Difference Methods for Initial-Value Problems. by Robert D. Richtmyer; K. W. MortonReview by: Burton WendroffSIAM Review, Vol. 10, No. 3 (Jul., 1968), pp. 381-383Published by: Society for Industrial and Applied MathematicsStable URL: http://www.jstor.org/stable/2027672 .

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SIAM REVIEW Vol. 10, No. 3, July, 1968

BOOK REVIEWS

EDITED BY ARTHUR WOUK

Publishers are invited to send books for review to Professor Arthur Wouk, Departmnent of Engineering Sciences, The Technological Institute, Northwestern University, Evanston, Illinois 60201.

Difference Methods for Initial-Value Problems. By ROBERT D. RICHTMYER and K. W. MORTON. Interscience Publishers, New York, 1967. xiv + 405 pp. $14.95. The first edition of this book appeared in 1957. It has had a profound influence

on research in the theory of stable difference operators, and it has been an in- valuable aid to workers in physical laboratories who have had to develop methods for specific and usually very difficult problems. This second edition includes many of the significant theoretical results obtained up to 1967, as well as some of the recent practical computing experience of numerical analysts, particularly in the field of hydrodynamics. The authors present theoretical and practical items with painstaking care and with mathematical honesty. If the art of solving partial differential equations numerically ever becomes a science, we shall surely be able to give much of the credit to this book.

Since this edition is such an extensive revision of the first, we present a chapter- by-chapter review without making any comparisons with the first edition.

Chapter 1. This contains an excellent introduction to the Fourier and energy methods. There is a dramatic illustration of instability and a succinct description of this phenomenon in the last paragraph of ?1.6.

Chapter 2. This is a very pleasant introduction to Banach spaces up to the uniform boundedness theorem.

Chapter 3. After defining well-posed problems the authors prove the Lax equivalence theorem; namely, that for well-posed problems stability and con- sistency are necessary and sufficient for convergence. The proof of sufficiency might have begun with the following heuristic argument: Let the difference equa- tion be vn+l = Cv?. Let u, the solution of the differential equation, satisfy un+1 = Cun + AtE. Let n+l = I Un-1 v 1n+ I. Then 3n+1 < ICn-1 I 50

+ Z==o I Cj I At I E 1. So if So = 0, J Cj I < const. (stability), and E >O (con- sistency), we have convergence.

In ?3.6 and ?3.7 the Lax theorem is extended to inhomogeneous problems. The authors warn in the preface that "proofs may be rough going for many readers." Indeed, ?3.6 is quite a jolt, but is only an indication of things to come.

The idea of equivalent norm is introduced in ?3.8. This idea is very important for the chapters which follow, so it seems strange that only half a page is given to it here.

In ?3.9 it is proved that a stable difference operator remains stable if it is perturbed by an operator of order At. It is stated that "certain alternative defi- nitions of stability are not equally insensitive to perturbations." The authors are referring to the stability definition of Forsythe-Wasow (weak stability). An ex-

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382 BOOK REVIEWS

armple is givTen in Chapter 5 of a weakly stable scheme which becomes unstable after a perturbation of order At. The implication, to the reviewer, of the above quot:e is that weak stability is not a good definition; in fact, weak stability is no better or worse than the concept of well-posedness in the sense of Hadamard (weak well-posedness). A weakly well-posed differential equation can become im- properly posed after the addition of a lower order term, so it is no surprise that a simnilar phenomenon exists for weakly stable difference equations.

Chapter 4. The Kreiss matrix theorem and associated stability conditions are derived. The role of Fourier analysis in the initial value problem with constant co- efficients is beautifully presented. There are also good examples of the use of various stability conditions.

Chapter 5. This chapter explores the relation between local and global stability for pure initial value problems with variable coefficients. We have Strang's elegant theorem to the effect that global stability implies local stability of the principal part. There are some comments on Forsythe-Wasow (weak) stability. Local criteria for parabolic equations and hyperbolic systems are given. There is also Strang's work on nonlinear problems with smooth solutions. Some indication is given of the practical difficulties involved in solving nonlinear problems.

