Modelling Physical Phenomena
|
|
- Gerald Pearson
- 6 years ago
- Views:
Transcription
1 Modelling Physical Phenomena Limitations and Challenges of the Differential Algebraic Equations Approach Olaf Trygve Berglihn Department of Chemical Engineering 30. June 2010
2 2 Outline Background Classification DAE-types, index and stiffness Model examples Index reduction strategies Modeling guidelines Initialization Discontinuities Industry example Where DAEs do not fit Current research and outlook Conclusion
3 3 Differential Algebraic Equations Ordinary differential equations with algebraic constraints on the variables. (Drawing by K. Wanner.)
4 4 Solution approach I ẏ = f(y, x, t), G(y, x, t) = 0 Nested approach Given y n, t n, solve G(y n, x, t n ) = 0, and evolve y n+1 using ODE-methods. Required approach if only explicit integrator is available (Euler, Runge-Kutta, etc). Can be expensive due to inner iterations.
5 5 Solution approach II F (ẏ, y, x, t) = 0 Simultaneous approach Solve the implicit or semi-explicit form simultaneously using an implicit solver and evolve both x and y in time. Requires an implicit solver. Is much more efficient. Provides for more flexible problem specification. This talk will focus on the simultaneous approach.
6 6 Typical models using DAE-formulation Chemical engineering processes with equilibrium conditions. Constrained mechanical systems, robots. Electrical circuits and power grids. Heating, ventilating and air-conditioning of buildings.
7 7 Evolution of DAE-solvers Gear LSODI (Hindmarch) DASSL (Petzold) IDA (Hindmarch, Taylor) ode15s (Shampine, Reichelt) GELDA (Kunkel, Merhmann) MANPACK (Rheinboldt) ode15i (Shampine, Reichelt) RADAU5 (Hairer, Wanner), COLDAE (Ascher, Spiteri) BzzDae (Manca, Buzzi-Ferraris) GENDA (Kunkel, Mehrmann, Seufer) A selection of the more commonly cited solvers.
8 8 Papers published containing Differential Algebraic Equations Papers per year (Source: Scopus)
9 9 Papers published by discipline Engineering Mathematics Computer Science Other Chemistry Biochemistry, Genetics and Molecular Biology Environmental Science Energy Physics and Astronomy Materials Science Chemical Engineering (Source: Scopus)
10 10 Classification I DAEs are primarily classified by type, index and stiffness. DAE-types Fully implicit: F (ẏ, y, x, t) = 0. Linearly implicit: A(y, x)ẏ = f(y, x, t), 0 = g(y, x, t) Semi-explicit: ẏ = f(y, x, t), 0 = g(y, x, t)
11 11 Classification II Differentiation index F (ẏ, y, x, t) = 0 has differentiaion index v = Ind(F ) if v is the minimal number of analytical differentiations F (ẏ, y, x, t) = 0, F (ẏ, y, x, t) x = 0,..., v F (ẏ, y, x, t) x v = 0 (1) such that equations (1) allow us to extract by algebraic manipulations an explicit ordinary differential equations system. Initial conditions must satisfy all intermediate algebraic relations.
12 12 Classification III Stiffness If both fast and slow processes are included in the model, the model becomes stiff. Consider the following system: ẋ 1 = 10 2 x 1 ẋ 2 = 10 2 x x 2 x 1 (0) = 1, x 2 (0) = 0 Reduction in x 2 is one million times faster than the reduction in x 1. Stability in integration is dictated by the fastest dynamics. Stiff systems require capable stiff solvers, or exceedingly small time stepping.
13 13 Hessenberg index 1-form ẏ = f(y, x) If g x is invertible, we can write ( ) x g 1 t = g x y f and have a system of only ODEs. (2a) 0 = g(y, x) (2b)
14 14 Hessenberg index 2-form I ẏ = f(y, x) (3a) 0 = g(y) (3b) We attempt to convert the set of equations to only ODEs by differentiation of (3b): 0 = g y y t = g y f (4) One differentiation does not do. We see that (4) is a hidden constraint of (3).
15 15 Hessenberg index 2-form II A second differentiation yields: 0 = 2 g y 2 f + g f y y f + g f x y x t (5) Assuming g y f x is invertible, this yields ( x g t = y ) f 1 ( 2 g x y 2 + g y ) f f. (6) y
16 16 Consequences of high index Index v > 1 imply constraints on the differential variables. Consider this semi-explicit index-2 system: ẏ = f(y, x) (7a) 0 = g(y) (7b) Initial conditions must satisfy g y f = 0 (8) Hidden constraints presents difficulties for DAE-solvers.
