementations of the discontinuous Galerkin (DG) method for solving a wide range of nonlinear partial diferential equations (PDEs). Although the C++ interfaces of Dune-Fem-DG are highly fexible 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 unifed form language are provided to open Dune-Fem-DG for a broader audience. The Python interfaces are demonstrated for both parabolic and frst-order hyperbolic PDEs.

Key words: Dune, Dune-Fem, Discontinuous Galerkin, Finite volume, Python, Advection-difusion, Euler, Navier-Stokes

中图分类号:

Andreas Dedner, Robert Klöfkorn. Extendible and Efcient Python Framework for Solving Evolution Equations with Stabilized Discontinuous Galerkin Methods[J]. Communications on Applied Mathematics and Computation, 2022, 4(2): 657-696.

参考文献

1.Alnæs, M.S., Logg, A., Ølgaard, K.B., Rognes, M.E., Wells, G.N.:Unifed form language:adomainspecifc language for weak formulations of partial diferential equations.CoRR abs/1211.4047 (2012).arxiv:1211.4047
2.Andersson, C., Führer, C., Åkesson, J.:Assimulo:a unifed framework for ODE solvers.Math.Comput.Simul.116, 26-43 (2015).https://doi.org/10.1016/j.matcom.2015.04.007
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 fnite element library.ACM Trans.Math.Softw.33(4), 24/1-24/27 (2007).https://doi.org/10.1145/1268776.1268779
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 scientifc computing.Part II:implementation and tests in DUNE.Computing 82(2/3), 121-138 (2008).https://doi.org/10.1007/s00607-008-0004-9
6.Bastian, P., Blatt, M., Dedner, M., Dreier, N.A., 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 (2021).https://doi.org/10.1016/j.camwa.2020.06.007
7.Birken, P., Gassner, G.J., Versbach, L.M.:Subcell fnite volume multigrid preconditioning for highorder discontinuous Galerkin methods.Int.J.Comput.Fluid Dyn.33(9), 353-361 (2019).https://doi.org/10.1080/10618562.2019.1667983
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).https://doi.org/10.1007/s00162-012-0264-z
9.Brdar, S., Dedner, A., Klöfkorn, R.:Compact and stable discontinuous Galerkin methods for convection-difusion problems.SIAM J.Sci.Comput.34(1), 263-282 (2012).https://doi.org/10.1137/10081 7528
10.Chen, L., Li, R.:An integrated linear reconstruction for fnite volume scheme on unstructured grids.J.Sci.Comput.68, 1172-1197 (2016).https://doi.org/10.1007/s10915-016-0173-1
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).https://doi.org/10.4208/csiam-am.2020-0003
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).https://doi.org/10.1016/j.jcp.2012.12.019
13.Cockburn, B., Shu, C.-W.:Runge-Kutta discontinuous Galerkin methods for convection-dominated problems.J.Sci.Comput.16(3), 173-261 (2001)
14.Dedner, A., Girke, S., Klöfkorn, R., Malkmus, T.:The DUNE-FEM-DG module.ANS (2017).https://doi.org/10.11588/ans.2017.1.28602
15.Dedner, A., Kane, B., Klöfkorn, R., Nolte, M.:Python framework for hp-adaptive discontinuous Galerkin methods for two-phase fow in porous media.AMM 67, 179-200 (2019).https://doi.org/10.1016/j.apm.2018.10.013
16.Dedner, A., Kloefkorn, R., Nolte, M.:Python bindings for the DUNE-FEM module (2020).https://doi.org/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).https://doi.org/10.1007/s10915-010-9448-0
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.:A Python framework for solving advection-difusion problems.In:Klöfkorn, R., Keilegavlen, E., Radu, F.A., Fuhrmann, J.(eds) Finite Volumes for Complex Applications IX-Methods, Theoretical Aspects, Examples, pp.695-703.Springer International Publishing, Cham (2020)
