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A Statically Condensed Discontinuous Galerkin Spectral Element Method on Gauss-Lobatto Nodes for the Compressible Navier-Stokes Equations [article]

Andrés M. Rueda-Ramírez and Esteban Ferrer and David A. Kopriva and Gonzalo Rubio and Eusebio Valero
<span title="2019-12-12">2019</span> <i > arXiv </i> &nbsp; <span class="release-stage" >pre-print</span>

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<span title="2019-11-04">2019</span> <span class="release-stage" >pre-print</span> <span class="external-identifiers"> <a target="_blank" rel="external noopener" href="https://arxiv.org/abs/1911.02366v1">arXiv:1911.02366v1</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/4czuvng6xvb7dhwltkchekw43y">fatcat:4czuvng6xvb7dhwltkchekw43y</a> </span>
We present a static-condensation method for time-implicit discretizations of the Discontinuous Galerkin Spectral Element Method on Gauss-Lobatto points (GL-DGSEM).  ...  To test it we solve the compressible Navier-Stokes equations with direct and Krylov subspace solvers, and we show for a selected problem that using the statically condensed GL-DGSEM leads to speed-ups  ...  This work was supported by a grant from the Simons Foundation (#426393, David Kopriva). Andrés Rueda-Ramírez developed.  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener" href="https://arxiv.org/abs/1911.02366v2">arXiv:1911.02366v2</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/vlpoqeeia5gatfxsx36islttly">fatcat:vlpoqeeia5gatfxsx36islttly</a> </span>
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Andrés M. Rueda-Ramírez and Esteban Ferrer and David A. Kopriva and Gonzalo Rubio and Eusebio Valero. "A Statically Condensed Discontinuous Galerkin Spectral Element Method on Gauss-Lobatto Nodes for the Compressible Navier-Stokes Equations." arXiv (2019)

A Statically Condensed Discontinuous Galerkin Spectral Element Methodon Gauss-Lobatto Nodes for the Compressible Navier-Stokes Equations

Andres Rueda, Eusebio Valero, Gonzalo Rubio, David A. Kopriva, Esteban Ferrer
<span title="2019-12-12">2019</span> <i title="Zenodo"> Zenodo </i> &nbsp;

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We present a static-condensation method for time-implicit discretizations of the Discontinuous GalerkinSpectral Element Method on Gauss-Lobatto points (GL-DGSEM).  ...  We show that, when solving the com-pressible Navier-Stokes equations, it is possible to reorganize the linear system that results from the implicittime-integration of the GL-DGSEM as a Schur complement  ...  This work was supported by a grant from the Simons Foundation (#426393, David Kopriva).  ... 
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Citation

Andres Rueda, Eusebio Valero, Gonzalo Rubio, David A. Kopriva, Esteban Ferrer. "A Statically Condensed Discontinuous Galerkin Spectral Element Methodon Gauss-Lobatto Nodes for the Compressible Navier-Stokes Equations." Zenodo (2019)

A Review of Element-Based Galerkin Methods for Numerical Weather Prediction: Finite Elements, Spectral Elements, and Discontinuous Galerkin

Simone Marras, James F. Kelly, Margarida Moragues, Andreas Müller, Michal A. Kopera, Mariano Vázquez, Francis X. Giraldo, Guillaume Houzeaux, Oriol Jorba
<span title="2015-05-19">2015</span> <i title="Springer Nature"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/wjer2chx7zdfnn3wbc3bburnzi" style="color: black;">Archives of Computational Methods in Engineering</a> </i> &nbsp;

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Operational and research NWP codes currently use a wide range of numerical methods: finite di erences, spectral transform, finite volumes and, increasingly, finite/spectral elements and discontinuous Galerkin  ...  , which constitute element-based Galerkin (EBG) methods.  ...  A convenient set is represented by the Legendre-Gauss-Lobatto (LGL) points.  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1007/s11831-015-9152-1">doi:10.1007/s11831-015-9152-1</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/tmjzwcxorfgt5fdcrlf6kb24ny">fatcat:tmjzwcxorfgt5fdcrlf6kb24ny</a> </span>
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Citation

Simone Marras, James F. Kelly, Margarida Moragues, Andreas Müller, Michal A. Kopera, Mariano Vázquez, Francis X. Giraldo, Guillaume Houzeaux, Oriol Jorba. "A Review of Element-Based Galerkin Methods for Numerical Weather Prediction: Finite Elements, Spectral Elements, and Discontinuous Galerkin." Archives of Computational Methods in Engineering 23.4 (2015) 673-722

A Conservative Discontinuous Galerkin Semi-Implicit Formulation for the Navier–Stokes Equations in Nonhydrostatic Mesoscale Modeling

