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
  Fall Semester, 2014

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.

There will be no class on Wed Nov. 26th (day before Thanksgiving), and also on Dec. 10th (Legislative Day).

1. (Sept 3rd) 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. (Sep 10th and 17th) 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. (Sept 17th and 24th) 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. Here are some additional notes about the difference between MOL and space-time schemes.

4. (Oct 1st) Temporal Integrators

We will finish the discussion of temporal integrators by first starting with more classical schemes such as Runge Kutta and multistep, and then discussing more general splitting methods. There is a homework relating to implicit time stepping schemes.

5. (Oct 8th, 15th and 22nd)  High-Resolution Advection Schemes

We will discuss limiters that ensure positivity and monotonicity (maximum principles) in advection schemes, based on an algebraic flux-limiting construction.

We will then discuss the basic elements for building high-resolution advection schemes from a more geometric/PDE perspective. My notes use a number of sources but the best source is chapters 6 and 10 in the book "Finite Volume Methods for Hyperbolic Problems" by Randall J. Leveque. I only cover 1D basics in class -- if you really want to understand how to do this in 2D and 3D choose it as a final project, either based on the book by Leveque or perhaps this article describing (and extending) the BDS algorithm by May, Nonaka, Almgren, and Bell.

There is a makeup homework in which the second problem asks you to look at Lax-Wendroff for advection-diffusion.

6. (Oct 29th) Multigrid

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.

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. (Nov 5th and 12th) Incompressible Navier-Stokes Equations

We will first discuss how to combine diffusion with (high-resolution) space-time methods for advection. This is an alternative to the MOL schemes we will use below for the full Navier-Stokes equations.

After reviewing the incompressible NS equations, 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. (Nov 12th) 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).

There will be no class on Wed Nov. 26th (day before Thanksgiving)

9. (Nov 19th) 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. (Dec 3rd) 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.

11. (Dec 17th) 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.

12. (Dec 17th) 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. Complex Fluids

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