20.Dedner, A., Klöfkorn, R., Nolte, M., Ohlberger, M.:A generic interface for parallel and adaptive scientifc computing:abstraction principles and the DUNE-FEM module.Computing 90(3/4), 165-196 (2010).https://doi.org/10.1007/s00607-010-0110-3
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).https://doi.org/10.1137/050624248
22.Dedner, A., Nolte, M.:Construction of local fnite element spaces using the generic reference elements.In:Dedner, A., Flemisch, B., Klöfkorn, R.(eds) Advances in DUNE, pp.3-16.Springer, Berlin, Heidelberg (2012).https://doi.org/10.1007/978-3-642-28589-9_1
23.Dedner, A., Nolte, M.:The Dune-Python Module.CoRR abs/1807.05252 (2018).arxiv:1807.05252
24.Discacciati, N., Hesthaven, J.S., Ray, D.:Controlling oscillations in high-order discontinuous Galerkin schemes using artifcial viscosity tuned by neural networks.J.Comput.Phys.409, 109-304 (2020).https://doi.org/10.1016/j.jcp.2020.109304
25.Dolejší, V., Feistauer, M., Schwab, C.:On some aspects of the discontinuous Galerkin fnite element method for conservation laws.Math.Comput.Simul.61(3), 333-346 (2003).https://doi.org/10.1016/S0378-4754(02)00087-3
26.Dumbser, M., Balsara, D.S., Toro, E.F., Munz, C.D.:A unifed framework for the construction of one-step fnite volume and discontinuous Galerkin schemes on unstructured meshes.J.Comput.Phys.227(18), 8209-8253 (2008).https://doi.org/10.1016/j.jcp.2008.05.025
27.The Feel++ Consortium:The Feel++ Book (2015).https://www.gitbook.com/book/feelpp/feelpp-book
28.Feistauer, M., Kučera, V.:A new technique for the numerical solution of the compressible Euler equations with arbitrary Mach numbers.In:Benzoni-Gavage, S., Serre, D.(eds) Hyperbolic Problems:Theory, Numerics and Applications, pp.523-531.Springer, Berlin, Heidelberg (2008)
29.Gottlieb, S., Shu, C.-W., Tadmor, E.:Strong stability-preserving high-order time discretization methods.SIAM Rev.43(1), 89-112 (2001).https://doi.org/10.1137/S003614450036757X
30.Guermond, J.L., Pasquetti, R., Popov, B.:Entropy viscosity method for nonlinear conservation laws.J.Comput.Phys.230(11), 4248-4267 (2011).https://doi.org/10.1016/j.jcp.2010.11.043.(Special issue High Order Methods for CFD Problems)
31.Hindenlang, F., Gassner, G.J., Altmann, C., Beck, A., Staudenmaier, M., Munz, C.D.:Explicit discontinuous Galerkin methods for unsteady problems.Comput.Fluids 61, 86-93 (2012).https://doi.org/10.1016/j.compfuid.2012.03.006
32.Hönig, J., Koch, M., Rüde, U., Engwer, C., Köstler, H.:Unifed generation of DG-kernels for diferent HPC frameworks.In:Foster, I., Joubert, G.R., Kucera, L., Nagel, W.E., Peters F.(eds) Advances in Parallel Computing, vol.36, pp.376-386.IOS Press BV (2020).https://doi.org/10.3233/APC200062
33.Houston, P., Sime, N.:Automatic symbolic computation for discontinuous Galerkin fnite element methods.SIAM J.Sci.Comput.40(3), C327-C357 (2018).https://doi.org/10.1137/17M1129751
34.Karniadakis, G., Sherwin, S.:Spectral/hp Element Methods for Computational Fluid Dynamics.Oxford University Press, New York (2005).http://www.nektar.info/
35.Ketcheson, D.I.:Highly efcient strong stability-preserving Runge-Kutta methods with low-storage implementations.SIAM J.Sci.Comput.30(4), 2113-2136 (2008).https://doi.org/10.1137/07070485X