17 17 Example: Tank with overflow I ˆN (f), z i ˆN (o) N j, x j, v j ˆN Mole flow, f =feed, o=overflow N i Number of moles x i, z i Mole fractions v Molar specific volume i Component index, i {a, b} N a = ˆN (f) z a ˆN (o) x a N b = ˆN (f) z b ˆN (o) x b N = N a + N b N a = x a N N b = x b N v = v a x a + v b x b V = vn INDEX 2
18 18 Example: Tank with overflow II ˆN (f), z i N a = ˆN (f) z a ˆN (o) x a N b = ˆN (f) z b ˆN (o) x b N j, x j, v j ˆN (o) N = N a + N b N a = x a N N b = x b N v = v a x a + v b x b V = vn ˆN (o) v = ˆN (f) (v a z a + v b z b ) INDEX-1
19 19 Example: Tank with overflow III ˆN (f), z i N a = ˆN (f) z a ˆN (o) x a N b = ˆN (f) z b ˆN (o) x b N j, x j, v j ˆN (o) N = N a + N b N a = x a N N b = x b N v = v a x a + v b x b V = vn ˆN (o) v = ˆN (f) (v a z a + v b z b ) INDEX-1
20 20 Example: Pressure vessel with variable volume I ˆN, h x N, U, p, T Ṅ = ˆN U = p V + ˆNh pv = NRT U = Nc v T Linearly-implicit index-1. Requires implicit DAE solver. pa = F x V = Ax F x = kx
21 21 Example: Pressure vessel with variable volume II V = s ˆN, h x N, U, p, T Ṅ = ˆN U = ps + ˆNh pv = NRT U = Nc v T Introduce dummy variable s. Semi-explicit index-2. Requires index-2 capable DAE solver. pa = F x V = Ax F x = kx
22 22 Example: Pressure vessel with variable volume III Symbolic differentiation of volume and elimination yields: U = Ṅ = ˆN ( prax 2 ) 1 prax 2 + pc v Ax 2 + V c v k pv = NRT U = Nc v T pa = F x V = Ax F x = kx Semi-explicit index-1. Not intuitive how to do this! ˆNh
23 23 Causes of high index Imposing constraints on differential variables is the major problem. Assumption of infinitely fast dynamics Pseudo-steady state approximations and equilibrium conditions. Specification on derived states p = ( ) U V S,N If U is a differential variable (in energy balance), specifying impose constraint on U. p = p spec
24 24 How to handle high index and implicit systems Implicit Use implicit solvers or rewrite to semi-explicit form. High index Use high-index capable solver or reformulate and use symbolic differentiation. Pitfalls Exchanging an algebraic constraint with a differential equation can cause drift. High index capable solvers are very sensitive to scaling. Symbolic derivation and manipulation is error prone. The change from implicit to semi-explicit increases the index.
25 25 Index reduction strategies a) Remodeling Model round the index problem. Include fast dynamics. Makes the model stiff. Include rate equations for flows and reactions. Transform into other variables (x = r cos θ, y = r sin θ). b) Remove variables and equations or lump control volumes. Implies loss of detail. c) Reduce index by index reduction algorithms Involves symbolic differentiation and manipulation. Several strategies reported in the literature.
26 26 Modeling guidelines Procedure to reach index-1 models: a) Conservation equations: Formulate dynamic differential equations for the conserved quantities. b) Constitutive equations: Express dependent state variables from the conserved quantities. c) Rate equations: Express rates by difference in potential.
27 27 Modeling guidelines: Example I Calculate pressure and temperature in a perfectly insulated gas pipe: Assumptions: Ideal gas, constant heat capacity. Adiabatic pipe, neglect heat capacity of walls, constant volume. Sub-sonic, turbulent flow. Simple valve equation for flow. Uniform pressure and temperature in pipe segment.
28 28 Modeling guidelines: Example II Constitutive equations: express T : express p: N and U U = N Z T pv = NRT T ref c vdt express p: p = 1 (p + p1) 2 Conservation on the number of moles and internal energy: express N: Ṅ = ˆN (0) ˆN (1) express U: U = ˆN (0) h (0) ˆN (1) h (1) ˆN (1) express h 1: h 1 = Z T T ref c pdt Rate equations: express ˆN (1) : ˆN (1) = k 1 q p(p p (1) ) p, p and T
29 29 Initialization Finding consistent initial conditions is a major obstacle with index > 1 models. Some possible approaches: Structural Successive Linear Programming (SLP) Gauss-Newton-Maquardt methods using singular value decomposition.
30 30 Initialization: Structural approach Develop derivative array equations: F (ẏ, y, x, t) = 0, F (ẏ, y, x, t) x. = 0, v F (ẏ, y, x, t) x v = 0 Analyze variable occurrence of the complete system, choose pivoting variables among (ẏ, y, x) from among all equations, and solve this set at t(0).
31 31 Initialization: Successive Linear Programming Develop derivative array equations: Solve G(y, y t,, n y t n, t) = 0 ( min G y, y ) t,, n y t n, 0 1 using Successive Linear Program (SLP).
32 32 Initialization: Gauss-Newton-Maquardt Develop derivative array equations: Solve G(y, y t,, n y t n, t) = 0 ( min G y, y ) t,, n y t n, 0 2 using Singular Value Decomposition (SVD).
33 33 Discontinuities Consider this definition of mass m: m(t) = ρ(x, t)dv (9) We write the time differential dm dt = d ρ(x, t)dv dt Ω Equation (10) is only valid if ρ is continuous over Ω. Remedy: Locate discontinuities. Integrate over piecewise smooth segments. Use smooth transition (tanh) Ω? Ω ρ(x, t)dv (10) t Some solvers can locate and traverse discontinuities.
