×

Finite difference computing with PDEs. A modern software approach. (English) Zbl 1377.65105

Texts in Computational Science and Engineering 16. Cham: Springer Open (ISBN 978-3-319-55455-6/hbk; 978-3-319-55456-3/ebook). xxiii, 507 p., open access (2017).
The aim of the authors is to introduce the basic concepts and a thorough understanding of the way how to think about applying numerical methods. The restriction to finite difference methods is helpful to this respect. Compared to finite element and finite volume methods the finite difference equations are simpler to be set up and coded and enable an understanding of the details involved as well as the solution procedure. The authors further underline their pedagogical philosophy by always applying a three-step process of simplifying, understanding and generalizing in the presentation of a new topic.
A typical example is Chapter 3 on the diffusion equation. The authors start with the forward Euler method for the 1D equation including (as in each section) implementation, verification and numerical experiments (11 p.). It follows a section on implicit methods (backward Euler, Crank-Nicolson, \(\theta\)-rule, leapfrog) (10 p.), then a section with the analysis of the schemes (23 p.). Next heterogeneous media and geometric symmetries are considered (9 p.) before turning to the 2D case where also algebraic solution methods are located (Jacobi, relaxed Jacobi, Gauss-Seidel and SOR, conjugate gradient, vectorization) (34 p.). As in all chapters the final pages are devoted to applications (diffusion of a substance, heat conduction, porous media, potential fluid flow, streamlines of 2D flow, potential of an electric field, flow between plates and in a straight tube, thin film flow, propagation of electric signals in the brain) (8 p.) and a considerable number of exercises (6 p.). Chapter 3 also contains a section “Random walk” (20 p.). As it is the book’s general intention, a deeper mathematical analysis is not given as well as e.g. more complex geometries than rectangles are not treated. But many lines of computer code are there.
The book’s list of contents reads as follows:
{} Chapter 1. Vibration ODEs (92 p.)
{} Chapter 2. Wave equations (113 p.)
{} Chapter 3. Diffusion euations (116 p.)
{} Chapter 4. Advection-dominated equations (30 p.)
{} Chapter 5. Nonlinear problems (55 p.)
{} Appendix A. Useful formulas (5 p.)
{} Appendix B. Truncation error analysis (35 p.)
{} Appendix C. Software engineering; Wave equation model (41 p.)
{} References. 18 items.
The way how the material of the book is presented consumes quite a lot of printing space and is perhaps a little bit too detailed for the already somewhat experienced reader. But it serves well the major purpose of the text “to help the practioner by providing all nuts and bolts necessary for savely going from the mathematics to a well-designed computer code” and that “it has a potential for use early in undergraduate student program”.

MSC:

65M06 Finite difference methods for initial value and initial-boundary value problems involving PDEs
65-01 Introductory exposition (textbooks, tutorial papers, etc.) pertaining to numerical analysis
65L12 Finite difference and finite volume methods for ordinary differential equations
65D05 Numerical interpolation
65N06 Finite difference methods for boundary value problems involving PDEs
65N12 Stability and convergence of numerical methods for boundary value problems involving PDEs
65H05 Numerical computation of solutions to single equations
65H10 Numerical computation of solutions to systems of equations
65Y05 Parallel numerical computation
35-04 Software, source code, etc. for problems pertaining to partial differential equations
76S05 Flows in porous media; filtration; seepage
76A20 Thin fluid films
60G50 Sums of independent random variables; random walks
65P10 Numerical methods for Hamiltonian systems including symplectic integrators
35K05 Heat equation
35L05 Wave equation
65Y15 Packaged methods for numerical algorithms
PDFBibTeX XMLCite
Full Text: DOI