Course Objectives
Students successfully completing 16.901 should have:
 A conceptual understanding of computational methods commonly used for analysis and design of aerospace systems.
 A working knowledge of computational methods including experience implementing them for model problems drawn from aerospace engineering applications.
 A basic foundation in theoretical techniques to analyze the behavior of computational methods.
Measurable Outcomes
The subject is divided into four sections:
 Integration of Systems of Ordinary Differential Equations (ODE's)
 Finite Volume and Finite Difference Methods for Partial Differential Equations (PDE's)
 Finite Element Methods for Partial Differential Equations
 Probabilistic Simulation Techniques
For each of these sections, the measurable outcomes are described below. Specifically, a student successfully completing 16.901 will be able to:
Integration Methods for ODE's
 (a) Describe the AdamsBashforth, AdamsMoulton, and Backwards Differentiation families of multistep methods;
(b) Describe the form of the RungeKutta family of multistage methods; and
(c) Explain the relative computational costs of multistep versus multistage methods.
 (a) Explain the concept of stiffness of a system of equations, and
(b) Describe how it impacts the choice of numerical method for solving the equations.
 (a) Explain the differences and relative advantages between explicit and implicit methods to integrate systems of ordinary differential equations; and
(b) For nonlinear systems of equations, explain how a NewtonRaphson can be used in the solution of an implicit method.
 (a) Define a convergent method;
(b) Define a consistent method;
(c) Explain what (zero) stability is; and
(d) Demonstrate an understanding of the Dahlquist Equivalence Theorem by describing the relationship between a convergent method, consistency, and stability.
 Determine if a multistep method is stable and consistent.
 (a) Define global and local order of accuracy for an ODE integration method,
(b) Describe the relationship between global and local order of accuracy, and
(c) Calculate the local order of accuracy for a given method using a Taylor series analysis.
 (a) Define eigenvalue stability, and
(b) Determine the stability boundary for a multistep or multistage method applied to a linear system of ODE's.
 Recommend an appropriate ODE integration method based on the features of the problem being solved.
 Implement multistep and multistage methods to solve a representative system of ODE's from an engineering application.
Finite Difference and Finite Volume Methods for PDE's
 (a) Define the physical domain of dependence for a problem,
(b) Define and determine the numerical domain of dependence for a discretization, and
(c) Explain the CFL condition and determine the timestep constraints resulting from the CFL conditions.
 Determine the local truncation error for a finite difference approximation of a PDE using a Taylor series analysis.
 Explain the difference between a centered and a onesided (e.g. upwind) discretization.
 Describe the Godunov finite volume discretization of twodimensional convection on an unstructured mesh.
 Perform an eigenvalue stability analysis of a finite difference approximation of a PDE using either Von Neumann analysis or a semidiscrete (method of lines) analysis.
 Implement a finite difference or finite volume discretization to solve a representative PDE (or set of PDE's) from an engineering application.
Finite Element Methods for PDE's
 (a) Describe how the Method of Weighted Residuals (MWR) can be used to calculate an approximate solution to a PDE,
(b) Describe the differences between MWR, the collocation method, and the leastsquares method for approximating a PDE, and
(c) Describe what a Galerkin MWR is.
 (a) Describe the choice of approximate solutions (i.e. the test functions or interpolants) used in the Finite Element Method, and
(b) Give examples of a basis for the approximate solutions in particular including a nodal basis for at least linear and quadratic solutions.
 (a) Describe how integrals are performed using a reference element,
(b) Explain how Gaussian quadrature rules are derived, and
(c) Describe how Gaussian quadrature is used to approximate an integral in the reference element.
 Explain how Dirichlet and Neumann boundary conditions are implemented for Laplace's equation discretized by FEM.
 (a) Describe how the FEM discretization results in a system of discrete equations and, for linear problems, gives rises to the stiffness matrix; and
(b) Describe the meaning of the entries (rows and columns) of the stiffness matrix and of the righthand side vector for linear problems.
