Runge-Kutta and Collocation Methods Florian Landis
|
|
- Sharyl Barrett
- 6 years ago
- Views:
Transcription
1 Runge-Kutta and Collocation Methods Florian Landis Geometrical Numetric Integration p.1
2 Overview Define Runge-Kutta methods. Introduce collocation methods. Identify collocation methods as Runge-Kutta methods. Find conditions to determine, of what order collocation methods are. Geometrical Numetric Integration p.2
3 Introduction General Goal: Find approximation to the solutions of using one step methods. ẏ(t) = f(t, y), y(t ) = y 3 Examples of one step methods (step size h = 1) for the Riccati equation t y = y 2 + t 2 : 1.5 Explicit Euler Rule 1.5 Explicit Trapezoidal Rule 1.5 Explicit 3 stage Runge Kutta method y y y t 1 t c2 c3 1 t Geometrical Numetric Integration p.3
4 Runga-Kutta Method Definition 1 (Runge-Kutta) Let b i, a ij (i,j = 1,...,s) be real numbers and let c i = s a ij. An s-stage Runge-Kutta method is given by ( k i = f t + c i h, y + h y 1 = y + h b i k i. a ij k j ), i = 1,...,s (1) Distinguish: explicit Runge-Kutta a ij = for j i implicit Runge-Kutta full matrix (a ij ) of non-zero coefficients allowed Implicit function theorem: for h small enough, (1) has a locally unique solution close to k i f(t,y ). Geometrical Numetric Integration p.4
5 Butcher Diagram The coefficients of the Runge-Kutta method are usually displayed in a Butcher diagram: c 1 a a 1s... c s a s1... a ss b 1... b s Example for explicit Runge-Kutta: 1.5 Runge Kutte method for Riccati equation t y=y 2 +t 2 b 1 b 2 b y 1.5 a 21 a 31 a 32 c2 c3 1 t Geometrical Numetric Integration p.5
6 Order of the Runge-Kutta method A general one-step method has order p, if By the Taylor expansions y 1 y(t + h) = O(h p+1 ) ash. y(t + h) = y(t ) + h f(t, y(t )) h2 d dt f(t, y(t)) t=t +... d ( ) ] y 1 = y + h b i [f(t, y ) + h dh f h= t + c i h, y + h a ij k j +... i=1 of y and y 1 of the Runge-Kutta method, one obtains the following conditions for the coefficients: i b i = 1 for order 1, i b ic i = 1/2 for order 2, i b ic 2 i = 1/3 and i,j b ia ij c j = 1/6 for order 3. Geometrical Numetric Integration p.6
7 The Collocation Method Definition 2 (Collocation Method) Let c 1,...,c s be distinct real numbers (usually c i 1). The collocation polynomial u(t) is a polynomial of degree s satisfying u(t ) = y (2) u(t + c i h) = f(t + c i h, u(t + c i h)), i = 1,...,s, (3) and the numerical solution of the collocation method is defined by Scetch of Collocation Polynomial of degree 3 y 1 = u(t + h). y 1.5 y u y 1 t c 1 c 2 c 3 Geometrical Numetric Integration p.7
8 The Collocation Method Theorem 1 (Guillou & Soulé 1969, Wright 197) The collocation method for c 1,...,c s is equivalent to the s-stage Runge-Kutta method with coefficients a ij = ci l j (τ) dτ, b i = 1 l i (τ) dτ, where l i (τ) is the Lagrange polynomial l i (τ) = l i (τ c l)/(c i c l ). Moreover: u(t + τh) = y + h τ k j l j (σ) dσ. Thus, the existence of the collocation polynomial depends on the existence of the k i (given for h ). Geometrical Numetric Integration p.8
9 Proof of Theorem 1 Proof. Let u(t) be the collocation polynomial and define k i := u(t + c i h). By the Lagrange interpolation formula we have u(t + τh) = s k j l j (τ), and by integration we get u(t + c i h) = y + h ci k j l j (τ) dτ. Inserted into the definition of the collocation polynomial u(t + c i h) = f(t + c i h, u(t + c i h)), this gives the first formula of the Runge-Kutta equation ( k i = f t + c i h, y + h a ij k j ). Integration from to 1 yields y 1 = y + h s b ik i. Geometrical Numetric Integration p.9
10 Collocation Coefficients If a Runge-Kutta methods corresponds to a collocation method of order s, a ij = ci l j (τ) dτ, b i = 1 l i (τ) dτ, leads to: C(q = s) : B(p = s) : a ij c k 1 j = ck i k, i, k = 1,...,q b i c k 1 i = 1, k = 1,...,p k since τ k 1 = s j 1 ck 1 j l j (τ) for k = 1,...,s. Note: B(p) y + s i=1 b if(t + hc i ) approximates the solution to ẏ = f(t), y(t ) = y with order p. Geometrical Numetric Integration p.1
11 Order of The Collocation Method Lemma 2 The collocation polynomial u(t) is an approximation of order s to the exact solution of ẏ = f(t,y), y(t ) = y on the whole interval, i.e., u(t) y(t) C h s+1 for t [t, t + h] and for sufficiently small h. Moreover, the derivatives of u(t) satisfy for t [t,t + h] u (k) (t) y (k) (t) C h s+1 k for k =,...,s. Geometrical Numetric Integration p.11