34 developing a model for predicting the pressure buildup, temperatures etc. in the pipe, such that 34 safe procedures for hydrate removal may be devised. This report documents the current status of the model (July 13, 2005), and documents the assumptions, model equations, numerics and Industry program structure. example: Direct electrical heating of gas hydrate plug Figure 1-1 shows the hydrate melting process schematically. There is a hydrate plug that blocks a pipe, and an external heat flux, Q 10 (t,x), is applied to the pipe in order to remove the plug. The external heat flux may vary with axial position and time. Model of direct heating of pipe line plugged by hydrate. r,z Q 10 θ x Figure 1-1. Melting hydrate plug Very stiff problem pipe elasticity. Discontinuous heating of hydrate vs. melting of hydrate. As a result of the external heating, the hydrate plug will start melting, and melted fluid leaves by axial flow in both directions. The present model assumes that all melted fluid leaves by flowing in the annular Assumptions gap that forms between can lead the pipe to high and the index. plug; i.e. there is no flow of melted fluid inside the plug (conservative assumption). Careful scaling is essential. As the plug is assumed to seal the pipe completely when the heating starts, there will be a
35 35 Industry example: results x 10 7 Pressure Pressure [Pa] Time [s] x 10 4
36 36 Limitations of DAEs Where DAEs do not fit: Purely discrete systems. Time discrete event. Cellular automaton Game of Life. Stochastic systems. Can the system be made continuous? Can jumps be made continuous transitions? Model on the continuous parameters of a distribution?
37 37 Current research and outlook A lot of specialized solvers published in recent years. Focus on model building software to aid modeling. Validated numerics for DAEs: give guaranteed interval for solution. Interval integration: given input set, what is output set? Will the process be stable?
38 38 Conclusion Modeling dynamic systems with constraints is challenging. Great care must be taken to avoid index problems. A good modeling methodology is essential. Index reduction can be applied, but is not trivial.
39 39 Selected bibliography I K. E. Brenan, S. L. Campell, and L. R. Petzold. Numerical solution of initial-value problems in differential-algebraic equations. North-Holland, New York, Charles. W. Gear. Simultaneous numerical solution of differential-algebraic equations. IEEE Trans. Circuit Theory, CT-18:89 95, E. Hairer, Norsett S. P., and G. Wanner. Solving Ordinary, Differential Equations II. Stiff and Differential-Algebraic Problems, volume 2. 2Ed. Springer-Verlag, 2002, ISBN Index.
40 40 Selected bibliography II Peter Kunkel. Differential-algebraic equations: analysis and numerical solution. European Mathematical Society, Volker Mehrmann and Lena Wunderlich. Hybrid systems of differential-algebraic equations - analysis and numerical solution. Journal of Process Control, 19(8): , ISSN Special Section on Hybrid Systems: Modeling, Simulation and Optimization. Håvard Ingvald Moe. Dynamic Process Simulation, Studies on Modeling and Index Reduction. PhD thesis, University of Trondheim, 1995.
41 41 Selected bibliography III Linda Petzold. Differential/algebraic equations are not ode s. SIAM Journal on Scientific and Statistical Computing, 3(3): , 1982.
Numerical Integration of Equations of Motion
GraSMech course 2009-2010 Computer-aided analysis of rigid and flexible multibody systems Numerical Integration of Equations of Motion Prof. Olivier Verlinden (FPMs) Olivier.Verlinden@fpms.ac.be Prof.
More informationDifferential-Algebraic Equations (DAEs)
Differential-Algebraic Equations (DAEs) L. T. Biegler Chemical Engineering Department Carnegie Mellon University Pittsburgh, PA 15213 biegler@cmu.edu http://dynopt.cheme.cmu.edu Introduction Simple Examples
More informationIndex Reduction and Discontinuity Handling using Substitute Equations
Mathematical and Computer Modelling of Dynamical Systems, vol. 7, nr. 2, 2001, 173-187. Index Reduction and Discontinuity Handling using Substitute Equations G. Fábián, D.A. van Beek, J.E. Rooda Abstract
More informationAn initialization subroutine for DAEs solvers: DAEIS
Computers and Chemical Engineering 25 (2001) 301 311 www.elsevier.com/locate/compchemeng An initialization subroutine for DAEs solvers: DAEIS B. Wu, R.E. White * Department of Chemical Engineering, Uniersity
More informationINRIA Rocquencourt, Le Chesnay Cedex (France) y Dept. of Mathematics, North Carolina State University, Raleigh NC USA
Nonlinear Observer Design using Implicit System Descriptions D. von Wissel, R. Nikoukhah, S. L. Campbell y and F. Delebecque INRIA Rocquencourt, 78 Le Chesnay Cedex (France) y Dept. of Mathematics, North
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 informationDifferential Equation Types. Moritz Diehl
Differential Equation Types Moritz Diehl Overview Ordinary Differential Equations (ODE) Differential Algebraic Equations (DAE) Partial Differential Equations (PDE) Delay Differential Equations (DDE) Ordinary
More informationAn Implicit Runge Kutta Solver adapted to Flexible Multibody System Simulation
An Implicit Runge Kutta Solver adapted to Flexible Multibody System Simulation Johannes Gerstmayr 7. WORKSHOP ÜBER DESKRIPTORSYSTEME 15. - 18. March 2005, Liborianum, Paderborn, Germany Austrian Academy
