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UNITEXT – La Matematica per il 3+2 Volume 65 Forfurthervolumes: http://www.springer.com/series/5418 Sandro Salsa • Federico M.G. Vegni Anna Zaretti (cid:129) Paolo Zunino A Primer on PDEs Models, Methods, Simulations SandroSalsa FedericoM.G.Vegni DepartmentofMathematics DepartmentofMathematics PolitecnicodiMilano PolitecnicodiMilano AnnaZaretti PaoloZunino DepartmentofMathematics MOX–DepartmentofMathematics PolitecnicodiMilano PolitecnicodiMilano,and DepartmentofMechanicalEngineeringand MaterialsScience UniversityofPittsburgh(USA) TranslatedandextendedversionoftheoriginalItalianedition: S.Salsa,F.M.G.Vegni,A.Zaretti,P.Zunino:Invitoalleequazioniaderivateparziali, ©Springer-VerlagItalia2009 UNITEXT–LaMatematicaperil3+2 ISSN2038-5722 ISSN2038-5757(electronic) ISBN978-88-470-2861-6 ISBN978-88-470-2862-3(eBook) DOI10.1007/978-88-470-2862-3 LibraryofCongressControlNumber:2012949463 SpringerMilanHeidelbergNewYorkDordrechtLondon ©Springer-VerlagItalia2013 Thisworkissubjecttocopyright.AllrightsarereservedbythePublisher,whetherthewholeorpartof thematerialisconcerned,specificallytherightsoftranslation,reprinting,reuseofillustrations,recita- tion,broadcasting,reproductiononmicrofilmsorinanyotherphysicalway,andtransmissionorin- formationstorageandretrieval,electronicadaptation,computersoftware,orbysimilarordissimilar methodologynowknownorhereafterdeveloped.Exemptedfromthislegalreservationarebriefex- cerptsinconnectionwithreviewsorscholarlyanalysisormaterialsuppliedspecificallyforthepurpose ofbeingenteredandexecutedonacomputersystem,forexclusiveusebythepurchaserofthework. 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Whiletheadviceandinformationinthisbookarebelievedtobetrueandaccurateatthedateofpub- lication,neithertheauthorsnortheeditorsnorthepublishercanacceptanylegalresponsibilityfor anyerrorsoromissionsthatmaybemade.Thepublishermakesnowarranty,expressorimplied,with respecttothematerialcontainedherein. 9 8 7 6 5 4 3 2 1 Cover-Design:BeatriceB,Milano TypesettingwithLATEX:PTP-Berlin,ProtagoTEX-ProductionGmbH,Germany (www.ptp-berlin.eu) PrintingandBinding:GrafichePorpora,Segrate(MI) Springer-VerlagItaliaS.r.l.,ViaDecembrio28,I-20137Milano SpringerisapartofSpringerScience+BusinessMedia(www.springer.com) Preface This book is designed as an advanced undergraduate or a first-year graduate courseforstudentsfromvariousdisciplineslikeappliedmathematics,physics, engineering.Ithasevolvedwhileteachingcoursesonpartialdifferentialequa- tions during the last decade at the Politecnico di Milano. The main purpose of these courses was twofold: on the one hand, to train the students to appreciate the interplay between theory and modelling in problemsarisingintheappliedsciencesandontheotherhandtogivethema solid background for numerical methods, such as finite differences and finite elements, also through numerical simulations for selected problems. Accord- ingly, this textbook is divided into two parts. The first one, Chapters 2 to 6, has a rather elementary character with the goal of developing and studying basic problems from the macro-areas of diffusion, propagation and transport, waves and vibrations. A knowledge of advanced calculus and ordinary differential equations is required to this part. Also, the repeated use of the method of separation of variables assumes some basic results from the theory of Fourier series. All this background material is summarized in the introductory Chapter 1 and in the Appendices. Chapter 2 is devoted to first order equations and in particular to first order scalar conservation laws. Simple models from traffic dynamics are used to introduce concepts as characteristics lines, rarefaction and shock waves. Chapters 3 and 5 deal with diffusion/reaction diffusion models, respec- tively. The heat and the Fisher-Kolmogoroff equations constitutes the refer- ence models to illustrate the qualitative properties of the solutions and the asymptotic behavior towards equilibria. In Chapter 4, the main properties of solutions to the Laplace/Poisson equation, Maximum principle, mean value properties, Green’s function and Newtonian potential are the main topics. InChapter6thefundamentalaspectsofwavespropagationareexamined, leading to the classical formulas of d’Alembert, Kirchhoff and Poisson. VI Preface The second part, Chapters 7,8 and 9, develops the Hilbert spaces methods for the variational formulation and the analysis of linear boundary and initial-boundary value problems. The understanding of these topics requires some basic knowledge of Lebesgue measure and integration, summarized in Chapter 7. This chapter contains tools from functional analysis in Hilbert spaces. The main theme is the solvability of abstract variational problems, leading to the Lax-Milgram Theorem.Then,wepresentabriefintroduction tothe theoryofdistributions of L. Schwarz and the most common Sobolev spaces, necessary for a correct variational formulation of the most common boundary value problems. Chapter 8 is devoted to the variational formulation of elliptic boundary value problems and their solvability. The development starts