12 Proof of Lemma 2 y u(t + τh) = ẏ(t + τh) = f ( t + c i h, u(t + c i h) ) l j (τ), f ( t + c i h, y(t + c i h) ) t l j (τ) + h s E(τ, h) y t + τh h s E y Σ f i l i u t E(τ, h) 2 max t [t,t +h] y (s+1) (t) s! Integrating the difference of the above two equations gives y(t + τh) u(t + τh) = h τ f i i=1 l i (σ) dσ + h s+1 τ E(σ, h) dσ with f i = f ( t + c i h,y(t + c i h) ) f ( t + c i h,u(t + c i h) ). Geometrical Numetric Integration p.12
13 Proof of Lemma 2 Using a Lipschitz condition for f(t, y) on the Integral y(t + τh) u(t + τh) = h τ f i i=1 l i (σ) dτ + h s+1 τ E(σ, h) dσ yields max y(t) u(t) h C L max y(t) u(t) + Const. t [t,t +h] t [t,t +h] hs+1, implying u(t) y(t) C h s+1 for sufficiently small h >. Geometrical Numetric Integration p.13
14 Superconvergence Theorem 3 (Superconvergence) If the condition B(p) holds for some p s, then the collocation method has order p. This means that the collocation method has the same order as the underlying quadrature formula. B(p) : b i c k 1 i = 1, k = 1,...,p k Note: B(p) cannot be met for p > 2s. Geometrical Numetric Integration p.14
15 Proof of Superconvergence Proof. We consider the collocation polynomial u(t) as the solution of a perturbed differential equation u = f(t, u) + δ(t) with defect δ(t) := u(t) f ( t,u(t) ). Subtracting ẏ(t) = f(t,y) from the above we get after linearization that u(t) ẏ(t) }{{} E(t) = f ( ) ( ) t, y(t) u(t) y(t) y }{{} E(t) where, for t t t + h, the remainder r(t) is of size O ( u(t) y(t) 2) = O(h 2s+2 ) by lemma 2. +δ(t) + r(t), Geometrical Numetric Integration p.15
16 Conclusion Collocation methods with polynomials of degree s are equivalent to s-stage Runge-Kutta methods: a ij = ci l j (τ) dτ, b i = 1 l i (τ) dτ, Collocation polynomials of degrees s lead to collocation methods of order s or better: If B(p) is met for p > s, the corresponding collocation method is of order p. B(p) : b i c k 1 i = 1, k = 1,...,p. k Geometrical Numetric Integration p.16
17 Geometrical Numetric Integration p.17
18 Proof of Lemma 2 The second statement follows from the first one: Taking the kth derivative of y(t + τh) u(t + τh) = h τ f i i=1 l i (σ) dτ + h s+1 τ E(σ, h) dσ gives h k( y (k) (t + τh) u (k) (t + τh) ) = h i=1 f i l (k 1) i (τ) + h s+1 E (k 1) (τ, h). With E (k 1) (τ, h) max t [t,t +h] y (s+1) (t) (s k + 1)! and a Lipschitz condition for f(t,y), u (k) y (k) C h s+1 k follows. Geometrical Numetric Integration p.18
19 Variation of Constants Formula For homogeneous systems of linear equations ẏ(t) = A(t)y(t) with initial condition y(t ) = y, the solution can be written as y(t) = R(t,t )y Ṙ(t,s) = A(t)R(t,s). Using this resolvent of the homogeneous differential system, the solution to inhomogeneous problems ẏ(t) = A(t)y(t) + f(t) can be found with the variation of constants formula: y(t) = R(t,t )y + t t R(t,s)f(s)ds. Geometrical Numetric Integration p.19
20 Proof of Superconvergence With Ṙ(t, s) = f(t, y(t))r(t, s) y The variation of constants formula then yields y 1 y(t + h) = E(t + h) = t +h t ( ) R(t + h, s) δ(s) + r(s) ds as the solution of The contribution of r(t): E(t + h) = f ( ) t, y(t) E(t) + δ(t) + r(t). y r(t) O(h 2s+2 ) t +h t R(t + h, s)r(s) ds O(h 2s+3 ) Geometrical Numetric Integration p.2
21 Proof of Superconvergence The main idea now is to apply the quadrature formula (b i,c i ) s i=1 integral of g(s) = R(t + h,s)δ(s): to the t +h t g(s) ds = b i g(t + hc i ) + quadrature Error. i=1 From δ(s) t +c i h = follows s i=1 b ig(t + hc i ) =. Thus, t +h t p+1 p g(s) ds = quadrature Error h s p g(s). p s p g(s) is bounded independently of h by Lemma 2. Therefore E(t + h) = t +h t ( ) R(t + h, s) δ(s) + r(s) ds O(h p+1 ). Geometrical Numetric Integration p.21
Lecture 4: Numerical solution of ordinary differential equations
Lecture 4: Numerical solution of ordinary differential equations Department of Mathematics, ETH Zürich General explicit one-step method: Consistency; Stability; Convergence. High-order methods: Taylor
More informationConsistency and Convergence
Jim Lambers MAT 77 Fall Semester 010-11 Lecture 0 Notes These notes correspond to Sections 1.3, 1.4 and 1.5 in the text. Consistency and Convergence We have learned that the numerical solution obtained
More informationThe collocation method for ODEs: an introduction
058065 - Collocation Methods for Volterra Integral Related Functional Differential The collocation method for ODEs: an introduction A collocation solution u h to a functional equation for example an ordinary
More informationMultistage Methods I: Runge-Kutta Methods
Multistage Methods I: Runge-Kutta Methods Varun Shankar January, 0 Introduction Previously, we saw that explicit multistep methods (AB methods) have shrinking stability regions as their orders are increased.
