Download or read book A Level Set Approach to the Numerical Simulation of the Stefan Problem on Non graded Adaptive Cartesian Grids written by Han Chen and published by . This book was released on 2007 with total page 198 pages. Available in PDF, EPUB and Kindle. Book excerpt: We present a level set approach to the numerical simulation of the Stefan problem on non-graded adaptive Cartesian grids, i.e. grids for which the size ratio between adjacent cells is not constrained. We use the quadtree (in 2D) and octree (in 3D) data structures to discretize the computational domain and a simple recursive algorithm to automatically generate the adaptive grid. We use the level set method to keep track of the evolving interface between the two phases. We first present finite difference schemes for solving the variable coefficient Poisson and heat equations on irregular domains with Dirichlet boundary conditions on the irregular boundary. These schemes produce solutions that are second-order accurate in both the L 1 and the Linfinity norms. In addition, the solution gradients are also second-order accurate in the L 1 and the Linfinity norms, a particularly desirable feature in the case of the Stefan problem since the evolution velocity of the interface is related to the jump of the solution gradients. Second, we use these schemes as the basis of a numerical algorithm to solve the moving boundary problem associated with the Stefan problem on non-graded adaptive Cartesian grids. We demonstrate the supra-convergence of our method in the L1 and the Linfinity norms. We also demonstrate that our method can simulate physical effects such as surface tension, crystalline anisotropy, molecular kinetics and undercoolings. Finally, we demonstrate the tremendous saving in computational time and memory usage, as well as the accuracy of our method, when compared with the state-of-the-art level set methods on uniform grids.
Download or read book Dissertation Abstracts International written by and published by . This book was released on 2007 with total page 960 pages. Available in PDF, EPUB and Kindle. Book excerpt:
Download or read book The Stefan Problem written by L. I. Rubinšteĭn and published by American Mathematical Soc.. This book was released on 2000-01-25 with total page 429 pages. Available in PDF, EPUB and Kindle. Book excerpt: Translations of Mathematical Monographs
Download or read book Sparse Grids and Applications written by Jochen Garcke and published by Springer Science & Business Media. This book was released on 2012-10-13 with total page 290 pages. Available in PDF, EPUB and Kindle. Book excerpt: In the recent decade, there has been a growing interest in the numerical treatment of high-dimensional problems. It is well known that classical numerical discretization schemes fail in more than three or four dimensions due to the curse of dimensionality. The technique of sparse grids helps overcome this problem to some extent under suitable regularity assumptions. This discretization approach is obtained from a multi-scale basis by a tensor product construction and subsequent truncation of the resulting multiresolution series expansion. This volume of LNCSE is a collection of the papers from the proceedings of the workshop on sparse grids and its applications held in Bonn in May 2011. The selected articles present recent advances in the mathematical understanding and analysis of sparse grid discretization. Aspects arising from applications are given particular attention.
Download or read book Sparse Grids and Applications Miami 2016 written by Jochen Garcke and published by Springer. This book was released on 2018-06-20 with total page 265 pages. Available in PDF, EPUB and Kindle. Book excerpt: Sparse grids are a popular tool for the numerical treatment of high-dimensional problems. Where classical numerical discretization schemes fail in more than three or four dimensions, sparse grids, in their different flavors, are frequently the method of choice. This volume of LNCSE presents selected papers from the proceedings of the fourth workshop on sparse grids and applications, and demonstrates once again the importance of this numerical discretization scheme. The articles present recent advances in the numerical analysis of sparse grids in connection with a range of applications including computational chemistry, computational fluid dynamics, and big data analytics, to name but a few.
Download or read book Sparse Grids and Applications Stuttgart 2014 written by Jochen Garcke and published by Springer. This book was released on 2016-03-16 with total page 348 pages. Available in PDF, EPUB and Kindle. Book excerpt: This volume of LNCSE is a collection of the papers from the proceedings of the third workshop on sparse grids and applications. Sparse grids are a popular approach for the numerical treatment of high-dimensional problems. Where classical numerical discretization schemes fail in more than three or four dimensions, sparse grids, in their different guises, are frequently the method of choice, be it spatially adaptive in the hierarchical basis or via the dimensionally adaptive combination technique. Demonstrating once again the importance of this numerical discretization scheme, the selected articles present recent advances on the numerical analysis of sparse grids as well as efficient data structures. The book also discusses a range of applications, including uncertainty quantification and plasma physics.
