Simulation of ODE/PDE Models with MATLAB , OCTAVE and SCILAB: Scientific and Engineering Applications

416 Pages·2016·11.65 MB·English

by  Vande Wouwer A.

Alain Vande Wouwer
Philippe Saucez
Carlos Vilas
Simulation of
ODE/PDE Models
with MATLAB®,
OCTAVE and SCILAB
Scientific and Engineering Applications
Simulation of ODE/PDE Models with MATLAB(cid:2),
OCTAVE and SCILAB
Alain Vande Wouwer Philippe Saucez
•
Carlos Vilas
Simulation of ODE/PDE
(cid:2)
Models with MATLAB ,
OCTAVE and SCILAB
Scientific and Engineering Applications
123
AlainVandeWouwer CarlosVilas
Service d’Automatique (Bio)Process EngineeringGroup
Université de Mons Institutode Investigaciones Marinas
Mons (CSIC)
Belgium Vigo
Spain
Philippe Saucez
Service deMathématique etRecherche
Opérationnelle
Université de Mons
Mons
Belgium
Additionalmaterial tothis bookcan bedownloaded from http://extras.springer.com
ISBN 978-3-319-06789-6 ISBN 978-3-319-06790-2 (eBook)
DOI 10.1007/978-3-319-06790-2
Springer ChamHeidelberg New YorkDordrecht London
LibraryofCongressControlNumber:2014939391
(cid:3)SpringerInternationalPublishingSwitzerland2014
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To V. L. E.
A.
To Daniel Druart
Philippe Saucez
Foreword
Withthe availabilityofcomputers ofincreasingpowerandlower cost, computer-
basedmodelingisnowawidespreadapproachtotheanalysisofcomplexscientific
andengineeringsystems.First-principlesmodelsandnumericalsimulationcanbe
used to investigate system dynamics, to perform sensitivity analysis, to estimate
unknown parameters or state variables, and to design model-based control
schemes. However, due to the usual gap between research and common practice,
scientists and engineers still often resort to conventional tools (e.g. low-order
approximate solutions), and do not make use of the full array of readily available
numerical methods.
Manysystemsfromscienceandengineeringaredistributedparametersystems,
i.e.,systemscharacterizedbystatevariables(ordependentvariables)intwoormore
coordinates (or independent variables). Time and space are the most frequent
combinationofindependentvariables,asisthecaseofthefollowing(time-varying,
transient,orunsteadystate)examples:
• temperature profiles in a heat exchanger
• concentration profiles in a sorptive packed column
• temperature and concentration profiles in a tubular reactor
• car density along a highway
• deflection profile of a beam subject to external forces
• shape and velocity of a water wave
• distribution of a disease in a population (spread of epidemics)
but other combinations of independent variables are possible as well. For
instance, time and individual size (or another characteristic such as age) occur in
population models used in ecology, or to describe some important industrial
processes such as polymerization, crystallization, or material grinding. In these
models, space can also be required to represent the distribution of individuals (of
various sizes)inaspatialregionorinanonhomogeneousreactor medium(dueto
nonideal mixing conditions in a batch reactor, or to continuous operation in a
tubular reactor).
The preceding examples show that there exists a great variety of distributed
parametersystems,arisingfrom different areas of science andengineering, which
are characterized by time-varying distributionsof dependent variables. In view of
vii
viii Foreword
the system complexity, a mathematical model, i.e., a mathematical description of
the physical (chemical, biological, mechanical, electrical, etc.) phenomena taking
place in the system, is often a prerequisite to system analysis and control. Such a
modelconsistsofpartialdifferentialequations(PDEs),boundaryconditions(BCs),
and initial conditions (ICs) describing the evolution of the state variables. In
addition, distributed parameter systems can interact with lumped parameter sys-
tems, whose state variables are described by ordinary differential equations
(ODEs), and supplementary algebraic equations (AEs) can be used to express
phenomenasuchasthermodynamicequilibria,heatandmasstransfer,andreaction
kinetics (combinations of AEs and ODEs are also frequently termed differential-
algebraic equations, or DAEs). Hence, a distributed parameter model is usually
described by a mixed set of nonlinear AE/ODE/PDEs or PDAEs. Most PDAEs
models are derived from first principles, i.e., conservation of mass, energy, and
momentum, and are given in a state space representation which is the basis for
system analysis.
