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Finite Difference Methods for Solving Differential
FINITE DIFFERENCE METHODS FOR SOLVING DIFFERENTIAL EQUATIONS I-Liang Chern Department of Mathematics National Taiwan University May 16, 2013 2 Contents 1 Introduction 3 1.1 Finite Difference Approximation . ........ 3 1.2 Basic Numerical Methods for Ordinary Differential Equations ........... 5 1.3 Runge-Kuttamethods..... ..... ...... ..... ...... ... ... 8 1.4 Multistepmethods ................................ 10 1.5 Linear difference equation . ...... 14 1.6 Stabilityanalysis ............................... 17 1.6.1 ZeroStability................................. 18 2 Finite Difference Methods for Linear Parabolic Equations 23 2.1 Finite Difference Methods for the Heat Equation . ........... 23 2.1.1 Some discretization methods . 23 2.1.2 Stability and Convergence for the Forward Euler method.......... 25 2.2 L2 Stability – von Neumann Analysis . 26 2.3 Energymethod .................................... 28 2.4 Stability Analysis for Montone Operators– Entropy Estimates ........... 29 2.5 Entropy estimate for backward Euler method . ......... 30 2.6 ExistenceTheory ................................. 32 2.6.1 Existence via forward Euler method . ..... 32 2.6.2 A Sharper Energy Estimate for backward Euler method . ......... 33 2.7 Relaxationoferrors.............................. 34 2.8 BoundaryConditions ..... ..... ...... ..... ...... ... 36 2.8.1 Dirichlet boundary condition . ..... 36 2.8.2 Neumann boundary condition . 37 2.9 The discrete Laplacian and its inversion . .......... 38 2.9.1 Dirichlet boundary condition . ..... 38 3 Finite Difference Methods for Linear elliptic Equations 41 3.1 Discrete Laplacian in two dimensions . ........ 41 3.1.1 Discretization methods . 41 3.1.2 The 9-point discrete Laplacian . ..... 42 3.2 Stability of the discrete Laplacian . ......... 43 3 4 CONTENTS 3.2.1 Fouriermethod ................................ 43 3.2.2 Energymethod ................................ 44 4 Finite Difference Theory For Linear Hyperbolic Equations 47 4.1 A review of smooth theory of linear hyperbolic equations ............. -
Exercises from Finite Difference Methods for Ordinary and Partial
Exercises from Finite Difference Methods for Ordinary and Partial Differential Equations by Randall J. LeVeque SIAM, Philadelphia, 2007 http://www.amath.washington.edu/ rjl/fdmbook ∼ Under construction | more to appear. Contents Chapter 1 4 Exercise 1.1 (derivation of finite difference formula) . 4 Exercise 1.2 (use of fdstencil) . 4 Chapter 2 5 Exercise 2.1 (inverse matrix and Green's functions) . 5 Exercise 2.2 (Green's function with Neumann boundary conditions) . 5 Exercise 2.3 (solvability condition for Neumann problem) . 5 Exercise 2.4 (boundary conditions in bvp codes) . 5 Exercise 2.5 (accuracy on nonuniform grids) . 6 Exercise 2.6 (ill-posed boundary value problem) . 6 Exercise 2.7 (nonlinear pendulum) . 7 Chapter 3 8 Exercise 3.1 (code for Poisson problem) . 8 Exercise 3.2 (9-point Laplacian) . 8 Chapter 4 9 Exercise 4.1 (Convergence of SOR) . 9 Exercise 4.2 (Forward vs. backward Gauss-Seidel) . 9 Chapter 5 11 Exercise 5.1 (Uniqueness for an ODE) . 11 Exercise 5.2 (Lipschitz constant for an ODE) . 11 Exercise 5.3 (Lipschitz constant for a system of ODEs) . 11 Exercise 5.4 (Duhamel's principle) . 11 Exercise 5.5 (matrix exponential form of solution) . 11 Exercise 5.6 (matrix exponential form of solution) . 12 Exercise 5.7 (matrix exponential for a defective matrix) . 12 Exercise 5.8 (Use of ode113 and ode45) . 12 Exercise 5.9 (truncation errors) . 13 Exercise 5.10 (Derivation of Adams-Moulton) . 13 Exercise 5.11 (Characteristic polynomials) . 14 Exercise 5.12 (predictor-corrector methods) . 14 Exercise 5.13 (Order of accuracy of Runge-Kutta methods) . -
THE FUTURE of SCREENS from James Stanton a Little Bit About Me