with one- dimensional problems, continues with Poisson’s equation and ends with general second order equations in divergence form. The last section contains an application to a simple control problem, with both distributed observation and control. The issue in Chapter 9 is the variational formulation of initial-boundary value problems for second order parabolic operators in divergence form. At the end of each chapter, a brief account of numerical methods is in- cluded, with a discussion of some particular case study, to complete a model- theory-simulation path. Also a number of exercises is presented. Some of them can be solved by a routine application of the theory or of the methods developed in the text. Other problems are intended as a completion of some arguments or proofs in thetext.Also,thereareproblemsinwhichthestudentisrequiredtobemore autonomous. Most problems are supplied with answers or hints at the end of the volume. In the first part the exposition if flexible enough to allow substantial changesintheorderofpresentationofthematerial,withoutcompromisingthe comprehension. All chapters are in practice mutually independent, with the exception of Chapter 5, which presumes the knowledge of Chapters 3 and 4. In the second part, which, in principle, may be presented independently of the first one, more attention has to be paid to the order of the arguments. A huge number of books on partial differential equation has been written. Attheendofthisvolumewehaveindicatedsomeofthemostpopularones,to whichthereadercanreferforamoreadvancedcomprehensionofthesubject. Acknowledgments. While writing this book we benefitted from comments, suggestions and criticisms of many colleagues. In particular, we express our gratitude to Cristina Cerutti, Michele Di Cristo, Maurizio Grasselli, Alessan- dro Veneziani and Gianmaria A. Verzini. Milan, September 2012 Sandro Salsa Federico M.G. Vegni Anna Zaretti Paolo Zunino Contents 1 Introduction............................................. 1 1.1 Mathematical Modelling .............................. 1 1.2 Partial Differential Equations .......................... 2 1.3 Well Posed Problems ................................. 5 1.4 Basic Notations and Facts............................. 7 1.5 Integration by Parts Formulas ......................... 10 1.6 Abstract Methods and Variational Formulation .......... 11 1.7 Numerical approximation methods ..................... 12 Part I Differential Models 2 Scalar Conservation Laws................................ 17 2.1 Introduction......................................... 17 2.2 Linear transport equation ............................. 20 2.2.1 Distributed source............................. 21 2.2.2 Extinction and localized source.................. 22 2.2.3 Inflow and outflow characteristics. A stability estimate...................................... 24 2.3 Traffic Dynamics ..................................... 26 2.3.1 A macroscopic model .......................... 26 2.3.2 The method of characteristics................... 28 2.3.3 The green light problem. Rarefaction waves....... 30 2.3.4 Traffic jam ahead. Shock waves. Rankine–Hugoniot condition..................................... 34 2.4 The method of characteristics revisited.................. 37 2.5 Generalized solutions. Uniqueness and entropy condition .. 40 2.6 The Vanishing Viscosity Method ....................... 44 VIII Contents 2.6.1 The viscous Burgers’ equation .................. 48 2.7 Numerical methods................................... 51 2.7.1 Finite difference approximation of scalar conservation laws.............................. 51 2.8 Exercises............................................ 53 2.8.1 Numerical approximation of a constant coefficient scalar conservation law......................... 55 2.8.2 Numerical approximation of Burgers equation..... 57 2.8.3 Numerical approximation of traffic dynamics...... 57 3 Diffusion................................................. 59 3.1 The Diffusion Equation ............................... 59 3.1.1 Introduction .................................. 59 3.1.2 The conduction of heat......................... 60 3.1.3 Well posed problems (n=1).................... 62 3.1.4 A solution by separation of variables............. 65 3.1.5 Problems in dimension n>1 ................... 73 3.2 Uniqueness .......................................... 76 3.2.1 Integral method............................... 76 3.2.2 Maximum principles ........................... 78 3.3 The Fundamental Solution ............................ 81 3.3.1 Invariant transformations....................... 81 3.3.2 Fundamental solution (n=1)................... 83 3.3.3 The Dirac distribution ......................... 85 3.3.4 Pollution in a channel. Diffusion, drift and reaction 88 3.3.5 Fundamental solution (n>1)................... 89 3.4 The Global Cauchy Problem (n=1).................... 90 3.4.1 The homogeneous case ......................... 90 3.4.2 Existence of a solution ......................... 91 3.4.3 The non homogeneous case. Duhamel’s method ... 92 3.5 An example of Nonlinear diffusion. The porous medium equation ............................................ 