Good books and articles on numerical methods for fluid dynamics

One of my current interests is recent advances in non-oscillatory central-upwind algorithms, such as in Kurganov & Tadmor, JCP 160 (200) 241 or Kurganov et al. SIAM J. Sci Comput. 23 (2001) 707, which are simpler and improved versions of Nessyahu & Tadmor JCP 87 (1990) 408. They are simple to implement, avoid the need for a Riemann solver, and appear to be quite robust. See

There are similarities with ENO/WENO algorithms at least for the reconstruction step (some of the later central-upwind papers use "Central Weno"). A nice review of ENO/WENO is:, "Essentially Non-Oscillatory and Weighted Essentially Non-Oscillatory Schemes for Hyperbolic Conservation Laws", Chi-Wang SHu, November 1997, ICAS Report No. 97-65, NASA-CR/97-206253.

Dale R. Durran, "Numerical Methods for Wave Equations in Geophysical Fluid Dynamics", Springer, 1999. (online within Princeton:

I learned about this book from Jeff Candy.  I like it because it covers a lot of the zoo of numerical methods I've picked up ad hoc over the years and puts them in context.  Different fields have developed methods independently, but they are often related.  The book discusses Flux-Corrected Transport algorithms (FCT) developed in plasma physics and relates them to methods developed in other areas, such as TVD, van Leer's flux limites, etc.  If you look for this in the Princeton Library online catalog, you'll see a link to an online version of this book at Providing books online is very useful.

Randall J. Leveque, Finite Volume Methods for Hyperbolic Problems, Cambridge Univ. Press 2002. Extensive discussion of shock-capturing methods, high-resolution Godunov methods, Reimann solvers, flux-limiters, flux-corrected transport, etc. Starts at a simple level and carrying you up to a level to understand current research papers in the Journal of Computational Physics, etc. Companion CLAWPACK software package.

Culbert B. Laney, Computational Gasdynamics, 1998.

An excellent book recommended to me by Bill Dorland, and which covers a lot of topics not really covered in Durran's book (such as elliptic solvers or more details on spectral methods) is:

Chebyshev and Fourier Spectral Methods, John P. Boyd

available for free at:

Trefethen numerical ODE/PDE Textbook online at Oxford.

Jim Stone and others involved in the development of Zeus (a widely used MHD code in Astrophysics), are working on a new higher-order Godunov code for astrophysical MHD and have written a number of interesting articles.

Two interesting articles pointed out by Eliot Quataert:

An MHD review article by Axel Brandenburg:

Another useful article was a review by Ue-Li Pen from CITA:

One comment:  It's kind of interesting that this 3D MHD code is only 500 lines.  It is even parallelized (though only with openMP) and uses a "relaxed TVD" algorithm. The relaxed TVD algorithm avoids the need for a Reimann solver, but does it necessarily assure positivity from advection? (I suppose it should if it is called TVD?)   But this code is written for a pretty simple limit and simple coordinate system.  The complexity of codes seems to grow exponentially with the number of features in the code.  The complexity of the code quickly increases with unusual geometries (in fusion devices such as tokamaks, a lot of effort goes into accurate evaluation of gradients along field lines, sometimes by using field-aligned coordinates, because the thermal conduction along field lines can be a million times faster than across field lines),  implicit methods to deal with irrelevant fast waves (fusion devices are often at relatively low beta, or have low beta regions where there is a very fast compresional Alfven wave that can be ordered out of the system, or if non-ideal MHD effects such as the Hall terms are kept then there are even faster waves...), i/o to set up a wide variety of problems, higher-order differencing algorithms, more advanced parallelization, graphics and diagnostics, and especially adaptive mesh refinement,..

Multidisciplinary computational science course at Princeton
Two 1995 articles by Antony Jameson on "Analysis and Design of Numerical Schemes for Gas Dynamics I: Artificial, Diffusion, Upwind Biasing, Limiters and Their Effect on Accuracy and Multigrid Convergence", recommended to me by various people: slip.pdf, slip2.pdf.
Computational Science Eductation Project at ORNL:
Some of it is out of date (for example it emphasizes PVM, not MPI, which has become the standard), and some of the derivations are incomplete, but it is interesting to just see how they have organized the field.

Useful starting points:
the table of contents for the online book
online lecture notes
Lecture link to ODE notes
Lecture link to PDE notes
Gear's Backward Differentiation Formulas for integrating stiff equations.

A comprehensive discussion of lots of iterative solvers (though it only mentions multigrid in passing):>
The Yale Multigrid web site