THE FUTURE OF SCREENS From james stanton A little bit about me. Hi I am James (Mckenzie) Stanton Thinker / Designer / Engineer / Director / Executive / Artist / Human / Practitioner / Gardner / Builder / and much more... Born in Essex, United Kingdom and survived a few hair raising moments and learnt digital from the ground up. Ok enough of the pleasantries I have been working in the design field since 1999 from the Falmouth School of Art and onwards to the RCA, and many companies. Ok. less about me and more about what I have seen… Today we are going to cover - SCREENS CONCEPTS - DIGITAL TRANSFORMATION - WHY ASSETS LIBRARIES - CODE LIBRARIES - COST EFFECTIVE SOLUTION FOR IMPLEMENTATION I know, I know, I know. That's all good and well, but what does this all mean to a company like mine? We are about to see a massive change in consumer behavior so let's get ready. DIGITAL TRANSFORMATION AS A USP Getting this correct will change your company forever. DIGITAL TRANSFORMATION USP-01 Digital transformation (DT) – the use of technology to radically improve performance or reach of enterprises – is becoming a hot topic for companies across the globe. VERY DIGITAL CHANGING NOT VERY DIGITAL DIGITAL TRANSFORMATION USP-02 Companies face common pressures from customers, employees and competitors to begin or speed up their digital transformation. However they are transforming at different paces with different results. VERY DIGITAL CHANGING NOT VERY DIGITAL DIGITAL TRANSFORMATION USP-03 Successful digital transformation comes not from implementing new technologies but from transforming your organisation to take advantage of the possibilities that new technologies provide. -
Xcode Package from App Store
KH Computational Physics- 2016 Introduction Setting up your computing environment Installation • MAC or Linux are the preferred operating system in this course on scientific computing. • Windows can be used, but the most important programs must be installed – python : There is a nice package ”Enthought Python Distribution” http://www.enthought.com/products/edudownload.php – C++ and Fortran compiler – BLAS&LAPACK for linear algebra – plotting program such as gnuplot Kristjan Haule, 2016 –1– KH Computational Physics- 2016 Introduction Software for this course: Essentials: • Python, and its packages in particular numpy, scipy, matplotlib • C++ compiler such as gcc • Text editor for coding (for example Emacs, Aquamacs, Enthought’s IDLE) • make to execute makefiles Highly Recommended: • Fortran compiler, such as gfortran or intel fortran • BLAS& LAPACK library for linear algebra (most likely provided by vendor) • open mp enabled fortran and C++ compiler Useful: • gnuplot for fast plotting. • gsl (Gnu scientific library) for implementation of various scientific algorithms. Kristjan Haule, 2016 –2– KH Computational Physics- 2016 Introduction Installation on MAC • Install Xcode package from App Store. • Install ‘‘Command Line Tools’’ from Apple’s software site. For Mavericks and lafter, open Xcode program, and choose from the menu Xcode -> Open Developer Tool -> More Developer Tools... You will be linked to the Apple page that allows you to access downloads for Xcode. You wil have to register as a developer (free). Search for the Xcode Command Line Tools in the search box in the upper left. Download and install the correct version of the Command Line Tools, for example for OS ”El Capitan” and Xcode 7.2, Kristjan Haule, 2016 –3– KH Computational Physics- 2016 Introduction you need Command Line Tools OS X 10.11 for Xcode 7.2 Apple’s Xcode contains many libraries and compilers for Mac systems. -
How to Access Python for Doing Scientific Computing
How to access Python for doing scientific computing1 Hans Petter Langtangen1,2 1Center for Biomedical Computing, Simula Research Laboratory 2Department of Informatics, University of Oslo Mar 23, 2015 A comprehensive eco system for scientific computing with Python used to be quite a challenge to install on a computer, especially for newcomers. This problem is more or less solved today. There are several options for getting easy access to Python and the most important packages for scientific computations, so the biggest issue for a newcomer is to make a proper choice. An overview of the possibilities together with my own recommendations appears next. Contents 1 Required software2 2 Installing software on your laptop: Mac OS X and Windows3 3 Anaconda and Spyder4 3.1 Spyder on Mac............................4 3.2 Installation of additional packages.................5 3.3 Installing SciTools on Mac......................5 3.4 Installing SciTools on Windows...................5 4 VMWare Fusion virtual machine5 4.1 Installing Ubuntu...........................6 4.2 Installing software on Ubuntu....................7 4.3 File sharing..............................7 5 Dual boot on Windows8 6 Vagrant virtual machine9 1The material in this document is taken from a chapter in the book A Primer on Scientific Programming with Python, 4th edition, by the same author, published by Springer, 2014. 7 How to write and run a Python program9 7.1 The need for a text editor......................9 7.2 Spyder................................. 10 7.3 Text editors.............................. 10 7.4 Terminal windows.......................... 11 7.5 Using a plain text editor and a terminal window......... 12 8 The SageMathCloud and Wakari web services 12 8.1 Basic intro to SageMathCloud................... -
