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879 results on '"VUIK, C."'

1. A comparison of Krylov methods for Shifted Skew-Symmetric Systems

Author

Idema, R. and Vuik, C.

Subjects
Mathematics - Numerical Analysis
Abstract

It is well known that for general linear systems, only optimal Krylov methods with long recurrences exist. For special classes of linear systems it is possible to find optimal Krylov methods with short recurrences. In this paper we consider the important class of linear systems with a shifted skew-symmetric coefficient matrix. We present the MRS3 solver, a minimal residual method that solves these problems using short vector recurrences. We give an overview of existing Krylov solvers that can be used to solve these problems, and compare them with the MRS3 method, both theoretically and by numerical experiments. From this comparison we argue that the MRS3 solver is the fastest and most robust of these Krylov method for systems with a shifted skew-symmetric coefficient matrix., Comment: 23 pages, 3 figures

Published
2023

2. A non-reflective boundary condition for LBM based on the assumption of non-equilibrium symmetry

Author

Euser, R. and Vuik, C.

Subjects
Physics - Computational Physics, Mathematics - Numerical Analysis
Abstract

In this study a new type of non-reflective boundary condition (NRBC) for the Lattice Boltzmann Method (LBM) is proposed; the Non-equilibrium Symmetry Boundary Condition (NSBC). The idea behind this boundary condition is to utilize the characteristics of the non-equilibrium distribution function to assign values to the incoming populations. A simple gradient based extrapolation technique and a far-field criterion are used to predict the macroscopic fluid variables. To demonstrate the non-reflective behaviour of the NSBC, two different tests have been carried out, examining the capability of the boundary to absorb acoustic waves respectively vortices. The results for both tests show that the amount of reflection generated by the NSBC is nearly zero., Comment: ENUMATH 2019 conference, 9 pages

Published
2020

3. A p-multigrid method enhanced with an ILUT smoother and its comparison to h-multigrid methods within Isogeometric Analysis

Author

Tielen, R., Möller, M., Göddeke, D., and Vuik, C.

Subjects
Mathematics - Numerical Analysis
Abstract

Over the years, Isogeometric Analysis has shown to be a successful alternative to the Finite Element Method (FEM). However, solving the resulting linear systems of equations efficiently remains a challenging task. In this paper, we consider a p-multigrid method, in which coarsening is applied in the approximation order p instead of the mesh width h. Since the use of classical smoothers (e.g. Gauss-Seidel) results in a p-multigrid method with deteriorating performance for higher values of p, the use of an ILUT smoother is investigated. Numerical results and a spectral analysis indicate that the resulting p-multigrid method exhibits convergence rates independent of h and p. In particular, we compare both coarsening strategies (e.g. coarsening in h or p) adopting both smoothers for a variety of two and threedimensional benchmarks.

Published
2019
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5. Inexact Subdomain Solves Using Deflated GMRES for Helmholtz Problems

Author

Bootland, N., Dwarka, V., Jolivet, P., Dolean, V., Vuik, C., Barth, Timothy J., Series Editor, Griebel, Michael, Series Editor, Keyes, David E., Series Editor, Nieminen, Risto M., Series Editor, Roose, Dirk, Series Editor, Schlick, Tamar, Series Editor, Brenner, Susanne C., editor, Chung, Eric, editor, Klawonn, Axel, editor, Kwok, Felix, editor, Xu, Jinchao, editor, and Zou, Jun, editor

Published
2022
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15. The Parallel Subdomain-Levelset Deflation Method in Reservoir Simulation

Author

van der Linden, J. H., Jönsthövel, T. B., Lukyanov, A. A., and Vuik, C.

Subjects
Mathematics - Numerical Analysis
Abstract

Extreme and isolated eigenvalues are known to be harmful to the convergence of an iterative solver. These eigenvalues can be produced by strong heterogeneity in the underlying physics. We can improve the quality of the spectrum by "deflating" the harmful eigenvalues. In this work, deflation is applied to linear systems in reservoir simulation. In particular, large, sudden differences in the permeability produce extreme eigenvalues. The number and magnitude of these eigenvalues is linked to the number and magnitude of the permeability jumps. Two deflation methods are discussed. Firstly, we state that harmonic Ritz eigenvector deflation, which computes the deflation vectors from the information produced by the linear solver, is unfeasible in modern reservoir simulation due to high costs and lack of parallelism. Secondly, we test a physics-based subdomain-levelset deflation algorithm that constructs the deflation vectors a priori. Numerical experiments show that both methods can improve the performance of the linear solver. We highlight the fact that subdomain-levelset deflation is particularly suitable for a parallel implementation. For cases with well-defined permeability jumps of a factor 104 or higher, parallel physics-based deflation has potential in commercial applications. In particular, the good scalability of parallel subdomain-levelset deflation combined with the robust parallel preconditioner for deflated system suggests the use of this method as an alternative for AMG.

Published
2015

25. Krylov Subspace Solvers and Preconditioners

Author

Vuik C.

Subjects
Applied mathematics. Quantitative methods, T57-57.97, Mathematics, QA1-939
Abstract

In these lecture notes an introduction to Krylov subspace solvers and preconditioners is presented. After a discretization of partial differential equations large, sparse systems of linear equations have to be solved. Fast solution of these systems is very urgent nowadays. The size of the problems can be 1013 unknowns and 1013 equations. Iterative solution methods are the methods of choice for these large linear systems. We start with a short introduction of Basic Iterative Methods. Thereafter preconditioned Krylov subspace methods, which are state of the art, are describeed. A distinction is made between various classes of matrices. At the end of the lecture notes many references are given to state of the art Scientific Computing methods. Here, we will discuss a number of books which are nice to use for an overview of background material. First of all the books of Golub and Van Loan [19] and Horn and Johnson [26] are classical works on all aspects of numerical linear algebra. These books also contain most of the material, which is used for direct solvers. Varga [50] is a good starting point to study the theory of basic iterative methods. Krylov subspace methods and multigrid are discussed in Saad [38] and Trottenberg, Oosterlee and Schüller [42]. Other books on Krylov subspace methods are [1, 6, 21, 34, 39].