More informationThe Plan. Initial value problems (ivps) for ordinary differential equations (odes) Review of basic methods You are here: Hamiltonian systems
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 2 Dianne P. O Leary c 2008 The Plan
More informationNumerical integration of DAE s
Numerical integration of DAE s seminar Sandra Allaart-Bruin sbruin@win.tue.nl seminar p.1 Seminar overview February 18 Arie Verhoeven Introduction to DAE s seminar p.2 Seminar overview February 18 Arie
More informationTechnische Universität Berlin
Technische Universität Berlin Institut für Mathematik M7 - A Skateboard(v1.) Andreas Steinbrecher Preprint 216/8 Preprint-Reihe des Instituts für Mathematik Technische Universität Berlin http://www.math.tu-berlin.de/preprints
More informationFour Point Gauss Quadrature Runge Kuta Method Of Order 8 For Ordinary Differential Equations
International journal of scientific and technical research in engineering (IJSTRE) www.ijstre.com Volume Issue ǁ July 206. Four Point Gauss Quadrature Runge Kuta Method Of Order 8 For Ordinary Differential
More informationSemi-implicit Krylov Deferred Correction Methods for Ordinary Differential Equations
Semi-implicit Krylov Deferred Correction Methods for Ordinary Differential Equations Sunyoung Bu University of North Carolina Department of Mathematics CB # 325, Chapel Hill USA agatha@email.unc.edu Jingfang
More information5. Coupling of Chemical Kinetics & Thermodynamics
5. Coupling of Chemical Kinetics & Thermodynamics Objectives of this section: Thermodynamics: Initial and final states are considered: - Adiabatic flame temperature - Equilibrium composition of products
More informationMODELLING OF RECIPROCAL TRANSDUCER SYSTEM ACCOUNTING FOR NONLINEAR CONSTITUTIVE RELATIONS
MODELLING OF RECIPROCAL TRANSDUCER SYSTEM ACCOUNTING FOR NONLINEAR CONSTITUTIVE RELATIONS L. X. Wang 1 M. Willatzen 1 R. V. N. Melnik 1,2 Abstract The dynamics of reciprocal transducer systems is modelled
More informationStabilität differential-algebraischer Systeme
Stabilität differential-algebraischer Systeme Caren Tischendorf, Universität zu Köln Elgersburger Arbeitstagung, 11.-14. Februar 2008 Tischendorf (Univ. zu Köln) Stabilität von DAEs Elgersburg, 11.-14.02.2008
More informationDelay Differential-Algebraic Equations
ECHNISCHE UNIVERSIÄ BERLIN Analysis and Numerical Solution of Linear Delay Differential-Algebraic Equations Ha Phi and Volker Mehrmann Preprint 214/42 Preprint-Reihe des Instituts für Mathematik echnische
More informationPartitioned Methods for Multifield Problems
C Partitioned Methods for Multifield Problems Joachim Rang, 6.7.2016 6.7.2016 Joachim Rang Partitioned Methods for Multifield Problems Seite 1 C One-dimensional piston problem fixed wall Fluid flexible
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 informationA note on the uniform perturbation index 1
Rostock. Math. Kolloq. 52, 33 46 (998) Subject Classification (AMS) 65L5 M. Arnold A note on the uniform perturbation index ABSTRACT. For a given differential-algebraic equation (DAE) the perturbation
More informationAstronomy 8824: Numerical Methods Notes 2 Ordinary Differential Equations
Astronomy 8824: Numerical Methods Notes 2 Ordinary Differential Equations Reading: Numerical Recipes, chapter on Integration of Ordinary Differential Equations (which is ch. 15, 16, or 17 depending on
More informationAIMS Exercise Set # 1
AIMS Exercise Set #. Determine the form of the single precision floating point arithmetic used in the computers at AIMS. What is the largest number that can be accurately represented? What is the smallest
More information10.34: Numerical Methods Applied to Chemical Engineering. Lecture 19: Differential Algebraic Equations
10.34: Numerical Methods Applied to Chemical Engineering Lecture 19: Differential Algebraic Equations 1 Recap Differential algebraic equations Semi-explicit Fully implicit Simulation via backward difference
More informationA New Block Method and Their Application to Numerical Solution of Ordinary Differential Equations
A New Block Method and Their Application to Numerical Solution of Ordinary Differential Equations Rei-Wei Song and Ming-Gong Lee* d09440@chu.edu.tw, mglee@chu.edu.tw * Department of Applied Mathematics/
More informationReview: control, feedback, etc. Today s topic: state-space models of systems; linearization
Plan of the Lecture Review: control, feedback, etc Today s topic: state-space models of systems; linearization Goal: a general framework that encompasses all examples of interest Once we have mastered
More informationNumerical Treatment of Unstructured. Differential-Algebraic Equations. with Arbitrary Index
Numerical Treatment of Unstructured Differential-Algebraic Equations with Arbitrary Index Peter Kunkel (Leipzig) SDS2003, Bari-Monopoli, 22. 25.06.2003 Outline Numerical Treatment of Unstructured Differential-Algebraic
More informationThree-Tank Experiment
Three-Tank Experiment Overview The three-tank experiment focuses on application of the mechanical balance equation to a transient flow. Three tanks are interconnected by Schedule 40 pipes of nominal diameter
More informationModeling and Experimentation: Compound Pendulum
Modeling and Experimentation: Compound Pendulum Prof. R.G. Longoria Department of Mechanical Engineering The University of Texas at Austin Fall 2014 Overview This lab focuses on developing a mathematical
More informationНаучный потенциал регионов на службу модернизации. Астрахань: АИСИ, с.
