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Extendible and Efficient Python Framework for Solving Evolution Equations with Stabilized Discontinuous Galerkin Method [article]

Andreas Dedner, Robert Klöfkorn
2021 arXiv   pre-print
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  ...  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.  ...  To our best knowledge this is the first package which combines high level scripting support with efficient stabilized Discontinuous Galerkin methods for solving the full range of pure hyperbolic systems  ... 
arXiv:2009.13416v2 fatcat:uzzbg2acvjehnmsyi3j7jbsvem

HORSES3D: a high-order discontinuous Galerkin solver for flow simulations and multi-physics applications [article]

E. Ferrer and G. Rubio and G. Ntoukas and W. Laskowski and O.A. Mariño and S. Colombo and A. Mateo-Gabín and F. Manrique de Lara and D. Huergo and J. Manzanero and A.M. Rueda-Ramírez and D.A. Kopriva and E. Valero
2022 arXiv   pre-print
We present the latest developments of our High-Order Spectral Element Solver (HORSES3D), an open source high-order discontinuous Galerkin framework, capable of solving a variety of flow applications, including  ...  Temporal discretisations include explicit, implicit, multigrid, and dual time-stepping schemes with efficient preconditioners.  ...  Spatial discretisation: discontinuous Galerkin HORSES3D discretises the Navier-Stokes equations using the Discontinuous Galerkin Spectral Element Method (DGSEM), which is a particularly efficient nodal  ... 
arXiv:2206.09733v1 fatcat:l4l66efdybbplaxurqcoxahhlm

Interface modeling in incompressible media using level sets in Escript

L. Gross, L. Bourgouin, A.J. Hale, H.-B. Mühlhaus
2007 Physics of the Earth and Planetary Interiors  
The implementation of the models is based on Escript which is a Python module for the solution of partial differential equations (PDEs) using spatial discretization techniques such as FEM.  ...  We begin with a brief outline of the Escript modules and then present illustrations of its usage for the numerical solutions of the problems mentioned above.  ...  Advanced Computing, The Queensland Cyberinfrastructure Foundation, and SGI Ltd.  ... 
doi:10.1016/j.pepi.2007.04.004 fatcat:obovbpsqovesterdcq63vimdju

Nektar++: Enhancing the capability and application of high-fidelity spectral/hp element methods

David Moxey, Chris D. Cantwell, Yan Bao, Andrea Cassinelli, Giacomo Castiglioni, Sehun Chun, Emilia Juda, Ehsan Kazemi, Kilian Lackhove, Julian Marcon, Gianmarco Mengaldo, Douglas Serson (+5 others)
2019 Computer Physics Communications  
Furthermore, we provide details on a newly developed Python interface that enables a more rapid introduction for new users unfamiliar with spectral/hp element methods, C++ and/or Nektar++.  ...  Spectral/hp element methods High-order finite element methods a b s t r a c t Nektar++ is an open-source framework that provides a flexible, high-performance and scalable platform for the development of  ...  Within this definition, we include continuous Galerkin (CG) and discontinuous Galerkin (DG) methods, along with their variants.  ... 
doi:10.1016/j.cpc.2019.107110 fatcat:cfkq2f7z3jbtzayhsnn5uwcswe

Numerical Stabilization of the Melt Front for Laser Beam Cutting [chapter]

Torsten Adolph, Willi Schönauer, Markus Niessen, Wolfgang Schulz
2010 Numerical Mathematics and Advanced Applications 2009  
Numerical experiments will be done for simple linear problems, the Burger's equation, and for the Euler equations with high Mach numbers. 24 This contribution deals with the modeling of soft biological  ...  The resulting linear systems are solved alternatively by the Multigrid method and Krylov subspace methods. We treat two types of boundary conditions: Dirichlet and natural boundary condition.  ...  interface with a discontinuous Galerkin method to solve the associated transport equation.  ... 
doi:10.1007/978-3-642-11795-4_6 fatcat:nx4nvuxaxfbcdjknopny53ck5e

