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Energy stable and conservative dynamical low-rank approximation for the Su-Olson problem

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arXiv 2023年

作者： Baumann, Lena Einkemmer, Lukas Klingenberg, Christian Kusch, Jonas University of Wuerzburg Department of Mathematics Wuerzburg Germany University of Innsbruck Numerical Analysis and Scientific Computing Innsbruck Austria Norwegian University of Life Sciences Scientific Computing Ås Norway

Computational methods for thermal radiative transfer problems exhibit high computational costs and a prohibitive memory footprint when the spatial and directional domains are finely resolved. A strategy to reduce such computational costs is dynamical low-rank approximation (DLRA), which represents and evolves the solution on a low-rank manifold, thereby significantly decreasing computational and memory requirements. Efficient discretizations for the DLRA evolution equations need to be carefully constructed to guarantee stability while enabling mass conservation. In this work, we focus on the Su-Olson closure leading to a linearized internal energy model and derive a stable discretization through an implicit coupling of internal energy and particle density. Moreover, we propose a rank-adaptive strategy to preserve local mass conservation. numerical results are presented which showcase the accuracy and efficiency of the proposed low-rank method compared to the solution of the full *** Codes 35L65, 35Q49, 65M12, 65M22 Copyright © 2023, The Authors. All rights reserved.

关键词： numerical methods

Matrix-free second-order methods in implicit time integration for compressible flows using automatic differentiation

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Matrix-free second-order methods in implicit time integratio...

作者： Bramkamp, Frank D. Pollul, Bernhard Rasch, Arno Schieffer, Gero Department for Computational Analysis of Technical Systems RWTH Aachen University D-52056 Aachen Germany Department for Numerical Mathematics RWTH Aachen University D-52056 Aachen Germany Institute for Scientific Computing RWTH Aachen University D-52056 Aachen Germany

In the numerical simulation of inviscid and viscous compressible fluid flow, implicit time integration schemes based on Newton-Krylov methods are frequently used. A crucial ingredient of Krylov subspace methods is the evaluation of the product of the Jacobian matrix of the spatial operator, e.g., fluxes, and a Krylov vector. In this article, we consider a matrix-free, consistently second-order accurate implementation of the Jacobian-vector product within the flow solver QUADFLOW using automatic differentiation. The convergences of the non-linear iteration using first- and second-order accurate Jacobian-vector products are compared. Different algorithmic parameters such as the CFL number and the choice of the preconditioner are taken into account. Investigations of inviscid flow around an airfoil and a wing of an aircraft, as well as the viscous laminar flow over a flat plate demonstrate that the convergence of the Newton-Krylov solver can significantly be improved using consistently second-order accurate matrix-vector products, compared to standard first-order approximations. It has been shown that a hybrid strategy using a blend between first-order and second-order accurate matrix-vector products is the best choice to obtain an efficient and robust algorithm. Copyright © 2014 Inderscience Enterprises Ltd.

关键词： Finite volume method

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

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arXiv 2018年

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

The present work is the second part of a pair of papers, considering Hybrid Discontinuous Galerkin methods with relaxed H(div)-conformity. The first part mainly dealt with presenting a robust analysis with respect to the mesh size h and the introduction of a reconstruction operator to restore divergence-conformity and pressure robustness using a non conforming right hand side. The aim of this part is the presentation of a high order polynomial robust analysis for the relaxed H(div)-conforming Hybrid Discontinuous Galerkin discretization of the two dimensional Stokes problem. It is based on the recently proven polynomial robust LBB-condition for BDM elements [P. L. Lederer, J. Schöberl, IMA Journal of numerical analysis, 2017] and is derived by a direct approach instead of using a best approximation Céa like result. We further treat the impact of the reconstruction operator on the hp analysis and present a numerical investigation considering polynomial robustness. We conclude the paper presenting an efficient operator splitting time integration scheme for the Navier–Stokes equations which is based on the methods recently presented in [C. Lehrenfeld, J. Schöberl, Comp. Meth. Appl. Mech. Eng., 361 (2016)] and includes the ideas of the reconstruction operator.35Q30, 65N12, 65N22, 65N30 Copyright © 2018, The Authors. All rights reserved.

关键词： Navier Stokes equations

Divergence-free tangential finite element methods for incompressible flows on surfaces

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arXiv 2019年

In this work we consider the numerical solution of incompressible flows on two-dimensional manifolds. Whereas the compatibility demands of the velocity and the pressure spaces are known from the flat case one further has to deal with the approximation of a velocity field that lies only in the tangential space of the given geometry. Abandoning H1-conformity allows us to construct finite elements which are – due to an application of the Piola transformation – exactly tangential. To reintroduce continuity (in a weak sense) we make use of (hybrid) discontinuous Galerkin techniques. To further improve this approach, H(divΓ)-conforming finite elements can be used to obtain exactly divergence-free velocity solutions. We present several new finite element discretizations. On a number of numerical examples we examine and compare their qualitative properties and accuracy. Copyright © 2019, The Authors. All rights reserved.