More informationmultistep methods Last modified: November 28, 2017 Recall that we are interested in the numerical solution of the initial value problem (IVP):
MATH 351 Fall 217 multistep methods http://www.phys.uconn.edu/ rozman/courses/m351_17f/ Last modified: November 28, 217 Recall that we are interested in the numerical solution of the initial value problem
More informationDefect-based a-posteriori error estimation for implicit ODEs and DAEs
1 / 24 Defect-based a-posteriori error estimation for implicit ODEs and DAEs W. Auzinger Institute for Analysis and Scientific Computing Vienna University of Technology Workshop on Innovative Integrators
More informationNumerical Methods for Differential Equations
Numerical Methods for Differential Equations Chapter 2: Runge Kutta and Linear Multistep methods Gustaf Söderlind and Carmen Arévalo Numerical Analysis, Lund University Textbooks: A First Course in the
More informationNumerical Methods for Differential Equations
Numerical Methods for Differential Equations Chapter 2: Runge Kutta and Multistep Methods Gustaf Söderlind Numerical Analysis, Lund University Contents V4.16 1. Runge Kutta methods 2. Embedded RK methods
More informationIntroduction to the Numerical Solution of IVP for ODE
Introduction to the Numerical Solution of IVP for ODE 45 Introduction to the Numerical Solution of IVP for ODE Consider the IVP: DE x = f(t, x), IC x(a) = x a. For simplicity, we will assume here that
More informationFourth Order RK-Method
Fourth Order RK-Method The most commonly used method is Runge-Kutta fourth order method. The fourth order RK-method is y i+1 = y i + 1 6 (k 1 + 2k 2 + 2k 3 + k 4 ), Ordinary Differential Equations (ODE)
More informationValidated Explicit and Implicit Runge-Kutta Methods
Validated Explicit and Implicit Runge-Kutta Methods Alexandre Chapoutot joint work with Julien Alexandre dit Sandretto and Olivier Mullier U2IS, ENSTA ParisTech 8th Small Workshop on Interval Methods,
More informationEuler s Method, cont d
Jim Lambers MAT 461/561 Spring Semester 009-10 Lecture 3 Notes These notes correspond to Sections 5. and 5.4 in the text. Euler s Method, cont d We conclude our discussion of Euler s method with an example
More informationDo not turn over until you are told to do so by the Invigilator.
UNIVERSITY OF EAST ANGLIA School of Mathematics Main Series UG Examination 216 17 INTRODUCTION TO NUMERICAL ANALYSIS MTHE612B Time allowed: 3 Hours Attempt QUESTIONS 1 and 2, and THREE other questions.
More informationAN OVERVIEW. Numerical Methods for ODE Initial Value Problems. 1. One-step methods (Taylor series, Runge-Kutta)
AN OVERVIEW Numerical Methods for ODE Initial Value Problems 1. One-step methods (Taylor series, Runge-Kutta) 2. Multistep methods (Predictor-Corrector, Adams methods) Both of these types of methods are
More informationNumerical Differential Equations: IVP
Chapter 11 Numerical Differential Equations: IVP **** 4/16/13 EC (Incomplete) 11.1 Initial Value Problem for Ordinary Differential Equations We consider the problem of numerically solving a differential
More informationDiscontinuous Collocation Methods for DAEs in Mechanics
Discontinuous Collocation Methods for DAEs in Mechanics Scott Small Laurent O. Jay The University of Iowa Iowa City, Iowa, USA July 11, 2011 Outline 1 Introduction of the DAEs 2 SPARK and EMPRK Methods
More informationJim Lambers MAT 772 Fall Semester Lecture 21 Notes
Jim Lambers MAT 772 Fall Semester 21-11 Lecture 21 Notes These notes correspond to Sections 12.6, 12.7 and 12.8 in the text. Multistep Methods All of the numerical methods that we have developed for solving
More informationPartitioned Runge-Kutta Methods for Semi-explicit Differential-Algebraic Systems of Index 2
Partitioned Runge-Kutta Methods for Semi-explicit Differential-Algebraic Systems of Index 2 A. Murua Konputazio Zientziak eta A. A. Saila, EHU/UPV, Donostia-San Sebastián email ander@si.ehu.es December
More informationOrdinary Differential Equations
Ordinary Differential Equations Introduction: first order ODE We are given a function f(t,y) which describes a direction field in the (t,y) plane an initial point (t 0,y 0 ) We want to find a function
More informationOrdinary Differential Equations
Chapter 7 Ordinary Differential Equations Differential equations are an extremely important tool in various science and engineering disciplines. Laws of nature are most often expressed as different equations.