Marco Restelli, Francis X. Giraldo
<span title="">2009</span> <i title="Society for Industrial &amp; Applied Mathematics (SIAM)"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/wbdvoluxebgjhn3wq3qsnldey4" style="color: black;">SIAM Journal on Scientific Computing</a> </i> &nbsp;

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A discontinuous Galerkin (DG) finite element formulation is proposed for the solution of the compressible Navier-Stokes equations for a vertically stratified fluid, which are of interest in mesoscale nonhydrostatic  ...  presence of the high order terms, and in [23, 26, 27] for the Navier-Stokes equations.  ...  The work of the first author (MR) was mainly done while he was a Ph.D. student at "Dipartimento di Matematica F.  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1137/070708470">doi:10.1137/070708470</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/r7wyqewgi5gknitsrcpr33yhai">fatcat:r7wyqewgi5gknitsrcpr33yhai</a> </span>
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Citation

Marco Restelli, Francis X. Giraldo. "A Conservative Discontinuous Galerkin Semi-Implicit Formulation for the Navier–Stokes Equations in Nonhydrostatic Mesoscale Modeling." SIAM Journal on Scientific Computing 31.3 (2009) 2231-2257

Nektar++: An open-source spectral/hp element framework

C.D. Cantwell, D. Moxey, A. Comerford, A. Bolis, G. Rocco, G. Mengaldo, D. De Grazia, S. Yakovlev, J.-E. Lombard, D. Ekelschot, B. Jordi, H. Xu (+7 others)
<span title="">2015</span> <i title="Elsevier BV"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/wfenstutwram7ghf2w6rgf5gyu" style="color: black;">Computer Physics Communications</a> </i> &nbsp;

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Nektar++ is an open-source software framework designed to support the development of highperformance scalable solvers for partial differential equations using the spectral/hp element method.  ...  Program summary Program title: Nektar++ Catalogue identifier: AEVV_v1_0 Program summary URL: Solution method: Spectral/hp element method Running time: The tests provided take a few minutes to run.  ...  In particular, we would like to acknowledge the contribution of Christian Roth for initial developments on the pulse-wave solver, Kilian Lackhove for work on extending the acoustic perturbation equations  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1016/j.cpc.2015.02.008">doi:10.1016/j.cpc.2015.02.008</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/34jsqpqhnnforcg34erg2nzory">fatcat:34jsqpqhnnforcg34erg2nzory</a> </span>
<span class="ui label basic orange small" title="Open Access">Open Access</span>
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Citation

C.D. Cantwell, D. Moxey, A. Comerford, A. Bolis, G. Rocco, G. Mengaldo, D. De Grazia, S. Yakovlev, J.-E. Lombard, D. Ekelschot, B. Jordi, H. Xu, Y. Mohamied, C. Eskilsson, B. Nelson, P. Vos, C. Biotto, R.M. Kirby, S.J. Sherwin. "Nektar++: An open-source spectral/hp element framework." Computer Physics Communications (2015) 205-219

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)
<span title="">2019</span> <i title="Elsevier BV"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/wfenstutwram7ghf2w6rgf5gyu" style="color: black;">Computer Physics Communications</a> </i> &nbsp;

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solvers for partial differential equations using the high-order spectral/hp element method.  ...  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  ...  Declaration of competing interest The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1016/j.cpc.2019.107110">doi:10.1016/j.cpc.2019.107110</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/cfkq2f7z3jbtzayhsnn5uwcswe">fatcat:cfkq2f7z3jbtzayhsnn5uwcswe</a> </span>
<span class="ui label basic orange small" title="Open Access">Open Access</span>
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Citation

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, Michael Turner, Hui Xu, Joaquim Peiró, Robert M. Kirby, Spencer J. Sherwin. "Nektar++: Enhancing the capability and application of high-fidelity spectral/hp element methods." Computer Physics Communications (2019) 107110

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)
<span title="2019-11-26">2019</span> <i > arXiv </i> &nbsp; <span class="release-stage" >pre-print</span>

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<span title="2019-06-08">2019</span> <span class="release-stage" >pre-print</span>
<span class="external-identifiers"> <a target="_blank" rel="external noopener" href="https://arxiv.org/abs/1906.03489v1">arXiv:1906.03489v1</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/3d7i6pityzgvhasue5wasdeuoq">fatcat:3d7i6pityzgvhasue5wasdeuoq</a> </span>
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++.  ...  This release also incorporates a number of numerical method developments - in particular: the method of moving frames, which provides an additional approach for the simulation of equations on embedded  ...  Equivalently, φ p (ξ) could also be defined by the Lagrange polynomials through the Gauss-Lobatto-Legendre quadrature points which would lead to a traditional spectral element method [12, §2.3.4] .  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener" href="https://arxiv.org/abs/1906.03489v2">arXiv:1906.03489v2</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/nlz3erpuzjenvbjetfbdv4nxc4">fatcat:nlz3erpuzjenvbjetfbdv4nxc4</a> </span>
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Citation