Ginga Documentation Release 2.5.20160420043855
Ginga Documentation Release 2.5.20160420043855 Eric Jeschke May 05, 2016 Contents 1 About Ginga 1 2 Copyright and License 3 3 Requirements and Supported Platforms5 4 Getting the source 7 5 Building and Installation 9 5.1 Detailed Installation Instructions for Ginga...............................9 6 Documentation 15 6.1 What’s New in Ginga?.......................................... 15 6.2 Ginga Quick Reference......................................... 19 6.3 The Ginga FAQ.............................................. 22 6.4 The Ginga Viewer and Toolkit Manual................................. 25 6.5 Reference/API.............................................. 87 7 Bug reports 107 8 Developer Info 109 9 Etymology 111 10 Pronunciation 113 11 Indices and tables 115 Python Module Index 117 i ii CHAPTER 1 About Ginga Ginga is a toolkit designed for building viewers for scientific image data in Python, visualizing 2D pixel data in numpy arrays. It can view astronomical data such as contained in files based on the FITS (Flexible Image Transport System) file format. It is written and is maintained by software engineers at the Subaru Telescope, National Astronomical Observatory of Japan. The Ginga toolkit centers around an image display class which supports zooming and panning, color and intensity mapping, a choice of several automatic cut levels algorithms and canvases for plotting scalable geometric forms. In addition to this widget, a general purpose “reference” FITS viewer is provided, based on a plugin framework. A fairly complete set of “standard” plugins are provided for features that we expect from a modern FITS viewer: panning and zooming windows, star catalog access, cuts, star pick/fwhm, thumbnails, etc. 1 Ginga Documentation, Release 2.5.20160420043855 2 Chapter 1. -
Curtains Up! Lights, Camera, Action! Documenting the Creation Of
Curtains Up! Lights, Camera, Action! Documenting the Creation of Theater and Opera Productions with Linked Data and Web Technologies Thomas Steiner, Rémi Ronfard, Pierre-Antoine Champin, Benoît Encelle, Yannick Prié To cite this version: Thomas Steiner, Rémi Ronfard, Pierre-Antoine Champin, Benoît Encelle, Yannick Prié. Curtains Up! Lights, Camera, Action! Documenting the Creation of Theater and Opera Productions with Linked Data and Web Technologies. International Conference on Web Engineering ICWE 2015, International Society for the Web Engineering, Jun 2015, Amsterdam, Netherlands. pp.10. hal-01159826 HAL Id: hal-01159826 https://hal.inria.fr/hal-01159826 Submitted on 22 Dec 2015 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. Curtains Up! Lights, Camera, Action! Documenting the Creation of Theater and Opera Productions with Linked Data and Web Technologies Thomas Steiner1?, R´emiRonfard2 Pierre-Antoine Champin1, Beno^ıtEncelle1, and Yannick Pri´e3 1CNRS, Universit´ede Lyon, LIRIS { UMR5205, Universit´eLyon 1, France ftsteiner, [email protected], [email protected] 2 Inria Grenoble Rh^one-Alpes / LJK Laboratoire J. Kuntzmann - IMAGINE, France [email protected] 3CNRS, Universit´ede Nantes, LINA { UMR 6241, France [email protected] Abstract. -
Polymer Libraries Photoresponsive Polymers II Volume Editors: Meier, M.A.R., Webster, D.C
