Numerical Integration
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Numerical Solution of Ordinary Differential Equations
NUMERICAL SOLUTION OF ORDINARY DIFFERENTIAL EQUATIONS Kendall Atkinson, Weimin Han, David Stewart University of Iowa Iowa City, Iowa A JOHN WILEY & SONS, INC., PUBLICATION Copyright c 2009 by John Wiley & Sons, Inc. All rights reserved. Published by John Wiley & Sons, Inc., Hoboken, New Jersey. Published simultaneously in Canada. No part of this publication may be reproduced, stored in a retrieval system, or transmitted in any form or by any means, electronic, mechanical, photocopying, recording, scanning, or otherwise, except as permitted under Section 107 or 108 of the 1976 United States Copyright Act, without either the prior written permission of the Publisher, or authorization through payment of the appropriate per-copy fee to the Copyright Clearance Center, Inc., 222 Rosewood Drive, Danvers, MA 01923, (978) 750-8400, fax (978) 646-8600, or on the web at www.copyright.com. Requests to the Publisher for permission should be addressed to the Permissions Department, John Wiley & Sons, Inc., 111 River Street, Hoboken, NJ 07030, (201) 748-6011, fax (201) 748-6008. Limit of Liability/Disclaimer of Warranty: While the publisher and author have used their best efforts in preparing this book, they make no representations or warranties with respect to the accuracy or completeness of the contents of this book and specifically disclaim any implied warranties of merchantability or fitness for a particular purpose. No warranty may be created ore extended by sales representatives or written sales materials. The advice and strategies contained herin may not be suitable for your situation. You should consult with a professional where appropriate. Neither the publisher nor author shall be liable for any loss of profit or any other commercial damages, including but not limited to special, incidental, consequential, or other damages. -
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. -
Vibration Odes 1
Vibration ODEs 1 Vibration problems lead to differential equations with solutions that oscillate in time, typically in a damped or undamped sinusoidal fashion. Such solutions put certain demands on the numerical methods compared to other phenomena whose solutions are monotone or very smooth. Both the frequency and amplitude of the oscillations need to be accurately handled by the numerical schemes. The forthcom- ing text presents a range of different methods, from classical ones (Runge-Kutta and midpoint/Crank-Nicolson methods), to more modern and popular symplectic (ge- ometric) integration schemes (Leapfrog, Euler-Cromer, and Störmer-Verlet meth- ods), but with a clear emphasis on the latter. Vibration problems occur throughout mechanics and physics, but the methods discussed in this text are also fundamen- tal for constructing successful algorithms for partial differential equations of wave nature in multiple spatial dimensions. 1.1 Finite Difference Discretization Many of the numerical challenges faced when computing oscillatory solutions to ODEs and PDEs can be captured by the very simple ODE u00 C u D 0. This ODE is thus chosen as our starting point for method development, implementation, and analysis. 1.1.1 A Basic Model for Vibrations The simplest model of a vibrating mechanical system has the following form: u00 C !2u D 0; u.0/ D I; u0.0/ D 0; t 2 .0; T : (1.1) Here, ! and I are given constants. Section 1.12.1 derives (1.1) from physical prin- ciples and explains what the constants mean. The exact solution of (1.1)is u.t/ D I cos.!t/ : (1.2) © The Author(s) 2017 1 H.P. -
A Position Based Approach to Ragdoll Simulation
LINKÖPING UNIVERSITY Department of Electrical Engineering Master Thesis A Position Based Approach to Ragdoll Simulation Fiammetta Pascucci LITH-ISY-EX- -07/3966- -SE June 2007 LINKÖPING UNIVERSITY Department of Electrical Engineering Master Thesis A position Based Approach to Ragdoll Simulation Fiammetta Pascucci LITH-ISY-EX- -07/3966- -SE June 2007 Supervisor: Ingemar Ragnemalm Examinator: Ingemar Ragnemalm Abstract Create the realistic motion of a character is a very complicated work. This thesis aims to create interactive animation for characters in three dimen- sions using position based approach. Our character is pictured from ragdoll, which is a structure of system particles where all particles are linked by equidistance con- straints. The goal of this thesis is observed the fall in the space of our ragdoll after creating all constraints, as structure, contact and environment constraints. The structure constraint represents all joint constraints which have one, two or three Degree of Freedom (DOF). The contact constraints are represented by collisions between our ragdoll and other objects in the space. Finally, the environment constraints are represented by means of the wall con- straint. The achieved results allow to have a realist fall of our ragdoll in the space. Keywords: Particle System, Verlet’s Integration, Skeleton Animation, Joint Con- straint, Environment Constraint, Skinning, Collision Detection. Acknowledgments I would to thanks all peoples who aid on making my master thesis work. In particular, thanks a lot to the person that most of all has believed in me. This person has always encouraged to go forward, has always given a lot love and help me in difficult movement. -