Nektar++: enhancing the capability and application of high-fidelity spectral/hp element methods [article]

David Moxey, Chris D. Cantwell, Yan Bao, Andrea Cassinelli, Giacomo Castiglioni, Sehun Chun, Emilia Juda, Ehsan Kazemi, Kilian Lackhove, Julian Marcon, Gianmarco Mengaldo, Douglas Serson (+5 others)
2019 arXiv   pre-print
Furthermore, we provide details on a newly developed Python interface that enables a more rapid introduction for new users unfamiliar with spectral/hp element methods, C++ and/or Nektar++.  ...  Nektar++ is an open-source framework that provides a flexible, high-performance and scalable platform for the development of solvers for partial differential equations using the high-order spectral/hp  ...  Within this definition, we include continuous Galerkin (CG) and discontinuous Galerkin (DG) methods, along with their variants.  ... 
arXiv:1906.03489v2 fatcat:nlz3erpuzjenvbjetfbdv4nxc4

Proceedings of the FEniCS Conference 2017

Jack S. Hale
2017 Figshare  
We argue that the demonstrated efficiency gains of the multi-level method make the approach competitive with alternatives to MC for eigenvalue problems.  ...  The techniques involve direct fusion, applied to mixed finite elements, and indirect fusion, applied to Discontinuous Galerkin finite elements.  ...  These projections remain local and can be efficiently combined with the hybridized Discontinuous Galerkin method for solving the constitutive equations [1] .  ... 
doi:10.6084/m9.figshare.5086369 fatcat:iwajrgqf3nfgzgyloge3f7dz74

Smoothed aggregation for Helmholtz problems

Luke N. Olson, Jacob B. Schroder
2010 Numerical Linear Algebra with Applications  
We consider standard 1D finite difference discretizations and 2D discontinuous Galerkin discretizations. The scalar Helmholtz problem is particularly difficult for algebraic multigrid (AMG) solvers.  ...  Our results for GMRES preconditioned with the proposed SA method exhibit consistent performance for fixed points-per-wavelength and decreasing mesh size.  ...  Raymond Tuminaro of Sandia National Laboratories for guidance in a CG version of Algorithm 3 that motivated the current state.  ... 
doi:10.1002/nla.686 fatcat:2iugbiqf5vddbcslv2wcajycj4

Local Fourier Analysis of a Space-Time Multigrid Method for DG-SEM for the Linear Advection Equation [article]

Lea M. Versbach, Philipp Birken, Viktor Linders, Gregor Gassner
2021 arXiv   pre-print
The space-time discretization is based on arbitrarily high order discontinuous Galerkin spectral element methods in time and a first order finite volume method in space.  ...  Asymptotic convergence factors for the smoother and the two-grid method for both coarsening strategies are presented.  ...  Dedner and R. Klöfkorn. Extendible and Efficient Python Framework for Solving Evolution Equations with Stabilized Discontinuous Galerkin Methods. Commun. Appl. Math.  ... 
arXiv:2112.03115v1 fatcat:sab3g4p6bbegphpj4mq2og5xzi

Thetis coastal ocean model: discontinuous Galerkin discretization for the three-dimensional hydrostatic equations

Tuomas Kärnä, Stephan C. Kramer, Lawrence Mitchell, David A. Ham, Matthew D. Piggott, António M. Baptista
2018 Geoscientific Model Development  
In this paper, we describe a novel discontinuous Galerkin (DG) finite element discretization for the hydrostatic equations.  ...  Monotonicity of the advection scheme is ensured by using a strong stability-preserving time integration method and slope limiters.  ...  The authors acknowledge the Texas Advanced Computing Center (TACC) at the University of Texas at Austin for providing HPC resources that have contributed to the research results reported within this paper  ... 
doi:10.5194/gmd-11-4359-2018 fatcat:lz46f2lrm5fofdq6y5y2vjs6xm