Thisbookisdedicatedtonumericalsimulationofdistributedparametersystems
described by mixed systems of PDAEs. Special attention is paid to the numerical
method of lines (MOL), a popular approach to the solution of time-dependent
PDEs, which proceeds in two basic steps. First, spatial derivatives are approxi-
mated using finite difference, element, or volume approximations. Second, the
resulting system of semi-discrete (discrete in space continuous in time) equations
is integrated in time using an available solver. Besides conventional finite dif-
ference, element, and volume techniques, which are of high practical value, more
advancedspatialapproximationtechniquesareexaminedinsomedetail,including
finite element and finite volume approaches.
Although the MOL has attracted considerable attention and several general-
purposelibrariesorspecificsoftwarepackageshavebeendeveloped,thereisstilla
need for basic, introductory, yet efficient, tools for the simulation of distributed
parameter systems, i.e., software tools that can be easily used by practicing sci-
entists and engineers, and that provide up-to-date numerical algorithms.
Consequently, a MOL toolbox has been developed within MATLAB/
OCTAVE/SCILAB.Theseenvironmentsconvenientlydemonstratetheusefulness
and effectiveness of the above-mentioned techniques and provide high-quality
mathematical libraries, e.g., ODE solvers that can be used advantageously in
combination with the proposed toolbox. In addition to a set of spatial approxi-
mations and time integrators, this toolbox includes a library of application
examples, in specific areas, which can serve as templates for developing new
programs. The idea here is that a simple code template is often more compre-
hensible and flexible than a software environment with specific user interfaces.
This way, various problems including coupled systems of AEs, ODEs, and PDEs
in one or more spatial dimensions can easily be developed, modified, and tested.
This text, which provides an introduction to some advanced computational
techniquesfordynamicsystemsimulation,issuitableasafinalyearundergraduate
course or at the graduate level. It can also be used for self-study by practicing
scientists and engineers.
Preface
Our initial objective in developing this book was to report on our experience in
numerical techniques for solving partial differential equation problems, using
simple programming environments such as MATLAB, OCTAVE, or SCILAB.
Computational tools and numerical simulation are particularly important for
engineers, but the specialized literature on numerical analysis is sometimes too
denseortoodifficulttoexploreduetoagapinthemathematicalbackground.This
book is intended to provide an accessible introduction to the field of dynamic
simulation, with emphasis on practical methods, yet including a few advanced
topicsthatfindanincreasingnumberofengineeringapplications.Attheoriginof
thisbookproject,someyearsago,wewereteamingupwithBillSchiesser(Lehigh
University) with whom we had completed a collective book on Adaptive Method
ofLines.Unfortunately,thispreviousworkhadtakentoomuchofourenergy,and
the project faded away, at least for the time being.
Time passed, and the book idea got a revival at the time of the post-doctoral
stay of Carlos in the Control Group of the University of Mons. Carlos had just
achieved a doctoral work at the University of Vigo, involving partial differential
equation models, finite element techniques, and the proper orthogonal decompo-
sition, ingredients, which all were excellent complements to our background
material.
Thethreeofusthendecidedtojoinourforcestodevelopamanuscriptwithan
emphasis on practical implementation of numerical methods for ordinary and
partial differential equation problems, mixing introductory material to numerical
methods, a variety of illustrative examples from science and engineering, and a
collection of codes that can be reused for the fast prototyping of new simulation
codes.
All in one, the book material is based on past research activities, literature
review,aswellascoursestaughtattheUniversityofMons,especiallyintroductory
numericalanalysiscoursesforengineeringstudents.Asacomplementtothetext,a
website (www.matmol.org) has been set up to provide a convenient platform for
downloading codes and method tutorials.