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, Michael Turner, Hui Xu, Joaquim Peiró, Robert M. Kirby, Spencer J. Sherwin. "Nektar++: enhancing the capability and application of high-fidelity spectral/hp element methods." arXiv (2019)

A Performance Comparison of Continuous and Discontinuous Galerkin Methods with Fast Multigrid Solvers

Martin Kronbichler, Wolfgang A. Wall
<span title="">2018</span> <i title="Society for Industrial &amp; Applied Mathematics (SIAM)"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/wbdvoluxebgjhn3wq3qsnldey4" style="color: black;">SIAM Journal on Scientific Computing</a> </i> &nbsp;

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<span title="2016-11-09">2016</span> <span class="release-stage" >pre-print</span> <span class="external-identifiers"> <a target="_blank" rel="external noopener" href="https://arxiv.org/abs/1611.03029v1">arXiv:1611.03029v1</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/pkbozmddtjejvjwiao2pzdvzp4">fatcat:pkbozmddtjejvjwiao2pzdvzp4</a> </span>

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<span title="2016-12-09">2016</span> <span class="release-stage">accepted</span>
<span class="external-identifiers"> <a target="_blank" rel="external noopener" href="https://arxiv.org/abs/1611.03029v2">arXiv:1611.03029v2</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/roudq5bbcbbzzcxhmjalwovnyy">fatcat:roudq5bbcbbzzcxhmjalwovnyy</a> </span>
For the hybridized discontinuous Galerkin method, a multigrid approach that combines a grid transfer from the trace space to the space of linear finite elements with algebraic multigrid on further levels  ...  This study presents a fair performance comparison of the continuous finite element method, the symmetric interior penalty discontinuous Galerkin method, and the hybridized discontinuous Galerkin method  ...  Acknowledgments The authors would like to thank Katharina Kormann and Niklas Fehn for discussions about the manuscript.  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1137/16m110455x">doi:10.1137/16m110455x</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/o7o7ddc7fnhstnzcgrwld7mpoy">fatcat:o7o7ddc7fnhstnzcgrwld7mpoy</a> </span>
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Martin Kronbichler, Wolfgang A. Wall. "A Performance Comparison of Continuous and Discontinuous Galerkin Methods with Fast Multigrid Solvers." SIAM Journal on Scientific Computing 40.5 (2018) A3423-A3448

Efficient exascale discretizations: High-order finite element methods

Tzanio Kolev, Paul Fischer, Misun Min, Jack Dongarra, Jed Brown, Veselin Dobrev, Tim Warburton, Stanimire Tomov, Mark S Shephard, Ahmad Abdelfattah, Valeria Barra, Natalie Beams (+18 others)
<span title="2021-06-08">2021</span> <i title="SAGE Publications"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/prmoui3bz5a2jktzh4pt7dtery" style="color: black;">The international journal of high performance computing applications</a> </i> &nbsp;

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One of the few viable approaches to achieve high efficiency in the area of PDE discretizations on unstructured grids is to use matrix-free/partially assembled high-order finite element methods, since these  ...  focused on the development of next-generation discretization software and algorithms to enable a wide range of finite element applications to run efficiently on future hardware.  ...  Department of Energy organizations (Office of Science and the National Nuclear Security Administration) responsible for the planning and preparation of a capable exascale ecosystem, including software,  ... 
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Tzanio Kolev, Paul Fischer, Misun Min, Jack Dongarra, Jed Brown, Veselin Dobrev, Tim Warburton, Stanimire Tomov, Mark S Shephard, Ahmad Abdelfattah, Valeria Barra, Natalie Beams, Jean-Sylvain Camier, Noel Chalmers, Yohann Dudouit, Ali Karakus, Ian Karlin, Stefan Kerkemeier, Yu-Hsiang Lan, David Medina, Elia Merzari, Aleksandr Obabko, Will Pazner, Thilina Rathnayake, Cameron W Smith, Lukas Spies, Kasia Swirydowicz, Jeremy Thompson, Ananias Tomboulides, Vladimir Tomov. "Efficient exascale discretizations: High-order finite element methods." The international journal of high performance computing applications 35.6 (2021) 109434202110208

Some aspects of adaptive grid technology related to boundary and interior layers

Graham F. Carey, M. Anderson, B. Carnes, B. Kirk
<span title="">2004</span> <i title="Elsevier BV"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/av75skmu6neednu63f27tpk7qu" style="color: black;">Journal of Computational and Applied Mathematics</a> </i> &nbsp;