More information2.161 Signal Processing: Continuous and Discrete Fall 2008
MIT OpenCourseWare http://ocw.mit.edu 2.6 Signal Processing: Continuous and Discrete Fall 2008 For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms. MASSACHUSETTS
More information2 Numerical Methods for Initial Value Problems
Numerical Analysis of Differential Equations 44 2 Numerical Methods for Initial Value Problems Contents 2.1 Some Simple Methods 2.2 One-Step Methods Definition and Properties 2.3 Runge-Kutta-Methods 2.4
More informationTheory, Solution Techniques and Applications of Singular Boundary Value Problems
Theory, Solution Techniques and Applications of Singular Boundary Value Problems W. Auzinger O. Koch E. Weinmüller Vienna University of Technology, Austria Problem Class z (t) = M(t) z(t) + f(t, z(t)),
More information8.1 Introduction. Consider the initial value problem (IVP):
8.1 Introduction Consider the initial value problem (IVP): y dy dt = f(t, y), y(t 0)=y 0, t 0 t T. Geometrically: solutions are a one parameter family of curves y = y(t) in(t, y)-plane. Assume solution
More informationEuler s Method, Taylor Series Method, Runge Kutta Methods, Multi-Step Methods and Stability.
Euler s Method, Taylor Series Method, Runge Kutta Methods, Multi-Step Methods and Stability. REVIEW: We start with the differential equation dy(t) dt = f (t, y(t)) (1.1) y(0) = y 0 This equation can be
More informationONE- AND TWO-STAGE IMPLICIT INTERVAL METHODS OF RUNGE-KUTTA TYPE
COMPUTATIONAL METHODS IN SCIENCE AND TECHNOLOGY 5,53-65 (1999) ONE- AND TWO-STAGE IMPLICIT INTERVAL METHODS OF RUNGE-KUTTA TYPE ANDRZEJ MARCINIAK 1, BARBARA SZYSZKA 2 1 Institute of Computing Science,
More informationOrdinary Differential Equations II
Ordinary Differential Equations II CS 205A: Mathematical Methods for Robotics, Vision, and Graphics Justin Solomon CS 205A: Mathematical Methods Ordinary Differential Equations II 1 / 29 Almost Done! No
More informationModule 4: Numerical Methods for ODE. Michael Bader. Winter 2007/2008
Outlines Module 4: for ODE Part I: Basic Part II: Advanced Lehrstuhl Informatik V Winter 2007/2008 Part I: Basic 1 Direction Fields 2 Euler s Method Outlines Part I: Basic Part II: Advanced 3 Discretized
More informationIdentification Methods for Structural Systems. Prof. Dr. Eleni Chatzi Lecture March, 2016
Prof. Dr. Eleni Chatzi Lecture 4-09. March, 2016 Fundamentals Overview Multiple DOF Systems State-space Formulation Eigenvalue Analysis The Mode Superposition Method The effect of Damping on Structural
More informationMath 216 Final Exam 14 December, 2012
Math 216 Final Exam 14 December, 2012 This sample exam is provided to serve as one component of your studying for this exam in this course. Please note that it is not guaranteed to cover the material that
More informationOptimal order a posteriori error estimates for a class of Runge Kutta and Galerkin methods
Numerische Mathematik manuscript No. (will be inserted by the editor) Optimal order a posteriori error estimates for a class of Runge Kutta and Galerkin methods Georgios Akrivis 1, Charalambos Makridakis
More informationNumerical Methods - Initial Value Problems for ODEs
Numerical Methods - Initial Value Problems for ODEs Y. K. Goh Universiti Tunku Abdul Rahman 2013 Y. K. Goh (UTAR) Numerical Methods - Initial Value Problems for ODEs 2013 1 / 43 Outline 1 Initial Value
More informationLecture 5: Single Step Methods
Lecture 5: Single Step Methods J.K. Ryan@tudelft.nl WI3097TU Delft Institute of Applied Mathematics Delft University of Technology 1 October 2012 () Single Step Methods 1 October 2012 1 / 44 Outline 1
More information2tdt 1 y = t2 + C y = which implies C = 1 and the solution is y = 1
Lectures - Week 11 General First Order ODEs & Numerical Methods for IVPs In general, nonlinear problems are much more difficult to solve than linear ones. Unfortunately many phenomena exhibit nonlinear
More informationYou may not use your books, notes; calculators are highly recommended.
Math 301 Winter 2013-14 Midterm 1 02/06/2014 Time Limit: 60 Minutes Name (Print): Instructor This exam contains 8 pages (including this cover page) and 6 problems. Check to see if any pages are missing.