Published
2018
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31. Two-Phase Flow Solved by High Order Discontinuous Galerkin Method

Author

van Zwieten, J. S. B., van der Heul, D. R., IJzermans, R. H. A., Henkes, R. A. W. M., Vuik, C., Barth, Timothy J., Series editor, Griebel, Michael, Series editor, Keyes, David E., Series editor, Roose, Dirk, Series editor, Schlick, Tamar, Series editor, Azaïez, Mejdi, editor, El Fekih, Henda, editor, and Hesthaven, Jan S., editor

Published
2014
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32. Computing three-dimensional two-phase flows with a mass-conserving level set method

Author

van der Pijl, S., Segal, A., Vuik, C., and Wesseling, P.

Abstract

Abstract: A method is described to compute three- dimensional two-phase flow, allowing large density ratios and coalescence and break-up of bubbles. The level set method is used to describe interfaces, and the volume-of-fluid method is used to ensure mass conservation. Efficiency in computing the interface dynamics is achieved by using a functional relation between the level set and volume-of-fluid functions. Difficulties and remedies in re-initialization of the level set function and inaccurate compution of surface tension are discussed. Test cases for validation are described, and demanding two-bubble computations to show the generality and the versatility of the method are presented.

Published
2024
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34. Numerical Methods for Ordinary Differential Equations

Author

Vuik, C., Vermolen, F.J., van Gijzen, M.B., and Vuik, M.J.

Subjects
numerical methods, numerical integration, nonlinear equations, differential equations, interpolation
Abstract

In this book we discuss several numerical methods for solving ordinary differential equations. We emphasize the aspects that play an important role in practical problems. We confine ourselves to ordinary differential equations with the exception of the last chapter in which we discuss the heat equation, a parabolic partial differential equation. The techniques discussed in the intro-ductory chapters, for instance interpolation, numerical quadrature and the solution to nonlinear equations, may also be used outside the context of differential equations. They have been in-cluded to make the book self-contained as far as the numerical aspects are concerned. Chapters, sections and exercises marked with a * are not part of the Delft Institutional Package.The numerical examples in this book were implemented in Matlab, but also Python or any other programming language could be used. A list of references to background knowledge and related literature can be found at the end of this book. Extra information about this course can be found at http://NMODE.ewi.tudelft.nl, among which old exams, answers to the exercises, and a link to an online education platform. We thank Matthias Moller for his thorough reading of the draft of this book and his helpful suggestions.

Published
2023

42. Block Preconditioners for the Incompressible Stokes Problem

Author

ur Rehman, M., Vuik, C., Segal, G., Hutchison, David, editor, Kanade, Takeo, editor, Kittler, Josef, editor, Kleinberg, Jon M., editor, Mattern, Friedemann, editor, Mitchell, John C., editor, Naor, Moni, editor, Nierstrasz, Oscar, editor, Pandu Rangan, C., editor, Steffen, Bernhard, editor, Sudan, Madhu, editor, Terzopoulos, Demetri, editor, Tygar, Doug, editor, Vardi, Moshe Y., editor, Weikum, Gerhard, editor, Lirkov, Ivan, editor, Margenov, Svetozar, editor, and Waśniewski, Jerzy, editor

Published
2010
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45. Computing Interfaces in Diverse Applications

Author

Javierre, E., Vermolen, F. J., Vuik, C., Wesseling, P., van der Zwaag, S., Oñate, E., editor, Eberhardsteiner, Josef, editor, Hellmich, Christian, editor, Mang, Herbert A., editor, and Périaux, Jacques, editor

Published
2009
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48. Acceleration of Preconditioned Krylov Solvers for Bubbly Flow Problems

Author

Tang, J. M., Vuik, C., Hutchison, David, editor, Kanade, Takeo, editor, Kittler, Josef, editor, Kleinberg, Jon M., editor, Mattern, Friedemann, editor, Mitchell, John C., editor, Naor, Moni, editor, Nierstrasz, Oscar, editor, Rangan, C. Pandu, editor, Steffen, Bernhard, editor, Sudan, Madhu, editor, Terzopoulos, Demetri, editor, Tygar, Doug, editor, Vardi, Moshe Y., editor, Weikum, Gerhard, editor, Shi, Yong, editor, van Albada, Geert Dick, editor, Dongarra, Jack, editor, and Sloot, Peter M. A., editor

Published
2007
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49. Numerical Methods for Reacting Gas Flow Simulations

Author

van Veldhuizen, S., Vuik, C., Kleijn, C. R., Hutchison, David, editor, Kanade, Takeo, editor, Kittler, Josef, editor, Kleinberg, Jon M., editor, Mattern, Friedemann, editor, Mitchell, John C., editor, Naor, Moni, editor, Nierstrasz, Oscar, editor, Pandu Rangan, C., editor, Steffen, Bernhard, editor, Sudan, Madhu, editor, Terzopoulos, Demetri, editor, Tygar, Dough, editor, Vardi, Moshe Y., editor, Weikum, Gerhard, editor, Alexandrov, Vassil N., editor, van Albada, Geert Dick, editor, Sloot, Peter M. A., editor, and Dongarra, Jack, editor

Published
2006
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