An operator-based local discontinuous Galerkin method compatible with the BSSN formulation of the Einstein equations

Jonah M Miller, Erik Schnetter
2016 Classical and quantum gravity  
Discontinuous Galerkin Finite Element (DGFE) methods offer a mathematically beautiful, computationally efficient, and efficiently parallelizable way to solve hyperbolic partial differential equations.  ...  To-date no one has been able to solve the full (3+1)-dimensional BSSN equations using DGFE methods.  ...  We use our discontinuous Galerkin scheme to solve the linear wave equation given arbitrary initial conditions and compare to the analytic solution.  ... 
doi:10.1088/1361-6382/34/1/015003 fatcat:uem2ep3eqrfjtoemdiucszy5f4

The DUNE Framework: Basic Concepts and Recent Developments [article]

Peter Bastian, Markus Blatt, Andreas Dedner, Nils-Arne Dreier, Christian Engwer, René Fritze, Carsten Gräser, Christoph Grüninger, Dominic Kempf, Robert Klöfkorn, Mario Ohlberger, Oliver Sander
2020 arXiv   pre-print
and node level performance, non-smooth multigrid methods, and multiscale methods.  ...  A brief discussion on current and future development directions of the framework concludes the paper.  ...  Acknowledgements We thank the Dune users and contributors for their continuous support, as only a vivid community unfolds the power of open source.  ... 
arXiv:1909.13672v3 fatcat:hc5a7al4xvhahdp7fs7f55cwg4

Towards a new multiscale air quality transport model using the fully unstructured anisotropic adaptive mesh technology of Fluidity (version 4.1.9)

J. Zheng, J. Zhu, Z. Wang, F. Fang, C. C. Pain, J. Xiang
2015 Geoscientific Model Development  
, which is based on a discontinuous Galerkin/control volume discretization on unstructured meshes.  ...  </strong> An integrated method of advanced anisotropic hr-adaptive mesh and discretization numerical techniques has been, for first time, applied to modelling of multiscale advection–diffusion problems  ...  XDB05030200), The authors acknowledge the reviewers and Editor for their in depth perspicacious comments that contributed to improving the presentation of this paper. Edited by: P. Jöckel  ... 
doi:10.5194/gmd-8-3421-2015 fatcat:t4bkreiiondofj5jamiy6dhxzi

Modelling of microstructures during in-situ alloying in additive manufacturing for efficient material qualification processes [article]

Patrick Zimbrod, Johannes Schilp
2021 arXiv   pre-print
In this work, a numerical simulation framework is presented based on the Phase Field Method that is able to capture the evolution of heterogeneous metallic microstructures during solidification.  ...  The resulting digital twin is expected to shorten development times of new materials as well as cut down on experimental resource needs considerably, therefore contributing to efficient material qualification  ...  One alternative route would be to transfer the presented phase field method to more efficient numerical schemes regarding parallelization, such as the Discontinuous Galerkin Finite Element Method (DG-FEM  ... 
arXiv:2110.10398v1 fatcat:3yck7imec5hpjelzg6zsolwip4

The Dune framework: Basic concepts and recent developments

Peter Bastian, Markus Blatt, Andreas Dedner, Nils-Arne Dreier, Christian Engwer, René Fritze, Carsten Gräser, Christoph Grüninger, Dominic Kempf, Robert Klöfkorn, Mario Ohlberger, Oliver Sander
2020 Computers and Mathematics with Applications  
Examples currently are Python bindings, a generic implementation of grid functions, and support for system testing. Application modules provide frameworks for applications.  ...  and node level performance, non-smooth multigrid methods, and multiscale methods.  ...  For example, adaptive discontinuous Galerkin computations of compressible flow, e.g. Euler equations [61] or atmospheric flow [62] .  ... 
doi:10.1016/j.camwa.2020.06.007 fatcat:37m6di54rbgo3m2y5y5m2hd234
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