More informationFinite Differences for Differential Equations 28 PART II. Finite Difference Methods for Differential Equations
Finite Differences for Differential Equations 28 PART II Finite Difference Methods for Differential Equations Finite Differences for Differential Equations 29 BOUNDARY VALUE PROBLEMS (I) Solving a TWO
More informationRunge-Kutta type methods for differentialalgebraic equations in mechanics
University of Iowa Iowa Research Online Theses and Dissertations Spring 211 Runge-Kutta type methods for differentialalgebraic equations in mechanics Scott Joseph Small University of Iowa Copyright 211
More information1 Ordinary Differential Equations
Ordinary Differential Equations.0 Mathematical Background.0. Smoothness Definition. A function f defined on [a, b] is continuous at ξ [a, b] if lim x ξ f(x) = f(ξ). Remark Note that this implies existence
More informationHigher Order Taylor Methods
Higher Order Taylor Methods Marcelo Julio Alvisio & Lisa Marie Danz May 6, 2007 Introduction Differential equations are one of the building blocks in science or engineering. Scientists aim to obtain numerical
More informationApplied Math for Engineers
Applied Math for Engineers Ming Zhong Lecture 15 March 28, 2018 Ming Zhong (JHU) AMS Spring 2018 1 / 28 Recap Table of Contents 1 Recap 2 Numerical ODEs: Single Step Methods 3 Multistep Methods 4 Method
More informationPart IB Numerical Analysis
Part IB Numerical Analysis Definitions Based on lectures by G. Moore Notes taken by Dexter Chua Lent 206 These notes are not endorsed by the lecturers, and I have modified them (often significantly) after
More informationOrdinary Differential Equations II
Ordinary Differential Equations II CS 205A: Mathematical Methods for Robotics, Vision, and Graphics Justin Solomon CS 205A: Mathematical Methods Ordinary Differential Equations II 1 / 33 Almost Done! Last
More informationNonlinear Control Systems
Nonlinear Control Systems António Pedro Aguiar pedro@isr.ist.utl.pt 3. Fundamental properties IST-DEEC PhD Course http://users.isr.ist.utl.pt/%7epedro/ncs2012/ 2012 1 Example Consider the system ẋ = f
More informationInitial-Value Problems for ODEs. Introduction to Linear Multistep Methods
Initial-Value Problems for ODEs Introduction to Linear Multistep Methods Numerical Analysis (9th Edition) R L Burden & J D Faires Beamer Presentation Slides prepared by John Carroll Dublin City University
More informationORBIT 14 The propagator. A. Milani et al. Dipmat, Pisa, Italy
ORBIT 14 The propagator A. Milani et al. Dipmat, Pisa, Italy Orbit 14 Propagator manual 1 Contents 1 Propagator - Interpolator 2 1.1 Propagation and interpolation............................ 2 1.1.1 Introduction..................................
More informationAdvanced methods for ODEs and DAEs
Lecture : Implicit Runge Kutta method Bojana Roić, 9. April 7 What you need to know before thi lecture numerical integration: Lecture from ODE iterative olver: Lecture 5-8 from ODE 9. April 7 Bojana Roić
More informationSolving Ordinary Differential equations
Solving Ordinary Differential equations Taylor methods can be used to build explicit methods with higher order of convergence than Euler s method. The main difficult of these methods is the computation
More informationTime Response of Systems
Chapter 0 Time Response of Systems 0. Some Standard Time Responses Let us try to get some impulse time responses just by inspection: Poles F (s) f(t) s-plane Time response p =0 s p =0,p 2 =0 s 2 t p =
More informationNUMERICAL SOLUTION OF ODE IVPs. Overview
NUMERICAL SOLUTION OF ODE IVPs 1 Quick review of direction fields Overview 2 A reminder about and 3 Important test: Is the ODE initial value problem? 4 Fundamental concepts: Euler s Method 5 Fundamental
More informationMath 128A Spring 2003 Week 11 Solutions Burden & Faires 5.6: 1b, 3b, 7, 9, 12 Burden & Faires 5.7: 1b, 3b, 5 Burden & Faires 5.
