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Annotated Bibliography

Table of Contents

See also Fairbairn and Macdonald's Bibliography of Fluid Dynamics Books, which lists a number of useful books within several domains of fluids.

1. Intermediate linear algebra

Linear algebra in infinite dimensions is basically functional analysis, which underlies a lot of partial differential equation machinery. So you need to be very comfortable with linear algebra.

  • Axler's Linear algebra done right
  • Strang's book on linear algebra

2. Partial differential equations

Fluids are famously nonlinear PDEs. So we'll need to get very comfortable with PDEs.

  • Walter Strauss,
    Partial Differential Equations: An Introduction.
    This is the book where I learned the basics of first-order PDEs, and the "big three" second-order PDEs (the heat equation, wave equation, Poisson's equation)
  • Peter J. Olver,
    Introduction to Partial Differential Equations.
    Springer, 2013. A great alternative to Strauss for people more concerned about rigor. Also Olver has great explanations.
  • Lawrence Evans,
    Partial Differential Equations.
    This has become the textbook for graduate level PDEs.
  • (I don't know where to put this, but Atkinson and Han's Theoretical Numerical Analysis: A Functional Analysis Framework discusses functional analysis and numerical differential equations in a unique way; it may be useful for people who want to learn functional analysis in an "applied" manner.)

3. Numerical Analysis

3.1. Basic Numerical Analysis

Learning floating-point arithmetic, root finding algorithms, polynomial interpolation, quadrature)

  • Burden and Faires' Numerical Analysis is the standard introductory text
  • Timothy Sauer's Numerical Analysis was the text I learned from, it's decent but not rigorous enough for my tastes.
  • Trefethen and Bau's Numerical Linear Algebra book does a great job discussing floating point in chapter 13

3.2. Numerical Linear Algebra

  • Lloyd N. Trefethen, David Bau III,
    Numerical Linear Algebra.
    Trefethen is a giant in numerical analysis, and the tone of this book is gentle and approachable.
  • James Demmel,
    Applied Numerical Linear Algebra.
    This complements Trefethen and Bau, a similar tone, and if possible should be checked out from the library.
  • Golub's Matrix Computations has been recommended, but it's an encyclopedia of results rather than an introduction to the field. I wouldn't recommend this as an introduction, but if you're serious about numerical linear algebra, then this is as necessary as a dictionary to a writer.

3.3. Numerical Differential Equations

This is trickier, because there are three basic toolkits to solve differential equations numerically (finite difference schemes, finite element method, and finite volume method)

3.3.1. Finite Difference Method

The "immediately obvious" method to numerically solve a differential equation is to replace the differential operator with finite differences.

  • R. LeVeque,
    Finite Difference Methods for Ordinary and Partial Differential Equations: Steady State and Time Dependent Problems.
    LeVeque writes beautifully, so I recommend whatever he writes; however, if you have the ability to check this out from the library, then check out the library's copy.
  • JW Thomas,
    Numerical Partial Differential Equations: Finite Differences.
    Springer. Thomas' book is an amazing experience, I heartily recommend it.
    • The sequel to this, Numerical Partial Differential Equations: Conservation Laws and Elliptic Equations, I have yet to read, but I am excited to begin it soon.

3.3.2. Finite Volume Method

Since continuity equations are usually in integral form, we can use a mesh to discretize the domain of integration, use the divergence theorem (or Stokes' theorem, or whatever) and sufficiently fine meshes to transform the continuity equations into sums of integrals of approximately piecewise-linear functions. This is the finite volume method (in a nutshell), and LeVeque writes beautifully about it.

  • R. LeVeque,
    Finite Volume Methods for Hyperbolic Problems.
    This specifically has fluids in mind, and discusses the algorithms used by LeVeque's CLAWPACK library.

3.3.3. Finite Element Method

I'm still looking for good references on finite element method. Currently I refer to Hughes, which is a very readable and enjoyable book. Basically, the finite element method discretizes the function space of solutions and test functions using locally smooth functions (usually piecewise linear functions like the "hat function", or some low order polynomial defined on some local domain).

  • Thomas J.R. Hughes,
    The Finite Element Method: Linear Static and Dynamic Finite Element Analysis.
    Published by Dover, it's the best (and only) book I've seen on finite element analysis. Basically, we transform a PDE into its weak form, which requires integrating against arbitrary test functions. The finite element method discretizes the space of test functions, which transforms the problem to one of linear algebra.

3.3.4. Spectral Methods

I have no good reference for this, but basically this discretizes the function space as a truncated Fourier series. More generally, instead of using \(\exp(\I nx)\) as the basis functions, we could use a different choice of basis. We could view this as a generalization of the finite element method, I think; the difference being the basis functions used in finite element method are local smooth functions (like the bump function), whereas the spectral method uses global smooth functions (like \(\E^{\I nx}\)).

Here are some references which have been recommended to me but I have not followed up reading yet…

  • John P. Boyd,
    Chebyshev and Fourier Spectral Methods.
    This is one of the only references I know for the spectral method, which basically uses a truncated Fourier series with unknown coefficients. For sufficiently smooth solutions, we only need a few coefficients to get a good approximation, which tends to work remarkably well with climate models. It also helps this book has been republished by Dover, so it's on the cheap side, and it's available on the author's website.
  • C. Canuto, M.Y. Hussaini, A. Quarteroni, and T.A. Zang,
    Spectral Methods in Fluid Dynamics.
    Springer Verlag, Berlin, Germany; Heidelberg, Germany, 1988.
  • C. Canuto, M.Y. Hussaini, A. Quarteroni, and T.A. Zang,
    Spectral Methods: Fundamentals in Single Domains.
    Springer Verlag, Berlin, Germany; Heidelberg, Germany, 2006).
  • C. Canuto, M.Y. Hussaini, A. Quarteroni, and T.A. Zang,
    Spectral Methods: Evolution to Complex Geometries and Applications to Fluid Dynamics.
    Springer Verlag, Berlin, Germany;Heidelberg, Germany, 2007.
  • B. Fornberg,
    Practical Guide to Pseudospectral Methods.
    Cambridge University Press, Cambridge, U.K., 1995.
  • D. Gottlieb and S.A. Orszag,
    Numerical Analysis of Spectral Methods: Theory and Applications.
    Society for Industrial and Applied Mathematics, Philadelphia, U.S.A., 1977.
  • J.S. Hesthaven, S. Gottlieb, and D. Gottlieb,
    Spectral Methods for Time-Dependent Problems.
    Cambridge University Press, Cambridge, U.K., 2007.