MODELING OF FLOWS IN PIPING TREES USING PROJECTION METHODS В.В. Войков, Астраханский инженерно-строительный институт, г. Астрахань, Россия Jason Mayes, Mihir Sen University of Notre Dame, Indiana, USA
More informationTheoretical Models of Chemical Processes
Theoretical Models of Chemical Processes Dr. M. A. A. Shoukat Choudhury 1 Rationale for Dynamic Models 1. Improve understanding of the process 2. Train Plant operating personnel 3. Develop control strategy
More informationOn The Numerical Solution of Differential-Algebraic Equations(DAES) with Index-3 by Pade Approximation
Appl. Math. Inf. Sci. Lett., No. 2, 7-23 (203) 7 Applied Mathematics & Information Sciences Letters An International Journal On The Numerical Solution of Differential-Algebraic Equations(DAES) with Index-3
More informationChapter 2 Optimal Control Problem
Chapter 2 Optimal Control Problem Optimal control of any process can be achieved either in open or closed loop. In the following two chapters we concentrate mainly on the first class. The first chapter
More informationNumerical Methods for Engineers
Numerical Methods for Engineers SEVENTH EDITION Steven C Chopra Berger Chair in Computing and Engineering Tufts University Raymond P. Canal Professor Emeritus of Civil Engineering of Michiaan University
More informationPart 1: Overview of Ordinary Dierential Equations 1 Chapter 1 Basic Concepts and Problems 1.1 Problems Leading to Ordinary Dierential Equations Many scientic and engineering problems are modeled by systems
More informationETNA Kent State University
Electronic Transactions on Numerical Analysis. Volume 34, pp. 14-19, 2008. Copyrigt 2008,. ISSN 1068-9613. ETNA A NOTE ON NUMERICALLY CONSISTENT INITIAL VALUES FOR HIGH INDEX DIFFERENTIAL-ALGEBRAIC EQUATIONS
More informationECE 422/522 Power System Operations & Planning/Power Systems Analysis II : 7 - Transient Stability
ECE 4/5 Power System Operations & Planning/Power Systems Analysis II : 7 - Transient Stability Spring 014 Instructor: Kai Sun 1 Transient Stability The ability of the power system to maintain synchronism
More informationOrdinary Differential Equations (ODEs)
Ordinary Differential Equations (ODEs) 1 Computer Simulations Why is computation becoming so important in physics? One reason is that most of our analytical tools such as differential calculus are best
More informationComputer Aided Design of Thermal Systems (ME648)
Computer Aided Design of Thermal Systems (ME648) PG/Open Elective Credits: 3-0-0-9 Updated Syallabus: Introduction. Basic Considerations in Design. Modelling of Thermal Systems. Numerical Modelling and
More informationCOMPARISON OF TWO METHODS TO SOLVE PRESSURES IN SMALL VOLUMES IN REAL-TIME SIMULATION OF A MOBILE DIRECTIONAL CONTROL VALVE
COMPARISON OF TWO METHODS TO SOLVE PRESSURES IN SMALL VOLUMES IN REAL-TIME SIMULATION OF A MOBILE DIRECTIONAL CONTROL VALVE Rafael ÅMAN*, Heikki HANDROOS*, Pasi KORKEALAAKSO** and Asko ROUVINEN** * Laboratory
More informationA relaxation of the strangeness index
echnical report from Automatic Control at Linköpings universitet A relaxation of the strangeness index Henrik idefelt, orkel Glad Division of Automatic Control E-mail: tidefelt@isy.liu.se, torkel@isy.liu.se
More informationThe one-dimensional equations for the fluid dynamics of a gas can be written in conservation form as follows:
Topic 7 Fluid Dynamics Lecture The Riemann Problem and Shock Tube Problem A simple one dimensional model of a gas was introduced by G.A. Sod, J. Computational Physics 7, 1 (1978), to test various algorithms
More informationAn Immersed Boundary Method for Restricted Diffusion with Permeable Interfaces
An Immersed Boundary Method for Restricted Diffusion with Permeable Interfaces Huaxiong Huang Kazuyasu Sugiyama Shu Takagi April 6, 009 Keywords: Restricted diffusion; Permeable interface; Immersed boundary
More informationFinite Element Decompositions for Stable Time Integration of Flow Equations
MAX PLANCK INSTITUT August 13, 2015 Finite Element Decompositions for Stable Time Integration of Flow Equations Jan Heiland, Robert Altmann (TU Berlin) ICIAM 2015 Beijing DYNAMIK KOMPLEXER TECHNISCHER
More informationMathematical Models with Maple
Algebraic Biology 005 151 Mathematical Models with Maple Tetsu YAMAGUCHI Applied System nd Division, Cybernet Systems Co., Ltd., Otsuka -9-3, Bunkyo-ku, Tokyo 11-001, Japan tetsuy@cybernet.co.jp Abstract
More informationChapter 17. Finite Volume Method The partial differential equation
Chapter 7 Finite Volume Method. This chapter focusses on introducing finite volume method for the solution of partial differential equations. These methods have gained wide-spread acceptance in recent
More informationVarious lecture notes for
Various lecture notes for 18311. R. R. Rosales (MIT, Math. Dept., 2-337) April 12, 2013 Abstract Notes, both complete and/or incomplete, for MIT s 18.311 (Principles of Applied Mathematics). These notes