Chapter 16: Differential Equations
✐ ✐ ✐ ✐ Chapter 16 Differential Equations A vast number of mathematical models in various areas of science and engineering involve differ- ential equations. This chapter provides a starting point for a journey into the branch of scientific computing that is concerned with the simulation of differential problems. We shall concentrate mostly on developing methods and concepts for solving initial value problems for ordinary differential equations: this is the simplest class (although, as you will see, it can be far from being simple), yet a very important one. It is the only class of differential problems for which, in our opinion, up-to-date numerical methods can be learned in an orderly and reasonably complete fashion within a first course text. Section 16.1 prepares the setting for the numerical treatment that follows in Sections 16.2–16.6 and beyond. It also contains a synopsis of what follows in this chapter. Numerical methods for solving boundary value problems for ordinary differential equations receive a quick review in Section 16.7. More fundamental difficulties arise here, so this section is marked as advanced. Most mathematical models that give rise to differential equations in practice involve partial differential equations, where there is more than one independent variable. The numerical treatment of partial differential equations is a vast and complex subject that relies directly on many of the methods introduced in various parts of this text. An orderly development belongs in a more advanced text, though, and our own description in Section 16.8 is downright anecdotal, relying in part on examples introduced earlier. 16.1 Initial value ordinary differential equations Consider the problem of finding a function y(t) that satisfies the ordinary differential equation (ODE) dy = f (t, y), a ≤ t ≤ b. -
Exponential Integration for Hamiltonian Monte Carlo
Exponential Integration for Hamiltonian Monte Carlo Wei-Lun Chao1 [email protected] Justin Solomon2 [email protected] Dominik L. Michels2 [email protected] Fei Sha1 [email protected] 1Department of Computer Science, University of Southern California, Los Angeles, CA 90089 2Department of Computer Science, Stanford University, 353 Serra Mall, Stanford, California 94305 USA Abstract butions that generate random walks in sample space. A fraction of these walks aligned with the target distribution is We investigate numerical integration of ordinary kept, while the remainder is discarded. Sampling efficiency differential equations (ODEs) for Hamiltonian and quality depends critically on the proposal distribution. Monte Carlo (HMC). High-quality integration is For this reason, designing good proposals has long been a crucial for designing efficient and effective pro- subject of intensive research. posals for HMC. While the standard method is leapfrog (Stormer-Verlet)¨ integration, we propose For distributions over continuous variables, Hamiltonian the use of an exponential integrator, which is Monte Carlo (HMC) is a state-of-the-art sampler using clas- robust to stiff ODEs with highly-oscillatory com- sical mechanics to generate proposal distributions (Neal, ponents. This oscillation is difficult to reproduce 2011). The method starts by introducing auxiliary sam- using leapfrog integration, even with carefully pling variables. It then treats the negative log density of selected integration parameters and precondition- the variables’ joint distribution as the Hamiltonian of parti- ing. Concretely, we use a Gaussian distribution cles moving in sample space. Final positions on the motion approximation to segregate stiff components of trajectories are used as proposals. the ODE. -
Simple Quantitative Tests to Validate Sampling from Thermodynamic