Download or read book A Multigrid Method on Non graded Adaptive Octree quadtree Cartesian Grids written by Maxime Theillard and published by . This book was released on 2010 with total page 126 pages. Available in PDF, EPUB and Kindle. Book excerpt:
Download or read book Sparse Grids and Applications Munich 2018 written by Hans-Joachim Bungartz and published by Springer Nature. This book was released on 2022-03-14 with total page 268 pages. Available in PDF, EPUB and Kindle. Book excerpt: Sparse grids are a popular tool for the numerical treatment of high-dimensional problems. Where classical numerical discretization schemes fail in more than three or four dimensions, sparse grids, in their different flavors, are frequently the method of choice. This volume of LNCSE presents selected papers from the proceedings of the fifth workshop on sparse grids and applications, and demonstrates once again the importance of this numerical discretization scheme. The articles present recent advances in the numerical analysis of sparse grids in connection with a range of applications including uncertainty quantification, plasma physics simulations, and computational chemistry, to name but a few.
Download or read book A New Fast and Accurate Grid Deformation Method for R adaptivity in the Context of High Performance Computing written by Matthias Grajewski and published by Logos Verlag Berlin GmbH. This book was released on 2008 with total page 158 pages. Available in PDF, EPUB and Kindle. Book excerpt: Thema der vorliegenden Arbeit ist die Entwicklung und Realisierung einer neuen Methode zur Gitterdeformation im Hinblick auf die Anwendung in r- und rh-adaptiven Verfahren. Das neu entwickelte Verfahren wird sowohl numerisch als auchtheoretisch analysiert. Die vorliegende Arbeit ist in sechs Kapitel gegliedert.Das erste Kapitel stellt den gegenwartigen Stand der Mathematik auf den Gebieten Fehlerkontrolle und Gittersteuerung dar. Besonderes Augenmerk liegt auf einer ubersicht gangiger Verfahren zur Gitterdeformation, also Methoden zur Neuanordnung der Punkte eines gegebenen Gitters unter Beibehaltung seiner Topologie. Hinzu kommt die Darstellung des Aspekts der hardware-orientierten Numerik. Hier geht es darum, durch ein geeignetes Design der numerischen Verfahren die Leistungsfahigkeit heutiger Computer voll auszuschopfen. Im zweiten Kapitel wird das Basisverfahren zur Gitterdeformation hergeleitet und grundlegende Eigenschaften der Methode bewiesen. Das neue Verfahren wird mit dem Vorlauferverfahren von Liao verglichen, wobei die weitaus grossere Flexibilitat der neuen Methode deutlich wird. Im letzten Teil dieses Kapitels wird die numerische Realisierung der Deformationsmethode mithilfe von FE-Ansatzen thematisiert. Das dritte Kapitel beinhaltet den Kern der Dissertation: Die theoretische und numerische Analyse des im zweiten Kapitel vorgestellten Basisverfahrens. Nach der Formulierung eines geeigneten Konvergenzbegriffes wird die Konvergenz der numerischen Realisierung des Deformationsverfahrens bewiesen. Sowohl der Konvergenzbegriff als auch die Konvergenzaussage sind neu und wurden bisher in der Literatur auch nicht in ahnlicher Weise formuliert. Ausfuhrliche numerische Tests bestatigen die theoretischen Ergebnisse. Das in Kapitel 2 eingefuhrte Basisverfahren wird hinsichtlich Genauigkeit und Robustheit weiterentwickelt. Durch geeignete Ausnutzung der Gitterhierarchie gelingt es mit der sog. Multilevel-Deformation ein Verfahren bereitzustellen, welches von optimaler Komplexitat ist. Im letzten Teil des Kapitels wird diese Multilevel-Deformation auf das L-Gebiet angewandt. Den Schwerpunkt im vierten Kapitel bildet die Anwendung der Gitterdeformation auf die Poisson-Gleichung auf dem L-Gebiet. Der Gradientenfehler wird auf a priori deformierten Gittern betrachtet. Die mit der Gitterdeformation erzeugten Gitter ermoglichen eine fur Q_1-Elemente optimale Konvergenzordnung. Nach einer numerischen Untersuchung des ZZ-Schatzers auf solchen Gittern wird ein voll r-adaptiver Algorithmus formuliert und getestet. Die Gitterdeformation wird nun vollautomatisch durch der geschatzten Fehlerverteilung gesteuert. Die so gewonnenen Ergebnisse sind in ihrer Genauigkeit mit den a priori gewonnen Resultaten vergleichbar. Das funfte Kapitel beinhaltet eine Erweiterung des zuvor eingefuhrten r-adaptiven Verfahrens um regulare Gitterverfeinerung und ihre Anwendung auf die Laplace-Gleichung auf dem L-Gebiet. Es werden zwei rh-adaptive Algorithmen betrachtet. Der zweite Algorithmus verzichtet auf hangende Knoten und erweist sich dem ersten als uberlegen; die Resultate entsprechen in ihrer Genauigkeit etwa den mit reiner r-Adaptivitat gewonnen Werten, erfordern aber eine erheblich geringere Rechenzeit.Im letzten Abschnitt wird die zuvor entwickelte rh-adaptive Methode auf Diffusionsprobleme mit anisotropem Diffusionstensor angewendet, die sich aus Problemen des Grundwasserflusses motivieren. Mithilfe des rh-adaptiven Verfahrens konnen nicht nur der Gradientenfehler, sondern auch abgeleitete Grossen wie der Punktfehler signifikant verringert werden. Hierbei wird die Verteilung des Punktfehlers mithilfe der DWR-Methode ermittelt. Das sechste und letzte Kapitel beinhaltet eine Zusammenfassung der Arbeit und zeigtAnsatzpunkte fur eine Fortentwicklung auf.
