# Computational Fluid Dynamics

Warren Weaver Hall, room 512, Thursdays, 9:00-10:50am
Courant Institute of Mathematical Sciences
New York University
Fall Semester, 2016

## 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 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 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 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 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 6th and 13th) 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 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 Leveque.

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 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.

### 8. (Friday Nov 4th at 1:30pm in WWH 312, Nov 10th) Incompressible Navier-Stokes Equations: Pseudospectral methods

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 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).

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.

### 10. (Dec 1st) 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.

### 11. (Dec 8th and 15th) 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. (Dec 8th) Adaptive Mesh Refinement

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.

### 13. 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).

### 14. Complex Fluids

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