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Associated error indicators that guide the reÿnement feedback control/grid optimization process are also covered and there is a brief commentary on the supporting data structure requirements.  ...  , and shockwaves in compressible gas dynamics.  ...  The compressible Navier-Stokes equations In this section, we consider the use of AMR for high-speed compressible ows governed by the Navier-Stokes equations.  ... 
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<span class="ui label basic orange small" title="Szcezepanski OA Journal List">Szczepanski</span>
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Graham F. Carey, M. Anderson, B. Carnes, B. Kirk. "Some aspects of adaptive grid technology related to boundary and interior layers." Journal of Computational and Applied Mathematics 166.1 (2004) 55-86

Statistical-mechanical predictions and Navier-Stokes dynamics of two-dimensional flows on a bounded domain

H. Brands, S. R. Maassen, H. J. H. Clercx
<span title="1999-09-01">1999</span> <i title="American Physical Society (APS)"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/hphsuvsjtvgjvnloxbomufglhe" style="color: black;">Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics</a> </i> &nbsp;

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Acknowledgements he struggled to get through his day, he was helplessly behind, he poured himself onto the page,writing for hours at a time.  ...  A major drawback of these methods is, however, that one deliberately alters the equations of motion, such that the resulting flow is no longer a pure Navier-Stokes solution.  ...  ) , for 0 ≤ i, j ≤ N. 1 shows such a non-equidistant Gauss-Lobatto collocation grid, for resolution N = 32.  ... 
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H. Brands, S. R. Maassen, H. J. H. Clercx. "Statistical-mechanical predictions and Navier-Stokes dynamics of two-dimensional flows on a bounded domain." Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics 60.3 (1999) 2864-2874

Dynamics of Nonlinear Waves on Bounded Domains [article]

Maciej Maliborski
<span title="2016-03-03">2016</span> <i > arXiv </i> &nbsp; <span class="release-stage" >pre-print</span>

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The main body of this dissertation is concerned with construction (by means of perturbative and numerical methods) of time-periodic solutions for various nonlinear wave equations on bounded domains.  ...  Either the solution behaves in an oscillatory, quasiperiodic manner or the nonlinear effects cause the energy to concentrate on smaller scales leading to a turbulent behaviour.  ...  As pointed out in [28] , the black hole formation provides a cut-off for the turbulent energy cascade for solutions of Einstein's equations (in analogy to viscosity in the case of the Navier-Stokes equation  ... 
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Maciej Maliborski. "Dynamics of Nonlinear Waves on Bounded Domains." arXiv (2016)

Fourier spectral/hp element method: investigation of time-stepping and parallelisation strategies

Alessandro Bolis, Spencer Sherwin, Engineering And Physical Sciences Research Council
<span title="2015-07-17">2015</span>

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This thesis focuses on algorithm optimization, to minimize run-time, when solving the incompressible Navier-Stokes equations.  ...  We initially describe the building blocks of a spectral/hp element and pure spectral method and how they can be encapsulated and combined to create a 3D discretisation, the Fourier spectral/hp element  ...  a static condensation approach.  ... 
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Alessandro Bolis, Spencer Sherwin, Engineering And Physical Sciences Research Council. "Fourier spectral/hp element method: investigation of time-stepping and parallelisation strategies." (2015)

Review of Lattice Boltzmann Method Applied to Computational Aeroacoustics

<span title="">2019</span>

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The Navier-Stokes equation-based solver faces the difficulty of computational efficiency when it has to satisfy the high-order of accuracy and spectral resolution.  ...  This paper presents the research studies carried out on the application of lattice Boltzmann method (LBM) to computational aeroacoustics (CAA).  ...  The authors would like to thank the anonymous reviewers for their helpful and constructive comments.  ... 
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"Review of Lattice Boltzmann Method Applied to Computational Aeroacoustics." (2019)

A discontinuous Galerkin method for the solution of compressible flows

Cristian Biotto, Joaquim Peiró
<span title="2011-03-17">2011</span>

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This thesis presents a methodology for the numerical solution of one-dimensional (1D) and two-dimensional (2D) compressible flows via a discontinuous Galerkin (DG) formulation.  ...  An Euler and a laminar Navier–Stokes solvers are presented and benchmark tests are used to assess their accuracy and performance.  ...  Outline This thesis presents a DG method for the solution of the compressible transonic Euler and Navier-Stokes equations in one-(1D) and two-dimensional (2D) domains using unstructured grids.  ... 
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Cristian Biotto, Joaquim Peiró. "A discontinuous Galerkin method for the solution of compressible flows." (2011)

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