95 3.6 Numerical methods................................... 99 3.6.1 Finite difference approximation of the heat equation 99 3.6.2 Stability analysis for Euler methods ............. 101 3.6.3 The solution of the heat equation as a probability density function............................... 102 3.7 Exercises............................................ 104 3.7.1 Application of Euler methods to the discretization of the Cauchy-Dirichlet problem................. 106 3.7.2 Application to the dynamics of chemicals......... 107 Contents IX 4 The Laplace Equation ................................... 109 4.1 Introduction......................................... 109 4.2 Well Posed Problems. Uniqueness ...................... 110 4.3 Harmonic Functions .................................. 112 4.3.1 Mean value properties ......................... 112 4.3.2 Maximum principles ........................... 114 4.3.3 The Dirichlet problem in a circle. Poisson’s formula 116 4.4 Fundamental Solution and Newtonian Potential .......... 120 4.4.1 The fundamental solution ...................... 120 4.4.2 The Newtonian potential ....................... 122 4.5 The Green Function .................................. 123 4.5.1 An integral identity............................ 123 4.5.2 The Green function for the Dirichlet problem ..... 125 4.5.3 Green’s representation formula .................. 127 4.5.4 The Neumann function......................... 129 4.6 Numerical methods................................... 130 4.6.1 The 5 point finite difference scheme for the Poisson problem...................................... 130 4.7 Exercises............................................ 134 4.7.1 Approximation of an elastic membrane using the 5 point scheme................................ 136 4.7.2 Numerical simulations for testing maximum principles..................................... 137 5 Reaction-diffusion models................................ 139 5.1 Reaction Models ..................................... 139 5.1.1 The mass action law ........................... 139 5.1.2 Inhibition, activation .......................... 142 5.2 Diffusion and linear reaction........................... 145 5.2.1 Pure diffusion. Asymptotic behavior ............. 145 5.2.2 Asymptotic behavior in general domains ......... 148 5.2.3 Linear reaction. Critical dimension .............. 151 5.2.4 Linear reaction and diffusion in two dimensions ... 153 5.2.5 An Example in dimension n=3................. 155 5.3 Diffusion and nonlinear reaction........................ 160 5.3.1 Monotone methods ............................ 160 5.3.2 The Fisher’s equation.......................... 163 5.3.3 Steady states, linearization and stability.......... 165 5.3.4 Application to Fisher’s equation (Dirichlet conditions) ................................... 168 X Contents 5.3.5 Application to Fisher’s equation (Neumann conditions) ................................... 173 5.4 Turing instability..................................... 175 5.5 Numerical methods................................... 181 5.5.1 Numerical approximation of a nonlinear reaction-diffusion problem ...................... 181 5.6 Exercises............................................ 182 5.6.1 Numerical simulation of Fisher’s equations........ 183 5.6.2 Numerical approximation of travelling wave solutions ..................................... 185 5.6.3 Numerical approximation of Turing instability and pattern formation ............................. 186 6 Waves and vibrations.................................... 189 6.1 General Concepts .................................... 189 6.1.1 Types of waves................................ 189 6.1.2 Group velocity and dispersion relation............ 191 6.2 Transversal Waves in a String.......................... 193 6.2.1 The model.................................... 193 6.2.2 Energy....................................... 195 6.3 The One-dimensional Wave Equation ................... 196 6.3.1 Initial and boundary conditions ................. 196 6.4 The d’Alembert Formula .............................. 198 6.4.1 The homogeneous equation ..................... 198 6.4.2 The nonhomogeneous equation. Duhamel’s method 202 6.4.3 Dissipation and dispersion...................... 203 6.5 Second Order Linear Equations ........................ 205 6.5.1 Classification ................................. 205 6.5.2 Characteristics and canonical form .............. 208 6.6 The Multi-dimensional Wave Equation (n>1)........... 213 6.6.1 Special solutions .............................. 213 6.6.2 Well posed problems. Uniqueness................ 215 6.6.3 Small vibrations of an elastic membrane. ......... 217 6.6.4 Small amplitude sound waves ................... 221 6.7 The Cauchy Problem ................................. 225 6.7.1 Fundamental solution (n=3) and strong Huygens’ principle ..................................... 225 6.7.2 The Kirchhoff formula ......................... 228 6.7.3 Cauchy problem in dimension 2 ................. 229 6.7.4 Non homogeneous equation. Retarded potentials .. 231