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Exa-Dune - Flexible PDE solvers, numerical methods and applications

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arXiv 2019年

作者： Bastian, Peter Altenbernd, Mirco Dreier, Nils-Arne Engwer, Christian Fahlke, Jorrit Fritze, René Geveler, Markus Göddeke, Dominik Iliev, Oleg Ippisch, Olaf Mohring, Jan Müthing, Steffen Ohlberger, Mario Ribbrock, Dirk Shegunov, Nikolay Turek, Stefan Interdisciplinary Center for Scientific Computing Heidelberg University Im Neuenheimer Feld 368 HeidelbergD-69120 Institute for Analysis and Numerics University of Münster Orleans-Ring 10 MünsterD-48149 Institute for Applied Analysis and Numerical Simulation University of Stuttgart Allmandring 5b StuttgartD-70569 Inst. f. Applied Mathematics TU Dortmund Vogelpothsweg 87 DortmundD-44227 Germany Fraunhofer Institute for Industrial Mathematics ITWM Fraunhofer-Platz 1 KaiserslauterD-67663 Germany Institut für Mathematik TU Clausthal-Zellerfeld Erzstr. 1 Clausthal-ZellerfeldD-38678 Germany

In the Exa-Dune project we have developed, implemented and optimised numerical algorithms and software for the scalable solution of partial differential equations (PDEs) on future exascale systems exhibiting a heterogeneous massively parallel architecture. In order to cope with the increased probability of hardware failures, one aim of the project was to add flexible, application-oriented resilience capabilities into the framework. Continuous improvement of the underlying hardware-oriented numerical methods have included GPU-based sparse approximate inverses, matrix-free sum-factorisation for high-order discontinuous Galerkin discretisations as well as partially matrix-free preconditioners. On top of that, additional scalability is facilitated by exploiting massive coarse grained parallelism offered by multiscale and uncertainty quantification methods where we have focused on the adaptive choice of the coarse/fine scale and the overlap region as well as the combination of local reduced basis multiscale methods and the multilevel Monte-Carlo algorithm. Finally, some of the concepts are applied in a land-surface model including subsurface flow and surface runoff. Copyright © 2019, The Authors. All rights reserved.

关键词： numerical methods

On reference solutions and the sensitivity of the 2D Kelvin–Helmholtz instability problem

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arXiv 2018年

作者： Schroeder, Philipp W. John, Volker Lederer, Philip L. Lehrenfeld, Christoph Lube, Gert Schöberl, Joachim Institute for Numerical and Applied Mathematics Georg-August-Universität Göttingen Göttingen37083 Germany Weierstrass Institute for Applied Analysis and Stochastics Berlin10117 Germany Department of Mathematics and Computer Science Freie Universität Berlin Berlin14195 Germany Institute for Analysis and Scientific Computing TU Wien 1040 Wien Austria

Two-dimensional Kelvin–Helmholtz instability problems are popular examples for assessing discretizations for incompressible flows at high Reynolds number. Unfortunately, the results in the literature differ considerably. This paper presents computational studies of a Kelvin–Helmholtz instability problem with high order divergence-free finite element methods. Reference results in several quantities of interest are obtained for three different Reynolds numbers up to the beginning of the final vortex pairing. A mesh-independent prediction of the final pairing is not achieved due to the sensitivity of the considered problem with respect to small perturbations. A theoretical explanation of this sensitivity to small perturbations is provided based on the theory of self-organization of 2D turbulence. Possible sources of perturbations that arise in almost any numerical simulation are discussed. Copyright © 2018, The Authors. All rights reserved.

关键词： numerical methods

Benchmark for numerical solutions of flow in heterogeneous groundwater formations

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arXiv 2019年

作者： Alecsa, Cristian D. Boros, Imre Frank, Florian Knabner, Peter Nechita, Mihai Prechtel, Alexander Rupp, Andreas Suciu, Nicolae Tiberiu Popoviciu Institute of Numerical Analysis Romanian Academy Fantanele 57 Cluj-Napoca400320 Romania Department of Mathematics Babeş-Bolyai University Mihail Kogalniceanu 1 Cluj-Napoca400084 Romania Mathematics Department Friedrich-Alexander University of Erlangen-Nuremberg Cauerstraße. 11 Erlangen91058 Germany Department of Mathematics University College London Gower Street LondonWC1E 6BT United Kingdom Interdisciplinary Center for Scientific Computing Ruprecht-Karls-University Im Neuenheimer Feld 205 Heidelberg69120 Germany