Math 128A Spring 2003 Week 11 Solutions Burden & Faires 5.6: 1b, 3b, 7, 9, 12 Burden & Faires 5.7: 1b, 3b, 5 Burden & Faires 5.8: 1b, 3b, 4 Burden & Faires 5.6. Multistep Methods 1. Use all the Adams-Bashforth
More informationReview Higher Order methods Multistep methods Summary HIGHER ORDER METHODS. P.V. Johnson. School of Mathematics. Semester
HIGHER ORDER METHODS School of Mathematics Semester 1 2008 OUTLINE 1 REVIEW 2 HIGHER ORDER METHODS 3 MULTISTEP METHODS 4 SUMMARY OUTLINE 1 REVIEW 2 HIGHER ORDER METHODS 3 MULTISTEP METHODS 4 SUMMARY OUTLINE
More informationGALERKIN AND RUNGE KUTTA METHODS: UNIFIED FORMULATION, A POSTERIORI ERROR ESTIMATES AND NODAL SUPERCONVERGENCE
GALERKIN AND RUNGE KUTTA METHODS: UNIFIED FORMULATION, A POSTERIORI ERROR ESTIMATES AND NODAL SUPERCONVERGENCE GEORGIOS AKRIVIS, CHARALAMBOS MAKRIDAKIS, AND RICARDO H. NOCHETTO Abstract. We unify the formulation
More informationBackward error analysis
Backward error analysis Brynjulf Owren July 28, 2015 Introduction. The main source for these notes is the monograph by Hairer, Lubich and Wanner [2]. Consider ODEs in R d of the form ẏ = f(y), y(0) = y
More informationLecture Notes to Accompany. Scientific Computing An Introductory Survey. by Michael T. Heath. Chapter 9
Lecture Notes to Accompany Scientific Computing An Introductory Survey Second Edition by Michael T. Heath Chapter 9 Initial Value Problems for Ordinary Differential Equations Copyright c 2001. Reproduction
More informationInitial value problems for ordinary differential equations
Initial value problems for ordinary differential equations Xiaojing Ye, Math & Stat, Georgia State University Spring 2019 Numerical Analysis II Xiaojing Ye, Math & Stat, Georgia State University 1 IVP
More informationSuboptimal feedback control of PDEs by solving Hamilton-Jacobi Bellman equations on sparse grids
Suboptimal feedback control of PDEs by solving Hamilton-Jacobi Bellman equations on sparse grids Jochen Garcke joint work with Axel Kröner, INRIA Saclay and CMAP, Ecole Polytechnique Ilja Kalmykov, Universität
More informationDepartment of Applied Mathematics Preliminary Examination in Numerical Analysis August, 2013
Department of Applied Mathematics Preliminary Examination in Numerical Analysis August, 013 August 8, 013 Solutions: 1 Root Finding (a) Let the root be x = α We subtract α from both sides of x n+1 = x
More informationKINGS COLLEGE OF ENGINEERING DEPARTMENT OF MATHEMATICS ACADEMIC YEAR / EVEN SEMESTER QUESTION BANK
KINGS COLLEGE OF ENGINEERING MA5-NUMERICAL METHODS DEPARTMENT OF MATHEMATICS ACADEMIC YEAR 00-0 / EVEN SEMESTER QUESTION BANK SUBJECT NAME: NUMERICAL METHODS YEAR/SEM: II / IV UNIT - I SOLUTION OF EQUATIONS
More informationOrdinary Differential Equations
Chapter 13 Ordinary Differential Equations We motivated the problem of interpolation in Chapter 11 by transitioning from analzying to finding functions. That is, in problems like interpolation and regression,
More informationNumerical Methods for the Solution of Differential Equations
Numerical Methods for the Solution of Differential Equations Markus Grasmair Vienna, winter term 2011 2012 Analytical Solutions of Ordinary Differential Equations 1. Find the general solution of the differential
More informationCHAPTER 10: Numerical Methods for DAEs
CHAPTER 10: Numerical Methods for DAEs Numerical approaches for the solution of DAEs divide roughly into two classes: 1. direct discretization 2. reformulation (index reduction) plus discretization Direct
More informationEnergy-Preserving Runge-Kutta methods
Energy-Preserving Runge-Kutta methods Fasma Diele, Brigida Pace Istituto per le Applicazioni del Calcolo M. Picone, CNR, Via Amendola 122, 70126 Bari, Italy f.diele@ba.iac.cnr.it b.pace@ba.iac.cnr.it SDS2010,
More informationNumerical solution of ODEs
Numerical solution of ODEs Arne Morten Kvarving Department of Mathematical Sciences Norwegian University of Science and Technology November 5 2007 Problem and solution strategy We want to find an approximation
More informationThe Initial Value Problem for Ordinary Differential Equations
Chapter 5 The Initial Value Problem for Ordinary Differential Equations In this chapter we begin a study of time-dependent differential equations, beginning with the initial value problem (IVP) for a time-dependent
More informationExponential Runge-Kutta methods for parabolic problems
Exponential Runge-Kutta methods for parabolic problems Marlis Hochbruck a,, Alexander Ostermann b a Mathematisches Institut, Heinrich-Heine Universität Düsseldorf, Universitätsstraße 1, D-4225 Düsseldorf,
More informationScientific Computing: An Introductory Survey
Scientific Computing: An Introductory Survey Chapter 9 Initial Value Problems for Ordinary Differential Equations Prof. Michael T. Heath Department of Computer Science University of Illinois at Urbana-Champaign
More informationMath Numerical Analysis Homework #4 Due End of term. y = 2y t 3y2 t 3, 1 t 2, y(1) = 1. n=(b-a)/h+1; % the number of steps we need to take
Math 32 - Numerical Analysis Homework #4 Due End of term Note: In the following y i is approximation of y(t i ) and f i is f(t i,y i ).. Consider the initial value problem, y = 2y t 3y2 t 3, t 2, y() =.
