Introduction to Partial Differential Equations: A Computational Approach Aslak Tveito Ragnar Winther Springer This page intentionally left blank Preface “It is impossible to exaggerate the extent to which modern applied mathematics has been shaped and fueled by the gen- eral availability of fast computers with large memories. Their impact on mathematics, both applied and pure, is comparable to the role of the telescopes in astronomy and microscopes in biology.” — Peter Lax, Siam Rev. Vol. 31 No. 4 Congratulations!Youhavechosentostudypartialdifferentialequations. That decision is a wise one; the laws of nature are written in the language of partial differential equations. Therefore, these equations arise as models in virtually all branches of science and technology. Our goal in this book is to help you to understand what this vast subject is about. The book is anintroductiontothefield.Weassumeonlythatyouarefamiliarwithba- sic calculus and elementary linear algebra. Some experience with ordinary differential equations would also be an advantage. Introductory courses in partial differential equations are given all over the world in various forms. The traditional approach to the subject is to introduce a number of analytical techniques, enabling the student to de- riveexactsolutionsofsomesimplifiedproblems.Studentswholearnabout viii Preface computational techniques on other courses subsequently realize the scope of partial differential equations beyond paper and pencil. Our approach is different. We introduce analytical and computational techniquesinthesamebookandthusinthesamecourse.Themainreason for doing this is that the computer, developed to assist scientists in solv- ing partial differential equations, has become commonly available and is currently used in all practical applications of partial differential equations. Therefore,amodernintroductiontothistopicmustfocusonmethodssuit- ableforcomputers.Butthesemethodsoftenrelyondeepanalyticalinsight into the equations. We must therefore take great care not to throw away basic analytical methods but seek a sound balance between analytical and computational techniques. Oneadvantageofintroducingcomputationaltechniquesisthatnonlinear problemscanbegivenmoreattentionthaniscommoninapurelyanalytical introduction. We have included several examples of nonlinear equations in addition to the standard linear models which are present in any introduc- tory text. In particular we have included a discussion of reaction-diffusion equations.Thereasonforthisistheirwidespreadapplicationasimportant models in various scientific applications. Our aim is not to discuss the merits of different numerical techniques. Thereareahugenumberofpapersinscientificjournalscomparingdifferent methodstosolvevariousproblems.Wedonotwanttoincludesuchdiscus- sions.Ouraimistodemonstratethatcomputationaltechniquesaresimple touseandoftengiveveryniceresults,nottoshowthatevenbetterresults can be obtained if slightly different methods are used. We touch briefly upon some such discussion, but not in any major way, since this really be- longs to the field of numerical analysis and should be taught in separate courses. Having said this, we always try to use the simplest possible nu- merical techniques. This should in no way be interpreted as an attempt to advocate certain methods as opposed to others; they are merely chosen for their simplicity. Simplicity is also our reason for choosing to present exclusively finite difference techniques. The entire text could just as well be based on finite element techniques, which definitely have greater potential from an appli- cation point of view but are slightly harder to understand than their finite difference counterparts. We have attempted to present the material at an easy pace, explaining carefully both the ideas and details of the derivations. This is particularly thecaseinthefirstchaptersbutsubsequentlylessdetailsareincludedand some steps are left for the reader to fill in. There are a lot of exercises included,rangingfromthestraightforwardtomorechallengingones.Some of them include a bit of implementation and some experiments to be done on the computer. We strongly encourage students not to skip these parts. In addition there are some “projects.” These are either included to refresh Preface ix the student’s memory of results needed in this course, or to extend the theories developed in the present text. Giventhefactthatweintroducebothnumericalandanalyticaltools,we have chosen to put little emphasis on modeling. Certainly, the derivation ofmodelsbasedonpartialdifferentialequationsisanimportanttopic,but it is also very large and can therefore not be covered in detail here. The first seven chapters of this book contain an elementary course in partialdifferentialequations.Topicslikeseparationofvariables,energyar- guments, maximum principles, and finite difference methods are discussed for the three basic linear partial differential equations, i.e. the heat equa- tion, the wave equation, and Poisson’s equation. In Chapters 8–10 more theoretical questions related to separation of variables and convergence of Fourier series are discussed. The purpose of Chapter 11 is to introduce nonlinear partial differential equations. In particular, we want to illustrate how easily finite difference methods adopt to such problems, even if these equationsmaybehardtohandlebyananalyticalapproach.InChapter12 wegiveabriefintroductiontotheFouriertransformanditsapplicationto partial differential equations. Some of the exercises in this text are small computer projects involving a bit of programming. This programming could be done in any language. In order to get started with these projects, you may find it useful to pick up some examples from our web site, http://www.ifi.uio.no/˜pde/, where you will find some Matlab code and some simple Java applets. Acknowledgments It is a great pleasure for us to thank our friends and colleagues for a lot of help and for numerous discussions throughout this project. In particular, we would like to thank Bent Birkeland and Tom Lyche, who both partici- pated in the development of the basic ideas underpinning this book. Also we would like to thank Are Magnus Bruaset, Helge Holden, Kenneth Hvis- tendahlKarlsen,JanOlavLangseth,HansPetterLangtangen,GlennTerje Lines, Knut Mørken, Bjørn Fredrik Nielsen, Gunnar Olsen, Klas Samuels- son,AchimSchroll,WenShen,JanSøreng,and˚AsmundØdeg˚ardforread- ingpartsofthemanuscript.Finally,wewouldliketothankHansBirkeland, TrulsFlatberg,RogerHansen,ThomasSkjønhaug,andFredrikTyvandfor doing an excellent job in typesetting most of this book. Oslo, Norway, April 1998. Aslak Tveito Ragnar Winther