225 Advances in Polymer Science Editorial Board: A. Abe · A.-C. Albertsson · K. Dušek · W.H. de Jeu H.-H. Kausch · S. Kobayashi · K.-S. Lee · L. Leibler T.E. Long · I. Manners · M. Möller · O. Nuyken E.M. Terentjev · M. Vicent · B. Voit G. Wegner · U. Wiesner Advances in Polymer Science Recently Published and Forthcoming Volumes Polymer Libraries Photoresponsive Polymers II Volume Editors: Meier, M.A.R., Webster, D.C. Volume Editors: Marder, S.R., Lee, K.-S. Vol. 225, 2010 Vol. 214, 2008 Polymer Membranes/Biomembranes Photoresponsive Polymers I Volume Editors: Meier, W.P., Knoll, W. Volume Editors: Marder, S.R., Lee, K.-S. Vol. 224, 2010 Vol. 213, 2008 Organic Electronics Polyfluorenes Volume Editors: Meller, G., Grasser, T. Volume Editors: Scherf, U., Neher, D. Vol. 223, 2010 Vol. 212, 2008 Inclusion Polymers Chromatography for Sustainable Polymeric Volume Editor: Wenz, G. Materials Vol. 222, 2009 Renewable, Degradable and Recyclable Volume Editors: Albertsson, A.-C., Advanced Computer Simulation Hakkarainen, M. Approaches for Soft Matter Sciences III Vol. 211, 2008 Volume Editors: Holm, C., Kremer, K. Vol. 221, 2009 Wax Crystal Control · Nanocomposites Stimuli-Responsive Polymers Self-Assembled Nanomaterials II Vol. 210, 2008 Nanotubes Volume Editor: Shimizu, T. Functional Materials and Biomaterials Vol. 220, 2008 Vol. 209, 2007 Self-Assembled Nanomaterials I Phase-Separated Interpenetrating Polymer Nanofibers Networks Volume Editor: Shimizu, T. Authors: Lipatov, Y.S., Alekseeva, T. Vol. 219, 2008 Vol. 208, 2007 Interfacial Processes and Molecular Aggregation of Surfactants Hydrogen Bonded Polymers Volume Editor: Narayanan, R. Volume Editor: Binder, W. Vol. 218, 2008 Vol. 207, 2007 · New Frontiers in Polymer Synthesis Oligomers Polymer Composites Volume Editor: Kobayashi, S. -
Finite Difference and Discontinuous Galerkin Methods for Wave Equations
Digital Comprehensive Summaries of Uppsala Dissertations from the Faculty of Science and Technology 1522 Finite Difference and Discontinuous Galerkin Methods for Wave Equations SIYANG WANG ACTA UNIVERSITATIS UPSALIENSIS ISSN 1651-6214 ISBN 978-91-554-9927-3 UPPSALA urn:nbn:se:uu:diva-320614 2017 Dissertation presented at Uppsala University to be publicly examined in Room 2446, Polacksbacken, Lägerhyddsvägen 2, Uppsala, Tuesday, 13 June 2017 at 10:15 for the degree of Doctor of Philosophy. The examination will be conducted in English. Faculty examiner: Professor Thomas Hagstrom (Department of Mathematics, Southern Methodist University). Abstract Wang, S. 2017. Finite Difference and Discontinuous Galerkin Methods for Wave Equations. Digital Comprehensive Summaries of Uppsala Dissertations from the Faculty of Science and Technology 1522. 53 pp. Uppsala: Acta Universitatis Upsaliensis. ISBN 978-91-554-9927-3. Wave propagation problems can be modeled by partial differential equations. In this thesis, we study wave propagation in fluids and in solids, modeled by the acoustic wave equation and the elastic wave equation, respectively. In real-world applications, waves often propagate in heterogeneous media with complex geometries, which makes it impossible to derive exact solutions to the governing equations. Alternatively, we seek approximated solutions by constructing numerical methods and implementing on modern computers. An efficient numerical method produces accurate approximations at low computational cost. There are many choices of numerical methods for solving partial differential equations. Which method is more efficient than the others depends on the particular problem we consider. In this thesis, we study two numerical methods: the finite difference method and the discontinuous Galerkin method. -
A High-Order Boris Integrator
A high-order Boris integrator Mathias Winkela,∗, Robert Speckb,a, Daniel Ruprechta aInstitute of Computational Science, University of Lugano, Switzerland. bJ¨ulichSupercomputing Centre, Forschungszentrum J¨ulich,Germany. Abstract This work introduces the high-order Boris-SDC method for integrating the equations of motion for electrically charged particles in an electric and magnetic field. Boris-SDC relies on a combination of the Boris-integrator with spectral deferred corrections (SDC). SDC can be considered as preconditioned Picard iteration to com- pute the stages of a collocation method. In this interpretation, inverting the preconditioner corresponds to a sweep with a low-order method. In Boris-SDC, the Boris method, a second-order Lorentz force integrator based on velocity-Verlet, is used as a sweeper/preconditioner. The presented method provides a generic way to extend the classical Boris integrator, which is widely used in essentially all particle-based plasma physics simulations involving magnetic fields, to a high-order method. Stability, convergence order and