This article presents numerical investigations on accuracy and convergence properties of several numerical approaches for simulating steady state flows in heterogeneous aquifers. Finite difference, finite element, discontinuous Galerkin, spectral, and random walk methods are tested on one- and two-dimensional benchmark flow problems. Realizations of log-normal hydraulic conductivity fields are generated by Kraichnan algorithms in closed form as finite sums of random periodic modes, which allow direct code verification by comparisons with manufactured reference solutions. The quality of the methods is assessed for increasing number of random modes and for increasing variance of the log-hydraulic conductivity fields with Gaussian and exponential correlation. Experimental orders of convergence are calculated from successive refinements of the grid. The numerical methods are further validated by comparisons between statistical inferences obtained from Monte Carlo ensembles of numerical solutions and theoretical first-order perturbation results. It is found that while for Gaussian correlation of the log-conductivity field all the methods perform well, in the exponential case their accuracy deteriorates and, for large variance and number of modes, the benchmark problems are practically not tractable with reasonably large computing resources, for all the methods considered in this *** Codes 65N06 Copyright © 2019, The Authors. All rights reserved.

关键词： Finite element method

Laplacian preconditioning of elliptic pdes: Localization of the eigenvalues of the discretized operator

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arXiv 2018年

作者： Gergelits, Tomáš Mardal, Kent-André Nielsen, Bjørn Fredrik Strakoš, Zdeněk Institute of Computer Sciences Czech Academy of Science Prague Czech Republic Faculty of Mathematics and Physics Charles University Prague Czech Republic Department of Mathematics University of Oslo Oslo0316 Norway Department of Numerical Analysis and Scientific Computing Simula Research Laboratory Lysaker1325 Norway Faculty of Science and Technology Norwegian University of Life Sciences ÅsNO-1432 Norway

In the paper Preconditioning by inverting the Laplacian;an analysis of the eigenvalues. IMA Journal of numerical analysis 29, 1 (2009), 24-42, Nielsen, Hackbusch and Tveito study the operator generated by using the inverse of the Laplacian as preconditioner for second order elliptic PDEs ∇ · (k(x)∇u) = f. They prove that the range of k(x) is contained in the spectrum of the preconditioned operator, provided that k is continuous. Their rigorous analysis only addresses mappings defined on infinite dimensional spaces, but the numerical experiments in the paper suggest that a similar property holds in the discrete case. Motivated by this investigation, we analyze the eigenvalues of the matrix L−1A, where L and A are the stiffness matrices associated with the Laplace operator and general second order elliptic operators, respectively. Without any assumption about the continuity of k(x), we prove the existence of a one-to-one pairing between the eigenvalues of L−1A and the intervals determined by the images under k(x) of the supports of the FE nodal basis functions. As a consequence, we can show that the nodal values of k(x) yield accurate approximations of the eigenvalues of L−1A. Our theoretical results are illuminated by several numerical experiments.65F08, 65F15, 65N12, 35J99 Copyright © 2018, The Authors. All rights reserved.

关键词： Eigenvalues and eigenfunctions

Hybrid discontinuous Galerkin methods with relaxed H(div)-conformity for incompressible flows. Part I

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arXiv 2017年

作者： Lederer, Philip L. Lehrenfeld, Christoph Schöberl, Joachim Institute for Analysis and Scientific Computing TU Wien Wien Austria Institute for Numerical and Applied Mathematics University of Göttingen Göttingen Germany

We propose a new discretization method for the Stokes equations. The method is an improved version of the method recently presented in [C. Lehrenfeld, J. Schöberl, Comp. Meth. Appl. Mech. Eng., 361 (2016)] which is based on an H(div)-conforming finite element space and a Hybrid Discontinuous Galerkin (HDG) formulation of the viscous forces. H(div)-conformity results in favourable properties such as pointwise divergence free solutions and pressure-robustness. However, for the approximation of the velocity with a polynomial degree k it requires unknowns of degree k on every facet of the mesh. In view of the superconvergence property of other HDG methods, where only unknowns of polynomial degree k−1 on the facets are required to obtain an accurate polynomial approximation of order k (possibly after a local post-processing) this is sub-optimal. The key idea in this paper is to slightly relax the H(div)-conformity so that only unknowns of polynomial degree k−1 are involved for normal-continuity. This allows for optimality of the method also in the sense of superconvergent HDG methods. In order not to loose the benefits of H(div)-conformity we introduce a cheap reconstruction operator which restores pressure-robustness and pointwise divergence free solutions and suits well to the finite element space with relaxed H(div)-conformity. We present this new method, carry out a thorough h-version error analysis and demonstrate the performance of the method on numerical examples.35Q30, 65N12, 65N22, 65N30 Copyright © 2017, The Authors. All rights reserved.

关键词： Navier Stokes equations

HYBRID DISCONTINUOUS GALERKIN METHODS WITH RELAXED H(DIV)-CONFORMITY FOR INCOMPRESSIBLE FLOWS. PART I∗

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arXiv 2017年

We propose a new discretization method for the Stokes equations. The method is an improved version of the method recently presented in [C. Lehrenfeld, J. Schöberl, Comp. Meth. Appl. Mech. Eng., 361 (2016)] which is based on an H(div)-conforming finite element space and a Hybrid Discontinuous Galerkin (HDG) formulation of the viscous forces. H(div)-conformity results in favourable properties such as pointwise divergence free solutions and pressure-robustness. However, for the approximation of the velocity with a polynomial degree k it requires unknowns of degree k on every facet of the mesh. In view of the superconvergence property of other HDG methods, where only unknowns of polynomial degree k−1 on the facets are required to obtain an accurate polynomial approximation of order k (possibly after a local post-processing) this is sub-optimal. The key idea in this paper is to slightly relax the H(div)-conformity so that only unknowns of polynomial degree k−1 are involved for normal-continuity. This allows for optimality of the method also in the sense of superconvergent HDG methods. In order not to loose the benefits of H(div)-conformity we introduce a cheap reconstruction operator which restores pressure-robustness and pointwise divergence free solutions and suits well to the finite element space with relaxed H(div)-conformity. We present this new method, carry out a thorough h-version error analysis and demonstrate the performance of the method on numerical *** Codes 35Q30, 65N12, 65N22, 65N30 Copyright © 2017, The Authors. All rights reserved.

关键词： Navier Stokes equations

Verification benchmarks for single-phase flow in three-dimensional fractured porous media

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arXiv 2020年

作者： Berre, Inga Boon, Wietse M. Flemisch, Bernd Fumagalli, Alessio Gläser, Dennis Keilegavlen, Eirik Scotti, Anna Stefansson, Ivar Tatomir, Alexandru Brenner, Konstantin Burbulla, Samuel Devloo, Philippe Duran, Omar Favino, Marco Hennicker, Julian Lee, I-Hsien Lipnikov, Konstantin Masson, Roland Mosthaf, Klaus Chiara Nestola, Maria Giuseppina Ni, Chuen-Fa Nikitin, Kirill Schädle, Philipp Svyatskiy, Daniil Yanbarisov, Ruslan Zulianp, Patrick Department of Mathematics University of Bergen Allégaten 41 Bergen5007 Norway Department of Mathematics KTH Royal Institute of Technology Lindstedtsvägen 25 Stockholm11428 Sweden Department of Hydromechanics and Modelling of Hydrosystems University of Stuttgart Pfaffenwaldring 61 Stuttgart70569 Germany Laboratory for Modeling and Scientific Computing MOX Politecnico di Milano p.za Leonardo da Vinci 32 Milano20133 Italy Department of Applied Geology Geosciences Center University of Göttingen Goldschmidtstrasse 3 Göttingen37077 Germany Department of Earth Sciences Uppsala University Villavägen 16 UppsalaS-75236 Sweden University of Côte d’Azur CNRS INRIA LJAD Nice France Institute of Applied Analysis and Numerical Simulation University of Stuttgart Pfaffenwaldring 57 Stuttgart70569 Germany FEC Universidade Estadual de Campinas Cidade Universitária R. Josiah Willard Gibbs 85 Campinas-SPCEP 13083-839 Brazil Institute of Earth Sciences University of Lausanne Building Geopolis UNIL-Mouline Lausanne1015 Switzerland Section de Mathématiques Université de Genève 2-4 rue du Lièvre CP 64 Genève1211 Switzerland Graduate Institute of Applied Geology National Central University Taiwan Center for Environmental Studies National Central University Taiwan Los Alamos National Laboratory New Mexico United States Department of Environmental Engineering Technical University of Denmark Bygningstorvet Building 115 Kgs. Lyngby2800 Denmark Numerical Simulation in Science Medicine and Engineering Group Institute of Computational Science Università della Svizzera italiana. Via G. Buffi 13 Lugano Ticino6900 Switzerland Marchuk Institute of Numerical Mathematics of Russian Academy of Sciences Moscow Russia ETH Zürich Geothermal Energy and Geofluids Group Institute of Geophysics Zürich8092 Switzerland

Flow in fractured porous media occurs in the earth’s subsurface, in biological tissues, and in man-made materials. Fractures have a dominating influence on flow processes, and the last decade has seen an extensive development of models and numerical methods that explicitly account for their presence. To support these developments, we present a portfolio of four benchmark cases for single-phase flow in three-dimensional fractured porous media. The cases are specifically designed to test the methods’ capabilities in handling various complexities common to the geometrical structures of fracture networks. Based on an open call for participation, results obtained with 17 numerical methods were collected. This paper presents the underlying mathematical model, an overview of the features of the participating numerical methods, and their performance in solving the benchmark cases. Copyright © 2020, The Authors. All rights reserved.

关键词： numerical methods

Convergence of finite difference methods for the wave equation in two space dimensions

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arXiv 2017年

作者： Wang, Siyang Nissen, Anna Kreiss, Gunilla Division of Scientific Computing Department of Information Technology Uppsala University UppsalaSE-751 05 Sweden Division of Numerical Analysis Department of Mathematics Royal Institute of Technology StockholmSE-100 44 Sweden Division of Scientific Computing Department of Information Technology Uppsala University UppsalaSE-751 05 Sweden

When using a finite difference method to solve an initial–boundary–value problem, the truncation error is often of lower order at a few grid points near boundaries than in the interior. Normal mode analysis is a powerful tool to analyze the effect of the large truncation error near boundaries on the overall convergence rate, and has been used in many previous literatures for different equations. However, existing work only concerns problems in one space dimension. In this paper, we extend the analysis to problems in two space dimensions. The two dimensional analysis is based on a diagonalization procedure that decomposes a two dimensional problem to many one dimensional problems of the same type. We present a general framework of analyzing convergence for such one dimensional problems, and explain how to obtain the result for the corresponding two dimensional problem. In particular, we consider two kinds of truncation errors in two space dimensions: the truncation error along an entire boundary, and the truncation error localized at a few grid points close to a corner of the computational domain. The accuracy analysis is in a general framework, here applied to the second order wave equation. numerical experiments corroborate our accuracy analysis.65M12, 65M06 Copyright © 2017, The Authors. All rights reserved.

关键词： Finite difference method

Preconditioned smoothers for the full approximation scheme for the RANS equations

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arXiv 2017年

作者： Birken, Philipp Bull, Jonathan Jameson, Antony Centre for the Mathematical Sciences Numerical Analysis Lund University Lund Sweden Division of Scientific Computing Dept. of Information Technology Uppsala University Box 337 Uppsala75105 Sweden Stanford University Department of Aeronautics and Astronautics StanfordCA94305 United States

We consider multigrid methods for finite volume discretizations of the Reynolds Averaged Navier-Stokes (RANS) equations for both steady and unsteady flows. We analyze the effect of different smoothers based on pseudo time iterations, such as explicit and additive Runge-Kutta (ARK) methods. Furthermore, we derive the new class of additive W (AW) methods from Rosenbrock smoothers. This gives rise to two classes of preconditioned smoothers, preconditioned ARK and additive W (AW), which are implemented the exact same way, but have different parameters and properties. The new derivation allows to choose some of these based on results for time integration methods. As preconditioners, we consider SGS preconditioners based on flux vector splitting discretizations with a cutoff function for small eigenvalues. We compare these methods based on a discrete Fourier analysis. numerical results on pitching and plunging airfoils identify AW3 as the best smoother regarding overall efficiency. Specifically, for the NACA 64A010 airfoil steady-state convergence rates of as low as 0.85 were achieved, or a reduction of 6 orders of magnitude in approximately 25 pseudo-time iterations. Unsteady convergence rates of as low as 0.77 were achieved, or a reduction of 11 orders of magnitude in approximately 70 pseudo-time iterations. Copyright © 2017, The Authors. All rights reserved.

关键词： Runge Kutta methods