Writingabookisdefinitelyadelicateexercise,andwewouldliketoseizethis
opportunitytothankBillforhissupportintheinitialphaseofthisproject.Manyof
his insightful suggestions are still present in the current manuscript, which has
definitely benefited from our discussions and nice collaboration.
ix
x Preface
Of course, we also would like to express our gratitude to our colleagues at
UMONS and at IMM-CSIC (Vigo), and particularly Marcel, Christine, Antonio,
Julio,Eva,andMíriam,andalltheformerandcurrentresearchteams,forthenice
working environment and for the research work achieved together, which was a
sourceofinspirationindevelopingthismaterial.Wearealsogratefultoanumber
of colleagues in other universities for the nice collaboration, fruitful exchanges at
several conferences, or insightful comments on some of our developments:
MichaelZeitz,AchimKienle,PaulZegeling,GerdSteinebach,KeithMiller,Skip
Thompson, Larry Shampine, Ken Anselmo, Filip Logist, to just name a few.
In addition, we acknowledge the support of the Belgian Science Policy Office
(BELSPO), which through the Interuniversity Attraction Program Dynamical
Systems,ControlandOptimization(DYSCO)supportedpartofthisresearchwork
and made possible several mutual visits and research stays at both institutions
(UMONS and IIM-CSIC) over the past several years.
Finally,wewouldliketostresstheexcellentcollaborationwithOliverJackson,
Editor in Engineering at Springer, with whom we had the initial contact for the
publication of this manuscript and who guided us in the review process and
selection of a suitable book series. In the same way, we would like to thank
Charlotte Cross, Senior editorial assistant at Springer, for the timely publication
process,andforherhelpandpatienceinthedifficultmanuscriptcompletionphase.
Mons, March 2014 Alain Vande Wouwer
Vigo Philippe Saucez
Carlos Vilas
Contents
1 An Introductory Tour . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.1 Some ODE Applications. . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 An ODE/DAE Application . . . . . . . . . . . . . . . . . . . . . . . . . . 32
1.3 A PDE Application . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
1.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44
2 More on ODE Integration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
2.1 A Basic Fixed Step ODE Integrator. . . . . . . . . . . . . . . . . . . . 45
2.2 A Basic Variable-Step Nonstiff ODE Integrator. . . . . . . . . . . . 51
2.3 A Basic Variable Step Implicit ODE Integrator. . . . . . . . . . . . 67
2.4 MATLAB ODE Suite. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85
2.5 Some Additional ODE Applications. . . . . . . . . . . . . . . . . . . . 86
2.5.1 Spruce Budworm Dynamics . . . . . . . . . . . . . . . . . . . 86
2.5.2 Liming to Remediate Acid Rain . . . . . . . . . . . . . . . . 93
2.6 On the Use of SCILAB and OCTAVE. . . . . . . . . . . . . . . . . . 109
2.7 How to Use Your Favorite Solvers in MATLAB? . . . . . . . . . . 115
2.7.1 A Simple Example: Matrix Multiplication. . . . . . . . . . 117
2.7.2 MEX-Files for ODE Solvers. . . . . . . . . . . . . . . . . . . 122
2.8 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123
3 Finite Differences and the Method of Lines. . . . . . . . . . . . . . . . . . 125
3.1 Basic Finite Differences . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126
3.2 Basic MOL. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127
3.3 Numerical Stability: Von Neumann and the Matrix Methods. . . 129
3.4 Numerical Study of the Advection Equation . . . . . . . . . . . . . . 136
3.5 Numerical Study of the Advection-Diffusion Equation. . . . . . . 142
3.6 Numerical Study of the Advection-Diffusion-Reaction
Equation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 150
3.7 Is it Possible to Enhance Stability? . . . . . . . . . . . . . . . . . . . . 151
3.8 Stiffness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153
3.9 Accuracy and the Concept of Differentiation Matrices . . . . . . . 157
xi