36.Klieber, W., Rivière, B.:Adaptive simulations of two-phase fow by discontinuous Galerkin methods.Comput.Methods Appl.Mech.Eng.196(1/2/3), 404-419 (2006).https://doi.org/10.1016/j.cma.2006.05.007
37.Klöckner, A., Warburton, T., Hesthaven, J.S.:Viscous shock capturing in a time-explicit discontinuous Galerkin method.Math.Model.Nat.Phenom.6(3), 57-83 (2011).https://doi.org/10.1051/mmnp/20116303
38.Klöfkorn, R.:Efcient matrix-free implementation of discontinuous Galerkin methods for compressible fow problems.In:Handlovicova, A.et al.(eds) Proceedings of the ALGORITMY 2012, pp.11-21.Slovak University of Technology in Bratislava, Publishing House of STU, Slovakia (2012)
39.Klöfkorn, R., Kvashchuk, A., Nolte, M.:Comparison of linear reconstructions for second-order fnite volume schemes on polyhedral grids.Comput.Geosci.21(5), 909-919 (2017).https://doi.org/10.1007/s10596-017-9658-8
40.Knoll, D.A., Keyes, D.E.:Jacobian-free Newton-Krylov methods:a survey of approaches and applications.J.Comput.Phys.193(2), 357-397 (2004)
41.Kopriva, D.A., Gassner, G.:On the quadrature and weak form choices in collocation type discontinuous Galerkin spectral element methods.J.Sci.Comput.44, 136-155 (2010).https://doi.org/10.1007/s10915-010-9372-3
42.Kopriva, D.A., Woodruf, S.L., Hussaini, M.Y.:Computation of electromagnetic scattering with a non-conforming discontinuous spectral element method.Int.J.Numer.Methods Eng.53(1), 105-122 (2002).https://doi.org/10.1002/nme.394
43.Krivodonova, L., Xin, J., Remacle, J.F., Chevaugeon, N., Flaherty, J.E.:Shock detection and limiting with discontinuous Galerkin methods for hyperbolic conservation laws.Appl.Numer.Math.48(3/4), 323-338 (2004).https://doi.org/10.1016/j.apnum.2003.11.002
44.Logg, A., Mardal, K.A., Wells, G.:Automated Solution of Diferential Equations by the Finite Element Method:the FEniCS Book.Springer Publishing Company Incorporated, Berlin (2012)
45.Mandli, K.T., Ahmadia, A.J., Berger, M., Calhoun, D., George, D.L., Hadjimichael, Y., Ketcheson, D.I., Lemoine, G.I., LeVeque, R.J.:Clawpack:building an open source ecosystem for solving hyperbolic PDEs.Peer J.Comput.Sci.2, e68 (2016).https://doi.org/10.7717/peerj-cs.68
46.May, S., Berger, M.:Two-dimensional slope limiters for fnite volume schemes on non-coordinatealigned meshes.SIAM J.Sci.Comput.35(5), A2163-A2187 (2013).https://doi.org/10.1137/12087 5624
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).https://doi.org/10.2514/6.2006-112
48.Rathgeber, F., Ham, D.A., Mitchell, L., Lange, M., Luporini, F., McRae, A.T.T., Bercea, G.T., Markall, G.R., Kelly, P.H.J.:Firedrake:automating the fnite element method by composing abstractions.ACM Trans.Math.Softw.43(3), 24/1-24/7 (2016).https://doi.org/10.1145/2998441
49.Schuster, D., Brdar, S., Baldauf, M., Dedner, A., Klöfkorn, R., Kröner, D.:On discontinuous Galerkin approach for atmospheric fow in the mesoscale with and without moisture.Meteorologische Zeitschrift 23(4), 449-464 (2014).https://doi.org/10.1127/0941-2948/2014/0565
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).https://doi.org/10.1016/j.jcp.2016.04.030
51.Wallwork, J.G., Barral, N., Kramer, S.C., Ham, D.A., Piggott, M.D.:Goal-oriented error estimation and mesh adaptation for shallow water modelling.SN Appl.Sci.2, 1053 (2020).https://doi.org/10.1007/s42452-020-2745-9
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).https://doi.org/10.1016/j.jcp.2010.08.016

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