More informatione st f (t) dt = e st tf(t) dt = L {t f(t)} s
Additional operational properties How to find the Laplace transform of a function f (t) that is multiplied by a monomial t n, the transform of a special type of integral, and the transform of a periodic
More informationExplicit One-Step Methods
Chapter 1 Explicit One-Step Methods Remark 11 Contents This class presents methods for the numerical solution of explicit systems of initial value problems for ordinary differential equations of first
More informationL2 gains and system approximation quality 1
Massachusetts Institute of Technology Department of Electrical Engineering and Computer Science 6.242, Fall 24: MODEL REDUCTION L2 gains and system approximation quality 1 This lecture discusses the utility
More informationCHAPTER 5: Linear Multistep Methods
CHAPTER 5: Linear Multistep Methods Multistep: use information from many steps Higher order possible with fewer function evaluations than with RK. Convenient error estimates. Changing stepsize or order
More informationOrdinary differential equations. Phys 420/580 Lecture 8
Ordinary differential equations Phys 420/580 Lecture 8 Most physical laws are expressed as differential equations These come in three flavours: initial-value problems boundary-value problems eigenvalue
More informationSPECIALIZED RUNGE-KUTTA METHODS FOR INDEX 2 DIFFERENTIAL-ALGEBRAIC EQUATIONS
MATHEMATICS OF COMPUTATION Volume 75, Number 254, Pages 641 654 S 0025-5718(05)01809-0 Article electronically published on December 19, 2005 SPECIALIZED RUNGE-KUTTA METHODS FOR INDEX 2 DIFFERENTIAL-ALGEBRAIC
More informationRunge-Kutta Theory and Constraint Programming Julien Alexandre dit Sandretto Alexandre Chapoutot. Department U2IS ENSTA ParisTech SCAN Uppsala
Runge-Kutta Theory and Constraint Programming Julien Alexandre dit Sandretto Alexandre Chapoutot Department U2IS ENSTA ParisTech SCAN 2016 - Uppsala Contents Numerical integration Runge-Kutta with interval
More informationMathematics for chemical engineers. Numerical solution of ordinary differential equations
Mathematics for chemical engineers Drahoslava Janovská Numerical solution of ordinary differential equations Initial value problem Winter Semester 2015-2016 Outline 1 Introduction 2 One step methods Euler
More informationNotes for Numerical Analysis Math 5466 by S. Adjerid Virginia Polytechnic Institute and State University (A Rough Draft) Contents Numerical Methods for ODEs 5. Introduction............................
More informationIntroduction to standard and non-standard Numerical Methods
Introduction to standard and non-standard Numerical Methods Dr. Mountaga LAM AMS : African Mathematic School 2018 May 23, 2018 One-step methods Runge-Kutta Methods Nonstandard Finite Difference Scheme
More informationReview. Numerical Methods Lecture 22. Prof. Jinbo Bi CSE, UConn
Review Taylor Series and Error Analysis Roots of Equations Linear Algebraic Equations Optimization Numerical Differentiation and Integration Ordinary Differential Equations Partial Differential Equations
More informationImproved Starting Methods for Two-Step Runge Kutta Methods of Stage-Order p 3
Improved Starting Methods for Two-Step Runge Kutta Methods of Stage-Order p 3 J.H. Verner August 3, 2004 Abstract. In [5], Jackiewicz and Verner derived formulas for, and tested the implementation of two-step
More informationInitial value problems for ordinary differential equations
AMSC/CMSC 660 Scientific Computing I Fall 2008 UNIT 5: Numerical Solution of Ordinary Differential Equations Part 1 Dianne P. O Leary c 2008 The Plan Initial value problems (ivps) for ordinary differential
More informationRunge - Kutta Methods for first order models
Runge - Kutta Methods for first order models James K. Peterson Department of Biological Sciences and Department of Mathematical Sciences Clemson University October 30, 2013 Outline 1 Runge - Kutte Methods
More informationPowerPoints organized by Dr. Michael R. Gustafson II, Duke University
Part 6 Chapter 20 Initial-Value Problems PowerPoints organized by Dr. Michael R. Gustafson II, Duke University All images copyright The McGraw-Hill Companies, Inc. Permission required for reproduction
More informationCOSC 3361 Numerical Analysis I Ordinary Differential Equations (II) - Multistep methods
COSC 336 Numerical Analysis I Ordinary Differential Equations (II) - Multistep methods Fall 2005 Repetition from the last lecture (I) Initial value problems: dy = f ( t, y) dt y ( a) = y 0 a t b Goal:
More informationA POSTERIORI ERROR ESTIMATES FOR THE BDF2 METHOD FOR PARABOLIC EQUATIONS
A POSTERIORI ERROR ESTIMATES FOR THE BDF METHOD FOR PARABOLIC EQUATIONS GEORGIOS AKRIVIS AND PANAGIOTIS CHATZIPANTELIDIS Abstract. We derive optimal order, residual-based a posteriori error estimates for
More informationCOMMENTS ON HIGH ORDER INTEGRATORS EMBEDDED WITHIN INTEGRAL DEFERRED CORRECTION METHODS
COMMENTS ON HIGH ORDER INTEGRATORS EMBEDDED WITHIN INTEGRAL DEFERRED CORRECTION METHODS ANDREW CHRISTLIEB, BENJAMIN ONG, AND JING-MEI QIU Abstract. Spectral deferred correction (SDC) methods for solving
More informationORDINARY DIFFERENTIAL EQUATIONS
ORDINARY DIFFERENTIAL EQUATIONS GABRIEL NAGY Mathematics Department, Michigan State University, East Lansing, MI, 48824. JANUARY 3, 25 Summary. This is an introduction to ordinary differential equations.