4. Basic Fluids

  • George Batchelor's An Introduction to Fluid Dynamics is the classic introduction to the field; the first few chapters are required reading for everyone
    • Simon J.A. Malham's Introductory fluid mechanics (pdf) is based on Batchelor's book, just as Batchelor's book was based on Lamb's Hydrodynamics
  • Stephen Childress,
    "An Introduction to Theoretical Fluid Dynamics".
    Lecture notes, pdf, 177 pages. This was polished then published as a book, I think it's wonderful.
  • Landau and Lifshitz,
    Fluid Mechanics, 2nd ed.
    This is a "classic", but assumes the reader knows thermodynamics, and is presented in a "reverse mathematical style" (instead of "definition–theorem–proof", they do "definition–proof–theorem" without any clear delineations). I admit I have not seen as elegant a derivation of consequences from Euler's flow equations.
  • Alexandre Chorin, Jerrold Marsden,
    A Mathematical Introduction to Fluid Mechanics.
    Third ed., Springer, 1993. This is probably the introduction most "pure mathematicians" prefer (your mileage may vary).
  • Frank White,
    Fluid Mechanics.
    Written with hydrodynamics in mind, it's a friendly introductory physics text.
  • Pijush K. Kundu and Ira M. Cohen,
    Fluid Mechanics.
    A standard introductory text, used by MIT.
  • D. J. Tritton,
    Physical fluid dynamics.
    An experimentally grounded text, written for physicists, but I'm including it to note (if mathematicians want to learn the empirical side of fluids) this is the resource for experiments.
  • M. Van Dyke,
    Album of Fluid Motion.
    Referenced by several syllabi, it's a picture book of various flows.

5. Numerical Methods for Fluids

  • R. LeVeque,
    Numerical Methods for Conservation Laws.
    A slim but beautiful book.
  • Roger Temam,
    Navier-Stokes equations.
    This book beautifully works through the numerical analysis (as of the mid 1980s) of the steady state Navier–Stokes, first the linearized version and then the full nonlinear stead-state, and then gives some discussion to the full problem.
    If interested, the reader may consult Galdi's 2-volume An introduction to the mathematical theory of the Navier-Stokes equations.

6. Navier–Stokes Equations

  • Pierre Gilles Lemarie-Rieusset,
    The Navier-Stokes problem in the 21st century.
    This is an amazingly thorough yet gentle introduction to the subject. It's written with the Clay millennium prize in mind.
  • Tai-Peng Tsai,
    Lectures on Navier-Stokes equations.
    AMS Graduate studies series. A good book after reading the first few chapters of Evans' PDEs, Tsai reviews the various aspects to Navier-Stokes in a very gentle manner. Also good is the citations to the literature (done immediately on page ix, literally the second sentence). The focus of this book is theory of the existence, uniqueness, and regularity of solutions to Navier-Stokes.

7. Syllabi

Whenever learning a subject, it's wise to examine what the top universities are doing when teaching the subject.

  • MIT
    • 1.63 "Advanced Fluid Dynamics of the Environment" syllabus
    • 2.06 "Fluid Dynamics" syllabus (studies the first three chapters of Frank White's Fluid Mechanics, 7th ed.)
    • 2.25 "Advanced fluid mechanics" syllabus
    • 2.26 "Compressible Fluid Dynamics" syllabus
    • 2.29 "Numerical Fluid Mechanics" syllabus
    • 12.003 "Atmosphere, Ocean and Climate Dynamics" syllabus
    • 12.800 "Fluid Dynamics of the Atmosphere and Ocean" syllabus
    • 13.024 "Numerical Marine Hydrodynamics" syllabus
  • Caltech
  • Georgia Tech
    • ME3340 Syllabus
      • Munson, Young, and Okiishi’s Fundamentals of Fluid Mechanics
    • ME6601 syllabus Introduction to Fluid Mechanics
      • Ronald Panton, Incompressible Flow, Panton; 2nd Edition, John Wiley; 1995.
      • G. M. Homsy, et al, Multi-Media Fluid Mechanics (w/CD), Cambridge University Press, 2000.
    • ME7751 Syllabus Computational Fluid Mechanics
      • Pieter Wesseling, Principles of Computational Fluid Dynamics; Springer-Verlag, 2000.
      • Joel H. Ferziger and Milovan Peric, Computational Methods of Fluid Dynamics, 3rd Edition, Springer-Verlag, 2001
      • C. Hirsch, Numerical Computation of Internal and External Flows, Vol. 1 and 2, John Wiley & Sons, Ltd., 1988.
      • G. F. Carey and Oden, J. T., Finite Elements: Fluid Mechanics, Vol. VI, Prentice-Hall, 1986.
      • C. Canuto, M. Y. Hussaini, A. Quarteroni, and T. A. Zang, Spectral Methods in Fluid Mechanics, Springer-Verlag, 1988.

Last Updated 2022-04-07 Thu 08:47.