Simple quantitative tests to validate sampling from thermodynamic ensembles Michael R. Shirts∗ Department of Chemical Engineering, University of Virginia, VA 22904 E-mail: [email protected] Abstract It is often difficult to quantitatively determine if a new molecular simulation algorithm or software properly implements sampling of the desired thermodynamic ensemble. We present some simple statistical analysis procedures to allow sensitive determination of whether a de- sired thermodynamic ensemble is properly sampled. We demonstrate the utility of these tests for model systems and for molecular dynamics simulations in a range of situations, includ- ing constant volume and constant pressure simulations, and describe an implementation of the tests designed for end users. 1 Introduction Molecular simulations, including both molecular dynamics (MD) and Monte Carlo (MC) tech- arXiv:1208.0910v1 [cond-mat.stat-mech] 4 Aug 2012 niques, are powerful tools to study the properties of complex molecular systems. When used to specifically study thermodynamics of such systems, rather than dynamics, the primary goal of molecular simulation is to generate uncorrelated samples from the appropriate ensemble as effi- ciently as possible. This simulation data can then be used to compute thermodynamic properties ∗To whom correspondence should be addressed 1 of interest. Simulations of several different ensembles may be required to simulate some thermo- dynamic properties, such as free energy differences between states. An ever-expanding number of techniques have been proposed to perform more and more sophisticated sampling from com- plex molecular systems using both MD and MC, and new software tools are continually being introduced in order to implement these algorithms and to take advantage of advances in hardware architecture and programming languages. -
A Spectral Deferred Correction Method Applied to the Shallow Water Equations on a Sphere
OCTOBER 2013 J I A E T A L . 3435 A Spectral Deferred Correction Method Applied to the Shallow Water Equations on a Sphere JUN JIA,JUDITH C. HILL,KATHERINE J. EVANS, AND GEORGE I. FANN Oak Ridge National Laboratory, Oak Ridge, Tennessee MARK A. TAYLOR Sandia National Laboratories, Albuquerque, New Mexico (Manuscript received 13 February 2012, in final form 7 December 2012) ABSTRACT Although significant gains have been made in achieving high-order spatial accuracy in global climate modeling, less attention has been given to the impact imposed by low-order temporal discretizations. For long-time simulations, the error accumulation can be significant, indicating a need for higher-order temporal accuracy. A spectral deferred correction (SDC) method is demonstrated of even order, with second- to eighth-order accuracy and A-stability for the temporal discretization of the shallow water equations within the spectral-element High-Order Methods Modeling Environment (HOMME). Because this method is stable and of high order, larger time-step sizes can be taken while still yielding accurate long-time simulations. The spectral deferred correction method has been tested on a suite of popular benchmark problems for the shallow water equations, and when compared to the explicit leapfrog, five-stage Runge–Kutta, and fully implicit (FI) second-order backward differentiation formula (BDF2) time-integration methods, it achieves higher accuracy for the same or larger time-step sizes. One of the benchmark problems, the linear advection of a Gaussian bell height anomaly, is extended to run for longer time periods to mimic climate-length simula- tions, and the leapfrog integration method exhibited visible degradation for climate length simulations whereas the second-order and higher methods did not. -
Advanced Programming in Engineering Lecture Notes
Advanced Programming in Engineering lecture notes P.J. Visser, S. Luding, S. Srivastava March 13, 2011 Changes 13/3/2011 Added Section 7.4 to 7.7. Added Section 4.4 7/3/2011 Added chapter Chapter 7 23/2/2011 Corrected the equation for the average temperature in terms of micro- scopic quantities, Eq. 3.14. Corrected the equation for the speed distribution in 2D, Eq. 3.19. Added links to sources of `true' random numbers available through the web as footnotes in Section 5.1. Added examples of choices for parameters of the LCG and LFG in Section 5.1.1 and 5.1.2, respectively. Added the chi-square test and correlation functions as ways for testing random numbers in Section 5.1.4. Improved the derivation of the average squared distance of particles in a random walk being linear with time, in Section 5.2.1. Added a discussion of the meaning of the terms in the master equation, in Section 5.2.5. Small changes. 19/2/2011 Corrected the Maxwell-Boltzmann distribution for the speed, Eq. 3.19. 14/2/2011 1 2 Added Chapter 6. 6/2/2011 Verified Eq. 3.15 and added footnote discussing its meaning in case of periodic boundary conditions. Added formula for the radial distribution function in Section 3.4.2, Eq. 3.21. 31/1/2011 Added Chapter 5. 9/1/2010 Corrected labels on the axes in Fig. 3.10, changed U · " and r · σ to U=" and r/σ, respectively. Added truncated version of the Lennard-Jones potential in Section 3.3.1. -
Differential Equations
