White, Robert E. Computational mathematics. Models, methods, and analysis with MATLAB and MPI. 2nd edition. (English) Zbl 1336.65002 Textbooks in Mathematics. Boca Raton, FL: CRC Press (ISBN 978-1-4822-3515-9/hbk; 978-1-4822-3517-3/ebook). xxii, 476 p. (2016). This is a textbook on computational science. There are ten chapters. The contents can be divided into two parts: the first six chapters and the last four chapters. The first part is a blend of applications, computations, and mathematical analysis. It mainly focuses on models that are nonlinear, 2D and 3D, nonrectangular domains, systems of partial differential equations. The emphasis is more modeling and computation and less analysis. Applications include heat diffusion to cooling fins and solar energy storage, pollutant transfer in streams and lakes, shallow water waves, fluid flow in a cavity, epidemic models with disperson, option contact models, and so on. Each section of this part contains a particular application, a numerical method, a computer implementation, and an assessment of what has been done. Most sections in this part have MATLAB codes. These codes provide a learning-by-doing environment. The exercises at the end of each section have three categories: routine computations, variation of models, and mathematical analysis. The “projects” in the exercises are very interesting. In fact, the six applications in Chapters 5 and 6 could also form a basis for additional student-advisor defined projects. This part is suitable to an undergraduate level topics course for students who have had basic physics, programming, linear algebra, and multivariable calculus. The second part focuses on multiprocessing algorithms, which are implemented using message passing interface (MPI). The contents in this part include “high-performance computing” (Chapter 7), “message passing interface” (Chapter 8), classical methods and Krylov subspace methods for systems of linear equations (Chapters 9 and 10). These chapters have elementary Fortran 9x codes to illustrate the basic MPI subroutines, and the applications of the first six chapters are revisited from a parallel implementation perspective. This part can be used in a graduate level course for students who are interested in high-performance computing. Compared with the first edition [Boca Raton, FL: Chapman and Hall/CRC (2004; Zbl 1042.65001)], the second edition has a new chapter, Chapter 5, with two sections on the finite element method, two sections on shallow water waves, and two sections on the driven cavity problem. The old Chapter 6 is the new Chapter 7, and it is rewritten to include introductions to multiprocessor/multicore computers, parallel MATLAB, and MPI. Most of the MATLAB codes in the first edition are rewritten to have a more uniform style and with better documentation. Overall, this is a very interesting textbook. It is also valuable for researchers who are interested in computational science. Reviewer: Liu Xinguo (Qingdao) Cited in 1 Document MSC: 65-01 Introductory exposition (textbooks, tutorial papers, etc.) pertaining to numerical analysis 65Fxx Numerical linear algebra 65Y05 Parallel numerical computation 65Y15 Packaged methods for numerical algorithms 35K05 Heat equation 35J05 Laplace operator, Helmholtz equation (reduced wave equation), Poisson equation 35L05 Wave equation 65M06 Finite difference methods for initial value and initial-boundary value problems involving PDEs 65N06 Finite difference methods for boundary value problems involving PDEs 76M20 Finite difference methods applied to problems in fluid mechanics 65D18 Numerical aspects of computer graphics, image analysis, and computational geometry 94A08 Image processing (compression, reconstruction, etc.) in information and communication theory 91G60 Numerical methods (including Monte Carlo methods) 68W30 Symbolic computation and algebraic computation 92D30 Epidemiology 76B15 Water waves, gravity waves; dispersion and scattering, nonlinear interaction Keywords:computational science; mathematical model; numerical algorithm; simulation; Matlab code; Fortran 9x code; high-performance computing; iterative method; message passing interface; instructional position; textbook; successive overrelaxation; conjugate gradient methods; heat diffusion; Gauss elimination; Poisson equation; Newton methods; epidemics; image restoration; option contracts; computer architectures; parallel computation; direct methods; Schur complement; domain decomposition; iterative methods; Krylov methods; generalized minimal residual method; preconditioners; exercises; Picard methods; epidemic model; shallow water wave; option contact model; multiprocessing algorithm; finite element method Citations:Zbl 1042.65001 Software:MPI; KELLEY; LAPACK; Matlab PDFBibTeX XMLCite \textit{R. E. White}, Computational mathematics. Models, methods, and analysis with MATLAB and MPI. 2nd edition. Boca Raton, FL: CRC Press (2016; Zbl 1336.65002)