Two deep results are given here: Kreiss' theorem on dissipative difference schemes, which takes nine pages and hardly seems "relatively straightfoward," and the Lax-Nirenberg theorem on the field of values of the amplification matrix. To the authors' statement that "the recent theoretical advances have generally used mathematical methods and concepts which are likely to be difficult or com- pletely inaccessible to persons who are not specialists in numerical analysis," we would add that even specialists will have some difficulty with this chapter.

Chapter 6. The energy method is developed in greater detail for a mixed prob- lem for the heat equation, and for coupled sound and heat flow. In the latter case a proof of well-posedness would have been instructive. Kreiss' work on semi- bounded operators is indicated, but the procedure that Kreiss developed for con- structing stable difference approximations to semibounded mixed problems is not mentioned. The stability criterion of Godunov and Ryabenkii is presented in detail, and Strang's use of the Wiener-Hopf technique is sketched.

Chapter 7. This extends the Lax equivalence theorem to multilevel schemes. The remaining chapters are devoted to applications. Chapter 8. Explicit and implicit schemes for heat flow and diffusion are given

along with some numerical results. The alternating direction and splitting methods for problems in several space variables are discussed, but not in great detail.

Chapter 9. This chapter describes a variety of methods for solving the trans- port equation: spherical harmonics, discrete ordinates, Carlson's Sn method, and the method of characteristic variables.

The particular problems presented in this part of the book are the subject of intensive research in physical laboratories throughout the world. Thousands of reports have been written, and it is not humanly possible to keep up with them all. For example, the authors refer here to a 1953 Los Alamos report on the Sn method by Bengt Carlson, when in fact many reports have since been written

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BOOK REVIEWS 383

by Carlson and others improving on the Sn method. Mlost of this work is surm- marized in Bengt G. Carlson and Kaye D. Lathrop, Transport theory, the method of discrete ordinates, Los Alamos Scientific Laboratory Report LA-3251-IMS, 1965.

Chapter 10. Specific difference equations are developed for the wave equation aind coupled sound and heat flow. A practical stability condition is offered: no Fourier component of the solution of the difference equation should grow faster than the most rapid possible growth of the exact solution. Here the authors are putting on a restriction as t -* oo. The restriction is perfectly reasonable, namely, that the solution of the difference equation behave like the solution of the differ- ential equation as t - oo; however, this involves a new definition of well-posed- ness and a corresponding stability definition. These ideas have been discussed by J. Gary, A generalization of the Lax-Richtmyer theorem on finite difference schemes, SIAM J. Numer. Anat., 3 (1966), pp. 467-473.

Chapter 11. Explicit and implicit difference equations are given for the vibra- tions of a thin beam.

Chapters 12 and 13. These two chapters almost form a handbook for hydro- dynamicists. Considerable space is given to methods of calculating flows with shock waves in one and two dimensions. The strength and weakness of the artificial viscosity methods of von Neumann-Richtmyer and Lax-Wendroff (with variants) are discussed in detail, as is the method of shock fitting and Godunov's method.

BURTON WENDROFF

University of Denver

Applications of Undergraduate Mathematics in Engineering. By BEN NOBLE. The Macmillan Co., New York, 1967. xvii + 364 pp. $9.00. This is an unusual book. It is the result of a joint effort by the Cornmittee

on the Undergraduate Program in Mathematics of the MNathematical Association of America and the Commission on Engineering Education. Engineers and mathe- maticians were asked to submit problems illustrating significant applications of mathematics in engineering. From those submitted 45 problems were selected, and Professor Noble was given the task of presenting them in a form instructive to undergraduate mathematicians and engineers.

The author has written a lively, readable book, presenting many novel ideas at an elementary level. The problems are examined from many points of view: various models for the physical situation, alternative mathematical procedures, effect of simplifying assumptions, role of computers. Each problem is pursued only as far as justified, routine calculations being suppressed.

The problems themselves are for the most part quite different from those found in textbooks and effectively show the extraordinary variety both of mathe- matical methods and of the practical problems which arise.

The following is a summary of the topics covered:

Optimum location problems, linear programming, steepest descent, applica- tions in chemical engineering and design of optical lenses.

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