Download or read book Multigrid Methods written by Stephen F. McCormick and published by SIAM. This book was released on 1987-12-01 with total page 292 pages. Available in PDF, EPUB and Kindle. Book excerpt: A thoughtful consideration of the current level of development of multigrid methods, this volume is a carefully edited collection of papers that addresses its topic on several levels. The first three chapters orient the reader who is familiar with standard numerical techniques to multigrid methods, first by discussing multigrid in the context of standard techniques, second by detailing the mechanics of use of the method, and third by applying the basic method to some current problems in fluid dynamics. The fourth chapter provides a unified development, complete with theory, of algebraic multigrid (AMG), which is a linear equation solver based on multigrid principles. The last chapter is an ambitious development of a very general theory of multigrid methods for variationally posed problems. Included as an appendix is the latest edition of the Multigrid Bibliography, an attempted compilation of all existing research publications on multigrid.
Download or read book Sparse Grids and Applications Munich 2012 written by Jochen Garcke and published by Springer Science & Business Media. This book was released on 2014-04-11 with total page 345 pages. Available in PDF, EPUB and Kindle. Book excerpt: Sparse grids have gained increasing interest in recent years for the numerical treatment of high-dimensional problems. Whereas classical numerical discretization schemes fail in more than three or four dimensions, sparse grids make it possible to overcome the “curse” of dimensionality to some degree, extending the number of dimensions that can be dealt with. This volume of LNCSE collects the papers from the proceedings of the second workshop on sparse grids and applications, demonstrating once again the importance of this numerical discretization scheme. The selected articles present recent advances on the numerical analysis of sparse grids as well as efficient data structures, and the range of applications extends to uncertainty quantification settings and clustering, to name but a few examples.
Download or read book Mathematical Aspects of Numerical Grid Generation written by Jos E. Castillo and published by SIAM. This book was released on 1991-01-01 with total page 171 pages. Available in PDF, EPUB and Kindle. Book excerpt: Numerical grid generation plays a critical role in any scientific computing problem when the geometry of the underlying region is complex or when the solution has a complex structure. The mathematical aspects of grid generation are discussed to provide a deeper understanding of the algorithms and their imitations. Variational methods are emphasized because they are more robust, but elliptic and transcendental algebraic methods are also considered.
Download or read book Numerical Grid Generation written by Joe F. Thompson and published by North Holland. This book was released on 1985 with total page 508 pages. Available in PDF, EPUB and Kindle. Book excerpt:
Download or read book The Stefan Problem written by L. I. Rubinshteĭn and published by . This book was released on 1971 with total page 429 pages. Available in PDF, EPUB and Kindle. Book excerpt:
Download or read book An Overset Adaptive Cartesian prism Grid Method for Moving Boundary Problems written by Ravishekar Kannan and published by . This book was released on 2005 with total page 138 pages. Available in PDF, EPUB and Kindle. Book excerpt:
Download or read book A Time accurate Adaptive Grid Method and the Numerical Simulation of a Shock vortex Interaction written by Michael J. Bockelie and published by . This book was released on 1990 with total page 24 pages. Available in PDF, EPUB and Kindle. Book excerpt:
Download or read book One dimensional Stefan Problems written by James M. Hill and published by Longman Scientific and Technical. This book was released on 1987 with total page 232 pages. Available in PDF, EPUB and Kindle. Book excerpt:
