DOCSLIB.ORG

High Order Strong Stability Preserving Time Integrators and Numerical Wave Propagation Methods for Hyperbolic Pdes

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text

Load more

High Order Strong Stability Preserving Time Integrators and Numerical Wave Propagation Methods for Hyperbolic PDEs David I. Ketcheson A dissertation submitted in partial fulfillment of the requirements for the degree of Doctor of Philosophy University of Washington 2009 Program Authorized to Offer Degree: Applied Mathematics University of Washington Graduate School This is to certify that I have examined this copy of a doctoral dissertation by David I. Ketcheson and have found that it is complete and satisfactory in all respects, and that any and all revisions required by the final examining committee have been made. Chair of the Supervisory Committee: Randall J. LeVeque Reading Committee: Randall J. LeVeque Bernard Deconinck Kenneth Bube Date: In presenting this dissertation in partial fulfillment of the requirements for the doctoral degree at the University of Washington, I agree that the Library shall make its copies freely available for inspection. I further agree that extensive copying of this dissertation is allowable only for scholarly purposes, consistent with “fair use” as prescribed in the U.S. Copyright Law. Requests for copying or reproduction of this dissertation may be referred to Proquest Information and Learning, 300 North Zeeb Road, Ann Arbor, MI 48106-1346, 1-800-521-0600, to whom the author has granted “the right to reproduce and sell (a) copies of the manuscript in microform and/or (b) printed copies of the manuscript made from microform.” Signature Date .. University of Washington Abstract High Order Strong Stability Preserving Time Integrators and Numerical Wave Propagation Methods for Hyperbolic PDEs David I. Ketcheson Chair of the Supervisory Committee: Professor Randall J. LeVeque Applied Mathematics Hyperbolic PDEs describe the great variety of physical phenomena governed by wave behavior. Numerical methods are necessary to provide approximate solutions to real wave propagation problems. High order accurate methods are often essential to achieve accurate solutions on computationally feasible grids. However, maintaining numerical stability and satisfying physical constraints becomes increasingly difficult as higher order methods are employed. In this thesis, high order numerical tools for hyperbolic PDEs are developed in a method of lines framework. Optimal high order accurate strong sta- bility preserving (SSP) time integrators, for both linear and nonlinear systems, are devel- oped. Improved SSP methods are found as a result of rewriting the relevant optimization problems in a form more amenable to efficient solution. A new, very general class of low-storage Runge-Kutta methods is proposed (based on the form of some optimal SSP methods) and shown to include methods with properties that cannot be achieved by existing classes of low-storage methods. A high order accurate semi-discrete wave propa- gation method is developed in one and two dimensions using wave propagation Riemann solvers and high order weighted essentially non-oscillatory (WENO) reconstruction. The space and time discretizations are combined to achieve a high order accurate method for general hyperbolic PDEs. This method is applied to model solitary waves in a nonlinear, non-dispersive periodic heterogeneous medium. TABLE OF CONTENTS Page List of Figures . vi List of Tables . viii Chapter 1: Introduction . .1 1.1 Motivation . .1 1.1.1 High Order Numerical Methods for Hyperbolic Conservation Laws1 1.1.2 Strong Stability Preserving Time Integration . .2 1.1.3 Flux Differencing and Wave Propagation . .3 1.1.4 Waves in Heterogeneous Media . .4 1.2 Outline . .4 Chapter 2: Numerical Methods for the Initial Value Problem . .6 2.1 The Method of Lines . .6 2.2 Linear Multistep Methods . .7 2.3 Runge-Kutta Methods . .8 2.4 General Linear Methods . .9 2.5 Additive Methods . 13 Chapter 3: Strong Stability Preserving Methods and Absolute Monotonicity . 15 3.1 Strong Stability Preservation . 17 i 3.1.1 TVD Time Integration . 17 3.1.2 The Shu-Osher Form . 19 3.1.3 Strong Stability Properties . 20 3.2 Absolute Monotonicity and Strong Stability Preservation for Linear IVPs . 23 3.3 Absolute Monotonicity of Runge-Kutta Methods . 26 3.3.1 The SSP Coefficient . 32 3.4 Unconditional Strong Stability Preservation . 33 3.5 Negative Coefficients and Downwinding . 33 3.6 Optimal SSP Methods . 35 3.6.1 Efficiency . 35 3.6.2 The Relation Between R and ...................... 36 C Chapter 4: Optimal Threshold Factors for Linear Initial Value Problems . 38 4.1 Threshold Factors for General Linear Methods . 39 4.2 ’Tall Tree’ Order Conditions for Explicit General Linear Methods . 40 4.3 An Upper Bound . 40 4.4 Solution Algorithm . 42 4.5 Optimal Threshold Factors for Explicit Methods . 43 4.5.1 One-step methods . 43 4.5.2 One-stage multistep methods . 43 4.5.3 Multistage multistep methods . 45 4.6 Threshold Factors for Methods with Downwinding . 48 4.6.1 Threshold Factors for Downwinded Explicit GLMs . 48 4.6.2 Optimal One-step Methods with Downwinding . 50 Chapter 5: Optimal SSP Linear Multistep Methods . 53 5.1 General Solution Algorithm . 54 5.2 Bounds on the SSP Coefficient . 54 5.2.1 Explicit Methods . 54 5.2.2 Implicit Methods . 55 5.3 Optimal Methods without Downwinding . 55 5.4 Optimal Methods with Downwinding . 56 Chapter 6: SSP Runge-Kutta Methods . 63 6.1 Bounds on the SSP Coefficient for Runge-Kutta Methods . 64 6.2 Formulation of the Optimization Problem . 67 ii 6.3 Implicit Runge-Kutta Methods . 69 6.3.1 Optimal Methods . 70 6.3.2 Numerical Experiments . 76 6.4 Explicit Runge-Kutta Methods . 82 6.4.1 Memory Considerations . 83 6.4.2 Optimal Methods . 84 6.4.3 Absolute Stability Regions . 91 6.4.4 Internal Stability . 91 6.4.5 Truncation Error Analysis . 94 6.4.6 Embedding optimal SSP methods . 95 6.4.7 Numerical Experiments . 96 6.5 Summary and Conjectures . 100 Chapter 7: Low-Storage Runge-Kutta Methods . 103 7.1 Introduction . 103 7.2 Two-register Methods . 105 7.2.1 Williamson (2N) methods . 106 7.2.2 van der Houwen (2R) methods . 107 7.3 Low-Storage Methods Have Sparse Shu-Osher Forms . 108 7.3.1 2N Methods . 108 7.3.2 2R Methods . 109 7.4 2S Methods . 109 7.4.1 2S* Methods . 110 7.4.2 2S Embedded Pairs . 111 7.4.3 3S* Methods . 112 7.5 Feasibility of Low-storage Assumptions . 114 7.5.1 2N Methods . 114 7.5.2 2R Methods . 114 7.5.3 2S Methods . 115 7.6 Improved low-storage methods . 115 7.7 Conclusions . 117 Chapter 8: Numerical Wave Propagation . 118 8.1 Linear Hyperbolic Systems . 118 8.2 The Semi-discrete Wave-Propagation form of Godunov’s Method . 119 8.3 Extension to Higher Order . 122 iii 8.4 Variable Coefficient Linear Systems . 123 8.5 Nonlinear Systems . 124 8.6 High Order Non-oscillatory Reconstruction of Scalar Functions . 125 8.6.1 Linear (Non-limited) Reconstruction . 125 8.6.2 TVD Reconstruction . 126 8.6.3 Weighted Essentially Non-Oscillatory Reconstruction . 127 8.7 Reconstruction of Vector-valued Functions . 128 8.7.1 Reconstruction of Eigencomponent Coefficients . 128 8.7.2 Characteristic-wise Reconstruction . 129 8.7.3 Wave-slope Reconstruction . 130 8.8 Extension to Two Dimensions . 130 Chapter 9: Numerical Tests . 132 9.1 Methods . 132 9.2 Acoustics . 133 9.2.1 Single Material Interface . 134 9.2.2 Several Interfaces . 138 9.2.3 A Sonic Crystal . 138 9.3 Fluid Dynamics . 142 Chapter 10: Stegotons . 144 10.1 Previous Work . ..
Recommended publications
  • Numerical Solution of Ordinary Differential Equations

    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.
  • Simulation Methods in Physics 1

    Simulation Methods in Physics 1

    ...script in development... Simulation Methods in Physics 1 Prof. Dr. Christian Holm Institute for Computational Physics University of Stuttgart WS 2012/2013 Contents 1 Introduction 3 1.1 Computer Science . 3 1.1.1 Historical developments . 3 1.1.2 TOP 500 Supercomputer list . 4 1.1.3 Moore's law (1970) . 5 1.2 Computer Simulations in Physics . 6 1.2.1 The New Trinity of Physics . 6 1.2.2 Optimization of simulation performance . 8 1.2.3 Optimization of simulation algorithms . 8 1.2.4 Computational approaches for different length scales . 9 2 Molecular Dynamics integrators 11 2.1 Principles of Molecular Dynamics . 11 2.1.1 Ergodicity . 11 2.2 Integration of equations of motion . 12 2.2.1 Algorithms . 12 2.2.2 Estimation of time step . 13 2.2.3 Verlet Algorithm . 13 2.2.4 Velocity Verlet Algorithm . 14 2.2.5 Leapfrog method . 15 2.2.6 Runge-Kutta method . 16 2.2.7 Predictor-Corrector methods . 17 2.2.8 Ljapunov-Characteristics . 17 3 Statistical mechanics - Quick & Dirty 20 3.1 Microstates . 20 3.1.1 Distribution functions . 20 3.1.2 Average values and thermodynamic limit . 22 3.2 Definition of Entropy and Temperature . 23 3.3 Boltzmann distribution . 24 3.3.1 Total energy and free energy . 25 3.4 Quantum mechanics and classcal partition functions . 26 3.5 Ensemble averages . 26 2 1 Introduction 1.1 Computer Science 1.1.1 Historical developments • The first calculating tool for performing arithmetic processes, the Abacus is used ∼ 1000 b.C.

  • Numerical Methods (Wiki) GU Course Work

    Numerical Methods (Wiki) GU course work PDF generated using the open source mwlib toolkit. See http://code.pediapress.com/ for more information. PDF generated at: Sun, 21 Sep 2014 15:07:23 UTC Contents Articles Numerical methods for ordinary differential equations 1 Runge–Kutta methods 7 Euler method 18 Numerical integration 25 Trapezoidal rule 32 Simpson's rule 35 Bisection method 40 Newton's method 44 Root-finding algorithm 57 Monte Carlo method 62 Interpolation 73 Lagrange polynomial 78 References Article Sources and Contributors 84 Image Sources, Licenses and Contributors 86 Article Licenses License 87 Numerical methods for ordinary differential equations 1 Numerical methods for ordinary differential equations Numerical methods for ordinary differential equations are methods used to find numerical approximations to the solutions of ordinary differential equations (ODEs). Their use is also known as "numerical integration", although this term is sometimes taken to mean the computation of integrals. Many differential equations cannot be solved using symbolic computation ("analysis"). For practical purposes, however – such as in engineering – a numeric approximation to the solution is often sufficient. The algorithms studied here can be used to compute such an approximation. An alternative method is to use techniques from calculus to obtain a series expansion of the solution. Ordinary differential equations occur in many scientific disciplines, for instance in physics, chemistry, biology, and economics. In addition, some methods in numerical partial differential equations convert the partial differential equation into an ordinary differential equation, which must then be solved. Illustration of numerical integration for the differential equation The problem Blue: the Euler method, green: the midpoint method, red: the exact solution, The A first-order differential equation is an Initial value problem (IVP) of step size is the form, where f is a function that maps [t ,∞) × Rd to Rd, and the initial 0 condition y ∈ Rd is a given vector.
  • An Analysis of the TR-BDF2 Integration Scheme Sohan

    An Analysis of the TR-BDF2 Integration Scheme Sohan

    An Analysis of the TR-BDF2 integration scheme by Sohan Dharmaraja Submitted to the School of Engineering in Partial Fulfillment of the Requirements for the degree of Master of Science in Computation for Design and Optimization at the MASSACHUSETTS INSTITUTE OF TECHNOLOGY September 2007 @ Massachusetts Institute of Technology 2007. All rights reserved. Author ............................ School of Engineering July 13, 2007 Certified by ...................... ...................... W. Gilbert Strang Professor of Mathematics Thesis Supervisor Accepted by I \\ Jaume Peraire Professor of Aeronautics and Astronautics MASSACHUSETTS INSTrTUTF Co-Director, Computation for Design and Optimization OF TECHNOLOGY SEP 2 7 2007 BARKER LIBRARIES 2 An Analysis of the TR-BDF2 integration scheme by Sohan Dharmaraja Submitted to the School of Engineering on July 13, 2007, in partial fulfillment of the requirements for the degree of Master of Science in Computation for Design and Optimization Abstract We intend to try to better our understanding of how the combined L-stable 'Trape- zoidal Rule with the second order Backward Difference Formula' (TR-BDF2) integra- tor and the standard A-stable Trapezoidal integrator perform on systems of coupled non-linear partial differential equations (PDEs). It was originally Professor Klaus- Jiirgen Bathe who suggested that further analysis was needed in this area. We draw attention to numerical instabilities that arise due to insufficient numerical damp- ing from the Crank-Nicolson method (which is based on the Trapezoidal rule) and demonstrate how these problems can be rectified with the TR-BDF2 scheme. Several examples are presented, including an advection-diffusion-reaction (ADR) problem and the (chaotic) damped driven pendulum. We also briefly introduce how the ideas of splitting methods can be coupled with the TR-BDF2 scheme and applied to the ADR equation to take advantage of the excellent modern day explicit techniques to solve hyperbolic equations.
  • Finite Difference Methods for Differential Equations

    Finite Difference Methods for Differential Equations

    Finite Difference Methods for Differential Equations Randall J. LeVeque DRAFT VERSION for use in the course AMath 585{586 University of Washington Version of September, 2005 WARNING: These notes are incomplete and may contain errors. They are made available primarily for students in my courses. Please contact me for other uses. [email protected] c R. J. LeVeque, 1998{2005 2 c R. J. LeVeque, 2004 | University of Washington | AMath 585{6 Notes Contents I Basic Text 1 1 Finite difference approximations 3 1.1 Truncation errors . 5 1.2 Deriving finite difference approximations . 6 1.3 Polynomial interpolation . 7 1.4 Second order derivatives . 7 1.5 Higher order derivatives . 8 1.6 Exercises . 8 2 Boundary Value Problems 11 2.1 The heat equation . 11 2.2 Boundary conditions . 12 2.3 The steady-state problem . 12 2.4 A simple finite difference method . 13 2.5 Local truncation error . 14 2.6 Global error . 15 2.7 Stability . 15 2.8 Consistency . 16 2.9 Convergence . 16 2.10 Stability in the 2-norm . 17 2.11 Green's functions and max-norm stability . 19 2.12 Neumann boundary conditions . 21 2.13 Existence and uniqueness . 23 2.14 A general linear second order equation . 24 2.15 Nonlinear Equations . 26 2.15.1 Discretization of the nonlinear BVP . 27 2.15.2 Nonconvergence . 29 2.15.3 Nonuniqueness . 29 2.15.4 Accuracy on nonlinear equations . 29 2.16 Singular perturbations and boundary layers . 31 2.16.1 Interior layers . 33 2.17 Nonuniform grids and adaptive refinement .
  • A General Linear Method Approach to the Design and Optimization of Efficient, Accurate, and Easily Implemented Time-Stepping Methods in CFD

    A General Linear Method Approach to the Design and Optimization of Efficient, Accurate, and Easily Implemented Time-Stepping Methods in CFD

    A general linear method approach to the design and optimization of efficient, accurate, and easily implemented time-stepping methods in CFD Victor DeCariaa,1,∗, Sigal Gottliebb, Zachary J. Granta,1, William J. Laytonc aOak Ridge National Laboratory, Computational and Applied Mathematics Group bMathematics Department, University of Massachusetts Dartmouth cDepartment of Mathematics, University of Pittsburgh Abstract In simulations of fluid motion time accuracy has proven to be elusive. We seek highly accurate methods with strong enough stability properties to deal with the richness of scales of many flows. These methods must also be easy to implement within current complex, possibly legacy codes. Herein we develop, analyze and test new time stepping methods addressing these two issues with the goal of accelerating the development of time accurate methods addressing the needs of applications. The new methods are created by introducing inexpensive pre-filtering and post-filtering steps to popular methods which have been implemented and tested within existing codes. We show that pre-filtering and post-filtering a multistep or multi-stage method results in new methods which have both multiple steps and stages: these are general linear methods (GLMs). We utilize the well studied properties of ∗Corresponding author Email addresses: [email protected] (Victor DeCaria), [email protected] (Sigal Gottlieb), [email protected] (Zachary J. Grant), [email protected] (William J. arXiv:2010.06360v1 [math.NA] 13 Oct 2020 Layton) 1This manuscript has been authored by UT-Battelle, LLC, under contract DE-AC05- 00OR22725 with the US Department of Energy (DOE). The US government retains and the publisher, by accepting the article for publication, acknowledges that the US govern- ment retains a nonexclusive, paid-up, irrevocable, worldwide license to publish or repro- duce the published form of this manuscript, or allow others to do so, for US government purposes.