Differential Equations • Overview of differential equation • Initial value problem • Explicit numeric methods • Implicit numeric methods • Modular implementation Physics-based simulation • It’s an algorithm that produces a sequence of states over time under the laws of physics • What is a state? Physics-based simulation xi xi+1 xi xi+1 = xi +∆x ∆x Physics-based simulation xi Newtonian laws gravity wind gust xi+1 elastic force . integrator xi xi+1 = xi +∆x ∆x Differential equations • What is a differential equation? • It describes the relation between an unknown function and its derivatives • Ordinary differential equation (ODE) • is the relation that contains functions of only one independent variable and its derivatives Ordinary differential equations An ODE is an equality equation involving a function and its derivatives known function x˙(t)=f(x(t)) time derivative of the unknown function that unknown function evaluates the state given time What does it mean to “solve” an ODE? Symbolic solutions • Standard introductory differential equation courses focus on finding solutions analytically • Linear ODEs can be solved by integral transforms • Use DSolve[eqn,x,t] in Mathematica Differential equation: x˙(t)= kx(t) − kt Solution: x(t)=e− Numerical solutions • In this class, we will be concerned with numerical solutions • Derivative function f is regarded as a black box • Given a numerical value x and t, the black box will return the time derivative of x Physics-based simulation xi Newtonian laws gravity wind gust xi+1 elastic force . integrator -
Introduction to Numerical Analysis
INTRODUCTION TO NUMERICAL ANALYSIS Cho, Hyoung Kyu Department of Nuclear Engineering Seoul National University 10. NUMERICAL INTEGRATION 10.1 Background 10.11 Local Truncation Error in Second‐Order 10.2 Euler's Methods Range‐Kutta Method 10.3 Modified Euler's Method 10.12 Step Size for Desired Accuracy 10.4 Midpoint Method 10.13 Stability 10.5 Runge‐Kutta Methods 10.14 Stiff Ordinary Differential Equations 10.6 Multistep Methods 10.7 Predictor‐Corrector Methods 10.8 System of First‐Order Ordinary Differential Equations 10.9 Solving a Higher‐Order Initial Value Problem 10.10 Use of MATLAB Built‐In Functions for Solving Initial‐Value Problems 10.1 Background Ordinary differential equation A differential equation that has one independent variable A first‐order ODE . The first derivative of the dependent variable with respect to the independent variable Example Rates of water inflow and outflow The time rate of change of the mass in the tank Equation for the rate of height change 10.1 Background Time dependent problem Independent variable: time Dependent variable: water level To obtain a specific solution, a first‐order ODE must have an initial condition or constraint that specifies the value of the dependent variable at a particular value of the independent variable. In typical time‐dependent problems . Initial condition . Initial value problem (IVP) 10.1 Background First order ODE statement General form Ex) . Flow lines Analytical solution In many situations an analytical solution is not possible! Numerical solution of a first‐order ODE A set of discrete points that approximate the function y(x) Domain of the solution: , N subintervals 10.1 Background Overview of numerical methods used/or solving a first‐order ODE Start from the initial value Then, estimate the value at a second nearby point third point … Single‐step and multistep approach . -
Transport Properties of Fluids: Symplectic Integrators and Their
Fluid Phase Equilibria 183–184 (2001) 351–361 Transport properties of fluids: symplectic integrators and their usefulness J. Ratanapisit a,b, D.J. Isbister c, J.F. Ely b,∗ a Chemical Engineering and Petroleum Refining Department, Colorado School of Mines, Golden, CO 80401, USA b Department of Chemical Engineering, Prince of Songla University, Hat-Yai, Songkla 90110, Thailand c School of Physics, University College, University of New South Wales, ADFA, Canberra, ACT 2600, Australia Abstract An investigation has been carried out into the effectiveness of using symplectic/operator splitting generated algorithms for the evaluation of transport coefficients in Lennard–Jones fluids. Equilibrium molecular dynamics is used to revisit the Green–Kubo calculation of these transport coefficients through integration of the appropriate correlation functions. In particular, an extensive series of equilibrium molecular dynamic simulations have been per- formed to investigate the accuracy, stability and efficiency of second-order explicit symplectic integrators: position Verlet, velocity Verlet, and the McLauchlan–Atela algorithms. Comparisons are made to nonsymplectic integrators that include the fourth-order Runge–Kutta and fourth-order Gear predictor–corrector methods. These comparisons were performed based on several transport properties of Lennard–Jones fluids: self-diffusion, shear viscosity and thermal conductivity. Because transport properties involve long time simulations to obtain accurate evaluations of their numerical values, they provide an excellent basis to study the accuracy and stability of the SI methods. To our knowledge, previous studies on the SIs have only looked at the thermodynamic energy using a simple model fluid. This study presents realistic, but perhaps the simplest simulations possible to test the effect of the integrators on the three main transport properties.