This is an archived course. A more recent version may be available at ocw.mit.edu.

Calendar

Below is the calendar for the course with detailed descriptions of the topics covered in each lecture as well and assignment due dates.

LEC # MATERIAL COVERED IN LECTURE ASSIGNMENTS
1 Introduction to Numerical Fluid Mechanics: Models to simulations, Error types. Approximation and round-off errors.  
  Recitation I: MATLAB Review  
2 Round-off errors. Number representations. Errors of numerical operations. Recursion. Truncation errors, Taylor Series and Error analysis. Error propagation and Estimation. Condition numbers. Homework 1 posted
3 Condition numbers. Roots of non-linear equations - Introduction and Bracketing Methods: Bisection/False Position. Open Methods: Open-Point Iteration/Newton-Raphson/Secant methods, Extension to systems of equations.  
4 Make-up Recitation: End of Roots of non-linear equations. Fluid flow modeling: the Navier-Stokes equations and their approximations: Conservation Laws, Material Derivative, Reynolds Transport Theorem.  
5 The Navier-Stokes equations and their approximations: Constitutive equations, compressible and incompressible, vorticity.  
  Student Holiday (no class)

Homework 1 due;

Homework 2 posted
6 Approximations of the Navier-Stokes equations: Euler's equations, Bernouilli's theorems, potential flows and (boundary) integral equations. Systems of Linear Equations: Motivations and Plans, Direct Methods, Gauss Elimination.  
7 Systems of linear equations. Gaussian elimination (special cases, multiple right hand sides). LU decomposition and factorization, Pivoting. Error analysis for linear systems. Operations counts.  
8 Systems of linear equations. Special Matrices: LU Decompositions Tri-diagonal systems, General Banded Matrices, Symmetric, positive-definite Matrices. Introduction to iterative Methods.

Homework 2 due;

Homework 3 posted
9 Systems of linear equations. Iterative Methods: Jacobi's method, Gauss-Seidel iteration, Convergence, Successive Over-Relaxation Methods, Gradient Methods, Stop Criteria, Examples.  
10 End of systems of linear equations: Gradient methods, Pre-conditioning. Krylov Methods. Finite-Differences (FD): Classification of PDEs and examples, Error Types and discretization properties, Finite difference based on Taylor Series.  
  Columbus Day Holiday (no class)  
11 FD schemes: Higher order accuracy differences and examples, Taylor tables or method of undetermined coefficients, Polynomial approximations (Newton, Lagrange, Hermite and Pade Schemes), Iterative improvements and extrapolations.

Homework 3 due;

Homework 4 posted
12 Finite-Differences: Boundary conditions, Non-Uniform Grids, Grid refinement, Fourier Analysis and Error Analysis.  
13 Finite-Differences: Fourier Error Analysis, Introduction to Stability: Heuristic, Energy and Von Neumann methods, Hyperbolic PDEs, Characteristics.

Homework 4 due;

Homework 5 posted
  Quiz 1  
14 Stability, Hyperbolic eqns. Revisited, CFL condition and Von Neumann stability, Elliptic Equations revisited and FD schemes.  
15 End of Elliptic/Hyperbolic equations, Special advection schemes (Donor Cell, Flux-corrected transport, WENO), Parabolic equations revisited and numerical FD schemes.  
16 Finite Volume Methods

Homework 5 due;

Homework 6 posted
17

Finite Volume Methods (cont.)

Methods for Unsteady Problems, Time Marching Methods.

Ordinary differential equations (ODEs). Initial value problems. Euler's method.

  
18 Methods for Unsteady Problems. Time Marching Methods. ODEs-IVP. Runge-Kutta methods. Higher order ODEs, Stiffness and multistep methods.  
19 Grid Generation and complex geometries

Homework 6 due;

Homework 7 posted
20

Finite Volume on complex geometries.

Finite Element methods: Introduction. Fluid Applications.

  
21 Finite Element methods (cont.): Continuous Galerkin and Discontinuous Galerkin Methods. Spectral Methods.  
22 Inviscid Flow equations: Boundary Element methods. Panel methods. Solutions of the Navier Stokes Equation, incompressible.

Homework 7 due;

Homework 8 posted
  Quiz 2  
23 Solutions of the Navier Stokes Equation: incompressible and compressible.  
24 Solutions of the Navier Stokes Equation: incompressible and compressible. Pressure-correction, Fractional step, Vorticity, Artificial compressibility and other methods.  
25

Solutions of the Navier Stokes Equation: incompressible and compressible, end.

Boundary Layer equations, Special Topics, ODEs – Boundary value problems.

 Homework 8 due
26 Turbulent flows: models and numerical simulations.  
  Final Project Presentations (Part I)  
  Final Project Presentations (Part II)