Lecture materialsI 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 final exam in this class, only a final project. The last class will be 12/15, and the final project will be due on Sunday 12/18 and is to be submitted electronically.
1. (Sept 8th) Advection-Diffusion EquationsA 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 15th and 22nd) 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. An example MATLAB code that does the first
assignment using a finite-volume scheme but with non-periodic
boundary conditions can be found in the codes BCs.m, SolveODE.m
and the core code AdvDiff.m.
Note that the first homework includes the lecture on spatial convergence, and that there is a new homework!
3. (Sept 29th and Oct 6th) Basic Spatio-Temporal DiscretizationsWe 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 6th and 13th) Temporal IntegratorsWe 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 20th and 27th) High-Resolution Schemes
We will introduce limiters that ensure positivity and
monotonicity (maximum principles) in advection schemes,
based on an algebraic flux-limiting construction. Here are some plots of a few
limiter functions from the main textbook and the book of
We will begin by discussing 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 BDS2D algorithm by May, Nonaka, Almgren, and Bell for 2D, and this paper on a BDS3D algorithm for 3D.
We will also discuss how to combine (high-resolution) advection with diffusion in space-time methods for advection-diffusion equations. This is an alternative to the MOL schemes we will use below for the full Navier-Stokes equations.
There is a homework that asks you to look at Lax-Wendroff/Fromm for advection-diffusion, similar to the first homework but now it is harder to use FFT-based schemes due to spatial inhomogeneity. The stability analys of these types of schemes in two dimensions can be found in the very readable and instructive paper "On the Stability of Godunov-Projection Methods for Incompressible Flow" by Michael Minion. Here is an example code to solve the homework.
7. (Nov 10th) 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
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.
8. (Friday Nov 4th at 1:30pm in WWH 312, Nov 10th)
Incompressible Navier-Stokes Equations: Pseudospectral
We will first 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.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.
9. (Nov 17th) Spatial Discretization of incompressible NS
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).
We will not discuss FEM discretizations of incompressible NS in this class in any detail, but for a brief peak into this rich topic you can look at the last 3 pages in the notes on series-expansion methods.