关键词： Finite element method

Performance comparison of the CRAY X-MP/24 with SDD and the CRAY-2

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The Journal of Supercomputing 1988年第4期1卷 409-419页

作者： Anderson, Richard E. Grimes, Roger G. Simon, Horst D. Scientific Computing and Analysis Division Boeing Computer Services M/S 7L-21 Seattle 98124 WA United States Numerical Aerodynamic Simulation (NAS) Systems Division NASA Ames Research Center Moffett Field 94035 CA Mail Stop 258-5 United States

The CRAY-2 is considered to be one of the most powerful supercomputers. Its state-of-the-art technology features a faster clock and more memory than any other supercomputer available today. In this report the single processor performance of the CRAY-2 is compared with the older, more mature CRAY X-MP. Benchmark results are included for both the slow and the fast memory DRAM MOS CRAY-2. Our comparison is based on a kernel benchmark set aimed at evaluating the performance of these two machines on some standard tasks in scientific computing. Particular emphasis is placed on evaluating the impact of the availability of large real memory on the CRAY-2 versus fast secondary memory on the CRAY X-MP with SSD. Our benchmark includes large linear equation solvers and FFT routines, which test the capabilities of the different approaches to providing large memory. We find that in spite of its higher processor speed the CRAY-2 does not perform as well as the CRAY X-MP on the Fortran kernel benchmark. We also find that for large-scale applications, which have regular and predictable memory access patterns, a high-speed secondary memory device such as the SSD can provide performance equal to the large real memory of the CRAY-2. © 1988 Kluwer Academic Publishers.

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A UNIFIED FRAMEWORK FOR THE ERROR analysis OF PHYSICS-INFORMED NEURAL NETWORKS

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arXiv 2023年

作者： Zeinhofer, Marius Masri, Rami Mardal, Kent-André Department of Numerical Analysis and Scientific Computing Simula Research Laboratory Oslo0164 Norway Department of Mathematics University of Oslo Norway

We prove a priori and a posteriori error estimates for physics-informed neural networks (PINNs) for linear PDEs. We analyze elliptic equations in primal and mixed form, elasticity, parabolic, hyperbolic and Stokes equations;and a PDE constrained optimization problem. For the analysis, we propose an abstract framework in the common language of bilinear forms, and we show that coercivity and continuity lead to error estimates. The obtained estimates are sharp and reveal that the L2 penalty approach for initial and boundary conditions in the PINN formulation weakens the norm of the error decay. Finally, utilizing recent advances in PINN optimization, we present numerical examples that illustrate the ability of the method to achieve accurate *** Codes 65M12, 65M15 Copyright © 2023, The Authors. All rights reserved.

关键词： Error analysis

Fast uncertainty quantification of tracer distribution in the brain interstitial fluid with multilevel and quasi Monte Carlo

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arXiv 2020年

作者： Croci, Matteo Vinje, Vegard Rognes, Marie E. Mathematical Institute University of Oxford Oxford United Kingdom Department for Numerical Analysis and Scientific Computing Simula Research Laboratory Lysaker Norway

Efficient uncertainty quantification algorithms are key to understand the propagation of uncertainty - from uncertain input parameters to uncertain output quantities - in high resolution mathematical models of brain physiology. Advanced Monte Carlo methods such as quasi Monte Carlo (QMC) and multilevel Monte Carlo (MLMC) have the potential to dramatically improve upon standard Monte Carlo (MC) methods, but their applicability and performance in biomedical applications is underexplored. In this paper, we design and apply QMC and MLMC methods to quantify uncertainty in a convection-diffusion model of tracer transport within the brain. We show that QMC outperforms standard MC simulations when the number of random inputs is small. MLMC considerably outperforms both QMC and standard MC methods and should therefore be preferred for brain transport models. Copyright © 2020, The Authors. All rights reserved.

关键词： Finite element method

A STUDY of multigrid smoothers used in compressible CFD based on the convection diffusion equation 7

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A STUDY of multigrid smoothers used in compressible CFD base...

7th European Congress on Computational Methods in Applied Sciences and Engineering, ECCOMAS Congress 2016

作者： Birken, Philipp Bull, Jonathan Jameson, Antony Lund University Centre for the Mathematical Sciences Numerical Analysis Box 118 Lund22100 Sweden Uppsala University Division of Scientific Computing Dept. of Information Technology Box 337 Uppsala75105 Sweden Stanford University Department of Aeronautics and Astronautics StanfordCA94305 United States

ISBN: (纸本)9786188284401

We look at multigrid methods for unsteady viscous compressible flows. We specifically target smoothers that can be used in parallel and without computation of a Jacobian, which are particlarly attractive candidates in the context of Discontinuous Galerkin discretizations. In CFD, a plethora of nonlinear smoothers have been suggested which are hard to analyze. Our methodology is to use a linear model problem, here the convection diffusion equation, to be able to classify and compare smoothers better. Specifically, we consider explicit and implicit pseudo time iterations, GMRES as a smoother, SGS and implicit line smoothers. We relate GMRES to explicit Runge-Kutta smoothers, identify implicit line smoothers as Block Jacobi and analyze the potential of implicit pseudo time iterations. Finally, we discuss the relation between methods for steady and unsteady flows. numerical results show that GMRES is a very attractive smoother in this context.

关键词： Unsteady flow

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