conserva- tion properties of the method are demonstrated for different simulation setups. Boris-SDC reproduces the expected high order of convergence for a single particle and for the center-of-mass of a particle cloud in a Penning trap and shows good long-term energy stability. Keywords: Boris integrator, time integration, magnetic field, high-order, spectral deferred corrections (SDC), collocation method 1. Introduction Often when modeling phenomena in plasma physics, for example particle dynamics in fusion vessels or particle accelerators, an externally applied magnetic field is vital to confine the particles in the physical device [1, 2]. In many cases, such as instabilities [3] and high-intensity laser plasma interaction [4], the magnetic field even governs the microscopic evolution and drives the phenomena to be studied. -
HP Chromebook 11 and 14 Notebook Pcs for Education Data Sheet
Data sheet HP Chromebook 11 and 14 A smart investment for any student, school, or district The new HP Chromebook 11 and HP Chromebook 14 for education combine essential HP innovations and support with the Chrome OS™ operating system. More students can access educational content than ever before with these simple-to-deploy and simple-to-manage Chromebook™ notebook computers. And schools and districts can cut down on IT costs and introduce new learning experiences into the classroom with an ever-expanding collection of educational apps. Data sheet | HP Chromebook 11 and 14 Collaborative learning in a flash Personalized learning HP Chromebooks allow you to provide By equipping students with their own your students and teachers with access to devices and IDs, schools can democratize an innovative, Web-based communication and personalize learning. Students can and collaboration platform that is free to explore the Web and all of its resources to all education accounts with no limit to the collect, review, and analyze data. Then, they number of users you can provision. Chrome™ can use Google™ Apps for Education to create devices are optimized for the Web’s vast and share their hypotheses and findings with educational resources. their peers, their teachers, and even their parents and guardians. Integrate rich content into lessons, inspire HP Chromebook 11 collaboration, and encourage students to Designed to engage create and share their own content with the The new HP Chromebook 11 and 14 include world. Chrome devices deliver it all without brilliant 11.6- and 14.0-inch diagonal HD lengthy startup times or tedious training. -
A Quasi-Static Particle-In-Cell Algorithm Based on an Azimuthal Fourier Decomposition for Highly Efficient Simulations of Plasma-Based Acceleration
A quasi-static particle-in-cell algorithm based on an azimuthal Fourier decomposition for highly efficient simulations of plasma-based acceleration: QPAD Fei Lia,c, Weiming And,∗, Viktor K. Decykb, Xinlu Xuc, Mark J. Hoganc, Warren B. Moria,b aDepartment of Electrical Engineering, University of California Los Angeles, Los Angeles, CA 90095, USA bDepartment of Physics and Astronomy, University of California Los Angeles, Los Angeles, CA 90095, USA cSLAC National Accelerator Laboratory, Menlo Park, CA 94025, USA dDepartment of Astronomy, Beijing Normal University, Beijing 100875, China Abstract The three-dimensional (3D) quasi-static particle-in-cell (PIC) algorithm is a very effi- cient method for modeling short-pulse laser or relativistic charged particle beam-plasma interactions. In this algorithm, the plasma response, i.e., plasma wave wake, to a non- evolving laser or particle beam is calculated using a set of Maxwell's equations based on the quasi-static approximate equations that exclude radiation. The plasma fields are then used to advance the laser or beam forward using a large time step. The algorithm is many orders of magnitude faster than a 3D fully explicit relativistic electromagnetic PIC algorithm. It has been shown to be capable to accurately model the evolution of lasers and particle beams in a variety of scenarios. At the same time, an algorithm in which the fields, currents and Maxwell equations are decomposed into azimuthal harmonics has been shown to reduce the complexity of a 3D explicit PIC algorithm to that of a 2D algorithm when the expansion is truncated while maintaining accuracy for problems with near azimuthal symmetry.