More informationApplication of Dual Time Stepping to Fully Implicit Runge Kutta Schemes for Unsteady Flow Calculations
Application of Dual Time Stepping to Fully Implicit Runge Kutta Schemes for Unsteady Flow Calculations Antony Jameson Department of Aeronautics and Astronautics, Stanford University, Stanford, CA, 94305
More informationEuropean Consortium for Mathematics in Industry E. Eich-Soellner and C. FUhrer Numerical Methods in Multibody Dynamics
European Consortium for Mathematics in Industry E. Eich-Soellner and C. FUhrer Numerical Methods in Multibody Dynamics European Consortium for Mathematics in Industry Edited by Leif Arkeryd, Goteborg Heinz
More informationc 2004 Society for Industrial and Applied Mathematics
SIAM J. SCI. COMPUT. Vol. 26, No. 2, pp. 359 374 c 24 Society for Industrial and Applied Mathematics A POSTERIORI ERROR ESTIMATION AND GLOBAL ERROR CONTROL FOR ORDINARY DIFFERENTIAL EQUATIONS BY THE ADJOINT
More informationAn Overly Simplified and Brief Review of Differential Equation Solution Methods. 1. Some Common Exact Solution Methods for Differential Equations
An Overly Simplified and Brief Review of Differential Equation Solution Methods We will be dealing with initial or boundary value problems. A typical initial value problem has the form y y 0 y(0) 1 A typical
More informationEfficient numerical methods for the solution of stiff initial-value problems and differential algebraic equations
10.1098/rspa.2003.1130 R EVIEW PAPER Efficient numerical methods for the solution of stiff initial-value problems and differential algebraic equations By J. R. Cash Department of Mathematics, Imperial
More informationOrdinary differential equations - Initial value problems
Education has produced a vast population able to read but unable to distinguish what is worth reading. G.M. TREVELYAN Chapter 6 Ordinary differential equations - Initial value problems In this chapter
More informationLogistic Map, Euler & Runge-Kutta Method and Lotka-Volterra Equations
Logistic Map, Euler & Runge-Kutta Method and Lotka-Volterra Equations S. Y. Ha and J. Park Department of Mathematical Sciences Seoul National University Sep 23, 2013 Contents 1 Logistic Map 2 Euler and
More informationODE - Problem ROBER. 0.04y y 2 y y y 2 y y y 2 2
ODE - Problem ROBER II-10-1 10 Problem ROBER 10.1 General information The problem consists of a stiff system of 3 non-linear ordinary differential equations. It was proposed by H.H. Robertson in 1966 [Rob66].
More informationSYMMETRIC PROJECTION METHODS FOR DIFFERENTIAL EQUATIONS ON MANIFOLDS
BIT 0006-3835/00/4004-0726 $15.00 2000, Vol. 40, No. 4, pp. 726 734 c Swets & Zeitlinger SYMMETRIC PROJECTION METHODS FOR DIFFERENTIAL EQUATIONS ON MANIFOLDS E. HAIRER Section de mathématiques, Université
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 informationTAU Solver Improvement [Implicit methods]
TAU Solver Improvement [Implicit methods] Richard Dwight Megadesign 23-24 May 2007 Folie 1 > Vortrag > Autor Outline Motivation (convergence acceleration to steady state, fast unsteady) Implicit methods
More informationImpact of the HP Preheater Bypass on the Economizer Inlet Header
Impact of the HP Preheater Bypass on the Economizer Inlet Header Dr.-Ing. Henning Zindler E.ON Kraftwerke Tresckowstrasse 5 30457 Hannover Germany henning.zindler@eon-energie.com Dipl.-Ing. Andreas Hauschke
More informationNORGES TEKNISK-NATURVITENSKAPELIGE UNIVERSITET. Singly diagonally implicit Runge-Kutta methods with an explicit first stage
NORGES TEKNISK-NATURVITENSKAPELIGE UNIVERSITET Singly diagonally implicit Runge-Kutta methods with an explicit first stage by Anne Kværnø PREPRINT NUMERICS NO. 1/2004 NORWEGIAN UNIVERSITY OF SCIENCE AND
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 informationResearch Article On the Numerical Solution of Differential-Algebraic Equations with Hessenberg Index-3
Discrete Dynamics in Nature and Society Volume, Article ID 474, pages doi:.55//474 Research Article On the Numerical Solution of Differential-Algebraic Equations with Hessenberg Inde- Melike Karta and
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 informationSolution: In standard form (i.e. y + P (t)y = Q(t)) we have y t y = cos(t)
Math 380 Practice Final Solutions This is longer than the actual exam, which will be 8 to 0 questions (some might be multiple choice). You are allowed up to two sheets of notes (both sides) and a calculator,
More informationModeling & Simulation 2018 Lecture 12. Simulations
Modeling & Simulation 2018 Lecture 12. Simulations Claudio Altafini Automatic Control, ISY Linköping University, Sweden Summary of lecture 7-11 1 / 32 Models of complex systems physical interconnections,
More informationNumerical Oscillations and how to avoid them
Numerical Oscillations and how to avoid them Willem Hundsdorfer Talk for CWI Scientific Meeting, based on work with Anna Mozartova (CWI, RBS) & Marc Spijker (Leiden Univ.) For details: see thesis of A.
