MATH-GA 2012.002 / CSCI-GA 2945.002

Advanced Topics in Numerical Analysis:

Computational Fluid Dynamics

Warren Weaver Hall, room 201, Mondays, 1:25 - 3:15 pm
Courant Institute of Mathematical Sciences
New York University
  Spring Semester, 2013

Lecture materials

I will post here some hand-written notes or presentation slides and links to relevant reading materials. Some of the links below may require being on the NYU network, which you can do from off-campus via a proxy server as well.

For background in fluid dynamics, you may consult the book by my colleague Stephen Childress.

1. (Jan 28th) Advection-Diffusion Equations

A nice book on the subject is "Numerical Solution of Time-Dependent Advection-Diffusion-Reaction Equation" by W. H. Hundsdorfer and J. G Verwe (not available electronically) and I will base the few first lectures on this book.

Here are notes on the continuum advection-diffusion equation, and also notes on finite-difference spatial discretization in one dimension.

Note that there is a homework!

2. (Feb 4th) Spatial Discretization

We will continue the discussion of advection-diffusion equations focusing on the analysis of convergence (stability+accuracy) of spatial discretizations. Here are some notes on spatial convergence and boundary condition treatment.

Note that the first homework includes the lecture on spatial convergence, and that there is a new homework!

3. (Feb 11th) Basic Spatio-Temporal Discretizations

We will continue discussing advection-diffusion equations but now consider how to discretize in time and not just in space (fully discrete). Here are some lecture notes and a related homework.

4. (Wednesday Feb 20th WWH.512) Higher Dimensions: Multigrid

Note that Monday Feb. 18th is a holiday and we will hold class on Feb 20th. There will be no class the following week.

We will discuss how to solve the types of linear systems that arise from implicit discretizations of diffusion, notably, Poisson and Helmholtz-type equations. Here are some lecture notes on iterative methods for solving linear systems. For multigrid I will rely on these multigrid lecture notes by William L. Briggs.

4. (Feb 25th) NO CLASS (Travel)

5. (March 4th) Positivity and Monotonicity

We will discuss limiters that ensure maximum principles in advection schemes, and wrap up the discussion of generalizations to dimensions larger than one.

6. (March 11th) Temporal Integrators

We will finish the discussion of temporal integrators by first starting with more classical schemes and then discussing more sophisticated splitting methods.

6. (March 18th) NO CLASS (Spring Break)

It is important that you start thinking about and working on your final projects. You should discuss your project with me and get it approved first.

7. (March 25th) Incompressible Navier-Stokes Equations

We will start by reviewing the incompressible NS equations. Then we will go through a simple pseudo-spectral method for two-dimensional flow. Also take a look at mit18336_spectral_ns2d.m (2D Navier-Stokes pseudo-spectral solver on the torus) and the new homework.

8. (April 1st) Spatial Discretization of NS equations

We will learn about using projection-type methods with a staggered or MAC spatial discretization, see also the Documentation  for the code mit18086_navierstokes.m (finite differences for the incompressible Navier-Stokes equations in a box).

9. (April 8th) Projection Methods

For temporal integration of the incompressible NS equations, we will discuss projection methods, based on a seminal paper by Brown, Minion and Cortez.

10. (April 15th) Spectral and Finite-Element Methods

We will discuss series-expansion methods, such as pseudo-spectral methods and briefly discuss finite-element methods, based on Ch. 4 in the book "Numerical Methods for Wave Equations" by Dale Durran.

We will then discuss Adaptive Mesh Refinement (AMR), including hyperbolic, elliptic and parabolic equations, based on lecture notes by my colleague John Bell, and software used to actually implement these algorithms in real life, particularly BoxLib, OpenFOAM, PETSc, IBAMR, and GPUs.

11. (April 22nd) Immersed-Boundary Methods

The best source for learning the foundations of the IB method is this review article by Charlie Peskin. For a more detailed computational paper, see this paper by Boyce Griffith and check out the IBAMR library.

12. (April 29th) Stokes flow

We will start by introducing boundary-Integral formulations, and then focus on the method of regularized Stokeslets based on this article by Cortez, Fauci and Medovikov (for two-dimensions, see this article).

13. (May 6th) Complex Fluids

We will discuss going beyond the Navier-Stokes equations and describing fluids with internal microstructure.

14. (May 13th) Final Presentations