More informationA SEMI-LAGRANGIAN RUNGE-KUTTA METHOD FOR TIME-DEPENDENT PARTIAL DIFFERENTIAL EQUATIONS
Journal of pplied nalysis and Computation Website:http://jaac-online.com/ Volume 3, Number 3, ugust 23 pp. 25 263 SEMI-LGRNGIN RUNGE-KUTT METHOD FOR TIME-DEPENDENT PRTIL DIFFERENTIL EQUTIONS Daniel X.
More informationNumerical Analysis of Differential Equations Numerical Solution of Parabolic Equations
Numerical Analysis of Differential Equations 215 6 Numerical Solution of Parabolic Equations 6 Numerical Solution of Parabolic Equations TU Bergakademie Freiberg, SS 2012 Numerical Analysis of Differential
More informationChapter 5 Exercises. (a) Determine the best possible Lipschitz constant for this function over 2 u <. u (t) = log(u(t)), u(0) = 2.
Chapter 5 Exercises From: Finite Difference Methods for Ordinary and Partial Differential Equations by R. J. LeVeque, SIAM, 2007. http://www.amath.washington.edu/ rjl/fdmbook Exercise 5. (Uniqueness for
More informationOrdinary Differential Equations
Ordinary Differential Equations Professor Dr. E F Toro Laboratory of Applied Mathematics University of Trento, Italy eleuterio.toro@unitn.it http://www.ing.unitn.it/toro September 19, 2014 1 / 55 Motivation
More informationPowerPoints organized by Dr. Michael R. Gustafson II, Duke University
Part 6 Chapter 20 Initial-Value Problems PowerPoints organized by Dr. Michael R. Gustafson II, Duke University All images copyright The McGraw-Hill Companies, Inc. Permission required for reproduction
More informationPowerPoints organized by Dr. Michael R. Gustafson II, Duke University
Part 5 Chapter 21 Numerical Differentiation PowerPoints organized by Dr. Michael R. Gustafson II, Duke University 1 All images copyright The McGraw-Hill Companies, Inc. Permission required for reproduction
More informationLinear Multistep Methods
Linear Multistep Methods Linear Multistep Methods (LMM) A LMM has the form α j x i+j = h β j f i+j, α k = 1 i 0 for the approximate solution of the IVP x = f (t, x), x(a) = x a. We approximate x(t) on
More informationCS 257: Numerical Methods
CS 57: Numerical Methods Final Exam Study Guide Version 1.00 Created by Charles Feng http://www.fenguin.net CS 57: Numerical Methods Final Exam Study Guide 1 Contents 1 Introductory Matter 3 1.1 Calculus
More informationName of the Student: Unit I (Solution of Equations and Eigenvalue Problems)
Engineering Mathematics 8 SUBJECT NAME : Numerical Methods SUBJECT CODE : MA6459 MATERIAL NAME : University Questions REGULATION : R3 UPDATED ON : November 7 (Upto N/D 7 Q.P) (Scan the above Q.R code for
More informationScientific Computing with Case Studies SIAM Press, Lecture Notes for Unit V Solution of
Scientific Computing with Case Studies SIAM Press, 2009 http://www.cs.umd.edu/users/oleary/sccswebpage Lecture Notes for Unit V Solution of Differential Equations Part 1 Dianne P. O Leary c 2008 1 The
More informationOn interval predictor-corrector methods
DOI 10.1007/s11075-016-0220-x ORIGINAL PAPER On interval predictor-corrector methods Andrzej Marcinia 1,2 Malgorzata A. Janowsa 3 Tomasz Hoffmann 4 Received: 26 March 2016 / Accepted: 3 October 2016 The
More informationRunga-Kutta Schemes. Exact evolution over a small time step: Expand both sides in a small time increment: d(δt) F (x(t + δt),t+ δt) Ft + FF x ( t)
Runga-Kutta Schemes Exact evolution over a small time step: x(t + t) =x(t)+ t 0 d(δt) F (x(t + δt),t+ δt) Expand both sides in a small time increment: x(t + t) =x(t)+x (t) t + 1 2 x (t)( t) 2 + 1 6 x (t)+
More informationExponential Integrators
Exponential Integrators John C. Bowman and Malcolm Roberts (University of Alberta) June 11, 2009 www.math.ualberta.ca/ bowman/talks 1 Outline Exponential Integrators Exponential Euler History Generalizations
More informationBasic Procedures for Common Problems
Basic Procedures for Common Problems ECHE 550, Fall 2002 Steady State Multivariable Modeling and Control 1 Determine what variables are available to manipulate (inputs, u) and what variables are available
More information