More informationDevelopment of Dynamic Models. Chapter 2. Illustrative Example: A Blending Process
Development of Dynamic Models Illustrative Example: A Blending Process An unsteady-state mass balance for the blending system: rate of accumulation rate of rate of = of mass in the tank mass in mass out
More informationDynamic Process Models
Dr. Simulation & Optimization Team Interdisciplinary Center for Scientific Computing (IWR) University of Heidelberg Short course Nonlinear Parameter Estimation and Optimum Experimental Design June 22 &
More informationSpectral Methods for Reaction Diffusion Systems
WDS'13 Proceedings of Contributed Papers, Part I, 97 101, 2013. ISBN 978-80-7378-250-4 MATFYZPRESS Spectral Methods for Reaction Diffusion Systems V. Rybář Institute of Mathematics of the Academy of Sciences
More informationTime integration. DVI and HHT time stepping methods in Chrono
Time integration DVI and HHT time stepping methods in Chrono Time Integration in Chrono Two classes of time stepping methods in Chrono Time steppers for smooth dynamics Classical multibody dynamics rigid
More informationChapter 3 Numerical Methods
Chapter 3 Numerical Methods Part 3 3.4 Differential Algebraic Systems 3.5 Integration of Differential Equations 1 Outline 3.4 Differential Algebraic Systems 3.4.1 Constrained Dynamics 3.4.2 First and Second
More informationDifferential Equations
Differential Equations Overview of differential equation! Initial value problem! Explicit numeric methods! Implicit numeric methods! Modular implementation Physics-based simulation An algorithm that
More informationNewton s Method and Efficient, Robust Variants
Newton s Method and Efficient, Robust Variants Philipp Birken University of Kassel (SFB/TRR 30) Soon: University of Lund October 7th 2013 Efficient solution of large systems of non-linear PDEs in science
More informationTime-adaptive methods for the incompressible Navier-Stokes equations
Time-adaptive methods for the incompressible Navier-Stokes equations Joachim Rang, Thorsten Grahs, Justin Wiegmann, 29.09.2016 Contents Introduction Diagonally implicit Runge Kutta methods Results with
More informationDevelopment of Dynamic Models. Chapter 2. Illustrative Example: A Blending Process
Development of Dynamic Models Illustrative Example: A Blending Process An unsteady-state mass balance for the blending system: rate of accumulation rate of rate of = of mass in the tank mass in mass out
More informationFinite volumes for complex applications In this paper, we study finite-volume methods for balance laws. In particular, we focus on Godunov-type centra
Semi-discrete central schemes for balance laws. Application to the Broadwell model. Alexander Kurganov * *Department of Mathematics, Tulane University, 683 St. Charles Ave., New Orleans, LA 708, USA kurganov@math.tulane.edu
More informationButcher tableau Can summarize an s + 1 stage Runge Kutta method using a triangular grid of coefficients
AM 205: lecture 13 Last time: ODE convergence and stability, Runge Kutta methods Today: the Butcher tableau, multi-step methods, boundary value problems Butcher tableau Can summarize an s + 1 stage Runge
More informationInvestigation on the Most Efficient Ways to Solve the Implicit Equations for Gauss Methods in the Constant Stepsize Setting
Applied Mathematical Sciences, Vol. 12, 2018, no. 2, 93-103 HIKARI Ltd, www.m-hikari.com https://doi.org/10.12988/ams.2018.711340 Investigation on the Most Efficient Ways to Solve the Implicit Equations
More informationUse of Differential Equations In Modeling and Simulation of CSTR
Use of Differential Equations In Modeling and Simulation of CSTR JIRI VOJTESEK, PETR DOSTAL Department of Process Control, Faculty of Applied Informatics Tomas Bata University in Zlin nám. T. G. Masaryka
More information16.7 Multistep, Multivalue, and Predictor-Corrector Methods
740 Chapter 16. Integration of Ordinary Differential Equations 16.7 Multistep, Multivalue, and Predictor-Corrector Methods The terms multistepand multivaluedescribe two different ways of implementing essentially
More informationDynamic Programming with Hermite Interpolation
Dynamic Programming with Hermite Interpolation Yongyang Cai Hoover Institution, 424 Galvez Mall, Stanford University, Stanford, CA, 94305 Kenneth L. Judd Hoover Institution, 424 Galvez Mall, Stanford University,
More informationChapter 1. Introduction to Nonlinear Space Plasma Physics
Chapter 1. Introduction to Nonlinear Space Plasma Physics The goal of this course, Nonlinear Space Plasma Physics, is to explore the formation, evolution, propagation, and characteristics of the large
More informationStrategies for Numerical Integration of Discontinuous DAE Models
European Symposium on Computer Arded Aided Process Engineering 15 L. Puigjaner and A. Espuña (Editors) 25 Elsevier Science B.V. All rights reserved. Strategies for Numerical Integration of Discontinuous
More informationApplied Numerical Analysis
Applied Numerical Analysis Using MATLAB Second Edition Laurene V. Fausett Texas A&M University-Commerce PEARSON Prentice Hall Upper Saddle River, NJ 07458 Contents Preface xi 1 Foundations 1 1.1 Introductory
More informationCS520: numerical ODEs (Ch.2)
.. CS520: numerical ODEs (Ch.2) Uri Ascher Department of Computer Science University of British Columbia ascher@cs.ubc.ca people.cs.ubc.ca/ ascher/520.html Uri Ascher (UBC) CPSC 520: ODEs (Ch. 2) Fall
More informationLocal Time Step for a Finite Volume Scheme I.Faille F.Nataf*, F.Willien, S.Wolf**
Controlled CO 2 Diversified fuels Fuel-efficient vehicles Clean refining Extended reserves Local Time Step for a Finite Volume Scheme I.Faille F.Nataf*, F.Willien, S.Wolf** *: Laboratoire J.L.Lions **:Université
More informationSolving Constrained Differential- Algebraic Systems Using Projections. Richard J. Hanson Fred T. Krogh August 16, mathalacarte.
Solving Constrained Differential- Algebraic Systems Using Projections Richard J. Hanson Fred T. Krogh August 6, 2007 www.vni.com mathalacarte.com Abbreviations and Terms ODE = Ordinary Differential Equations
More informationNumerical Methods for Engineers. and Scientists. Applications using MATLAB. An Introduction with. Vish- Subramaniam. Third Edition. Amos Gilat.
Numerical Methods for Engineers An Introduction with and Scientists Applications using MATLAB Third Edition Amos Gilat Vish- Subramaniam Department of Mechanical Engineering The Ohio State University Wiley
More informationThe family of Runge Kutta methods with two intermediate evaluations is defined by
AM 205: lecture 13 Last time: Numerical solution of ordinary differential equations Today: Additional ODE methods, boundary value problems Thursday s lecture will be given by Thomas Fai Assignment 3 will
More informationNumerical Solutions to Partial Differential Equations
Numerical Solutions to Partial Differential Equations Zhiping Li LMAM and School of Mathematical Sciences Peking University The Implicit Schemes for the Model Problem The Crank-Nicolson scheme and θ-scheme
More informationCOMP 558 lecture 18 Nov. 15, 2010
Least squares We have seen several least squares problems thus far, and we will see more in the upcoming lectures. For this reason it is good to have a more general picture of these problems and how to
More informationGEOMETRIC INTEGRATION OF ORDINARY DIFFERENTIAL EQUATIONS ON MANIFOLDS
BIT 0006-3835/01/4105-0996 $16.00 2001, Vol. 41, No. 5, pp. 996 1007 c Swets & Zeitlinger GEOMETRIC INTEGRATION OF ORDINARY DIFFERENTIAL EQUATIONS ON MANIFOLDS E. HAIRER Section de mathématiques, Université
More informationE. KOFMAN. Latin American Applied Research 36: (2006)
Latin American Applied Research 36:101-108 (006) A THIRD ORDER DISCRETE EVENT METHOD FOR CONTINUOUS SYSTEM SIMULATION E. KOFMAN Laboratorio de Sistemas Dinámicos. FCEIA UNR CONICET. Riobamba 45 bis (000)
More informationEntrance Exam, Differential Equations April, (Solve exactly 6 out of the 8 problems) y + 2y + y cos(x 2 y) = 0, y(0) = 2, y (0) = 4.
Entrance Exam, Differential Equations April, 7 (Solve exactly 6 out of the 8 problems). Consider the following initial value problem: { y + y + y cos(x y) =, y() = y. Find all the values y such that the
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 informationAC&ST AUTOMATIC CONTROL AND SYSTEM THEORY SYSTEMS AND MODELS. Claudio Melchiorri
C. Melchiorri (DEI) Automatic Control & System Theory 1 AUTOMATIC CONTROL AND SYSTEM THEORY SYSTEMS AND MODELS Claudio Melchiorri Dipartimento di Ingegneria dell Energia Elettrica e dell Informazione (DEI)
More informationSolving PDEs with PGI CUDA Fortran Part 4: Initial value problems for ordinary differential equations
Solving PDEs with PGI CUDA Fortran Part 4: Initial value problems for ordinary differential equations Outline ODEs and initial conditions. Explicit and implicit Euler methods. Runge-Kutta methods. Multistep
More informationINTEGRATION OF THE EQUATIONS OF MOTION OF MULTIBODY SYSTEMS USING ABSOLUTE NODAL COORDINATE FORMULATION
INTEGRATION OF THE EQUATIONS OF MOTION OF MULTIBODY SYSTEMS USING ABSOLUTE NODAL COORDINATE FORMULATION Grzegorz ORZECHOWSKI *, Janusz FRĄCZEK * * The Institute of Aeronautics and Applied Mechanics, The
More informationA Stand Alone Quantized State System Solver. Part I
A Stand Alone Quantized State System Solver. Part I Joaquín Fernández and Ernesto Kofman CIFASIS-CONICET Facultad de Cs. Exactas, Ingeniería y Agrim., UNR, Argentina fernandez@cifasis-conicet.gov.ar -
More informationNumerical Analysis. A Comprehensive Introduction. H. R. Schwarz University of Zürich Switzerland. with a contribution by
Numerical Analysis A Comprehensive Introduction H. R. Schwarz University of Zürich Switzerland with a contribution by J. Waldvogel Swiss Federal Institute of Technology, Zürich JOHN WILEY & SONS Chichester
More informationDifferential equations, comprehensive exam topics and sample questions
Differential equations, comprehensive exam topics and sample questions Topics covered ODE s: Chapters -5, 7, from Elementary Differential Equations by Edwards and Penney, 6th edition.. Exact solutions
More information