Probabilistic Methods
Note: all students are expected to have a thorough understanding of probability, random variables, PDF's, CDF's, mean (expectation), variance, standard deviation, percentiles, uniform distributions, normal distributions, and x^{2}distributions from the prerequisite coursework.
 Describe how Monte Carlo sampling from multivariable, uniform distributions works.
 Describe how to modify Monte Carlo sampling from uniform distributions to general distributions.
 (a) Describe what an unbiased estimator is;
(b) State unbiased estimators for mean, variance, and probability; and
(c) State the distributions of these unbiased estimators.
 (a) Define standard error;
(b) Give standard errors for mean, variance and probability;
(c) Place confidence intervals for estimates of the mean, variance, and probability; and
(d) Demonstrate the dependence of Monte Carlo convergence on the number of random inputs and the number of samples using the above error estimates.
 (a) Describe stratified sampling for single input and multiple inputs,
(b) Describe Latin Hypercube Sampling (LHS), and
(c) Describe the benefits of LHS for nearly linear outputs in terms of the standard error convergence of the mean with the number of samples.
 (a) Describe the Response Surface Method (RSM);
(b) Describe the construction of a response surface through Taylor series, Design of Experiments with the leastsquare regression, and random sampling with leastsquares regression; and
(c) Describe the R^{2} metric, its use in measuring the quality of a response surface, and its potential problems.
Homework Problems
A homework problem will be given at the end of most regular lectures and will be due at the beginning of the next class. These homework problems are intended to take 12 hours to complete. The individual homework sets will be graded on the following scale:
3: A complete solution demonstrating an excellent understanding of the concepts.
2: A complete solution demonstrating an adequate understanding of the concepts, though some minor mistakes may have been made.
1: A complete or nearlycomplete solution demonstrating some understanding of the concepts, though major mistakes may have been made.
0: A largely incomplete solution or no solution at all.
Note: the individual homework grades will only be integer values. At the end of the semester, the highest 2/3's of the grades received in the homeworks will be averaged to determine an overall homework letter grade. Roughly, the following ranges will be used. A: 2.53; B: 22.5; C: 1.52; D: 11.5; F: 01.
Projects
Currently, three programming projects are planned for this semester (one for each section of the course except the ODE section). The projects will focus on applying numerical algorithms to aerospace applications. The programming is highly recommended to be done in Matlab®. The expected due dates for the projects are as follows.
Projects Table
PROJECTS 
DUE DATES 
Project 1 
Lecture 16 
Project 2 
Lecture 30 
Project 3 
Lecture 38 
The project assignments will be distributed at least one week prior to the due dates. No homeworks will be given during the week the projects are due. Each project will be assigned a letter grade based on the standard MIT letter grade descriptions (see Course Grade).
Homework and Project Collaboration
While discussion of the homework and projects is encouraged among students, the work submitted for grading must represent your understanding of the subject matter. Significant help from other sources should be noted.
Oral Exams
There will be a midterm and final oral exam. The midterm oral exam will be held between Lecture 20 and Lecture 21. The final oral exam will be held during Final Exam Week. I will schedule the midterm oral exam by the end of February based on preferences from each student. I will schedule the final oral exam once the final exam schedule for the institute has been published. Each oral exam will be assigned a letter grade based on the standard MIT letter grade descriptions (see Course Grade).
Course Grade
The subject total grade will be based on the letter grades from the homework, projects, and oral exams. Roughly, the weighting of the individual letters grade is as follows:
Grading Criteria
ACTIVITIES 
BREAKDOWN 
Homework Letter Grades 
1/8 of the Subject Total Grade 
Project Letter Grades 
Each Project is 1/8 of the Total Grade 
Oral Exam Letter Grades 
Each Exam is 1/4 of the Total Grade 
For the subject letter grade, I adhere to the MIT grading guidelines which give the following description of the letter grades:
A: Exceptionally good performance demonstrating a superior understanding of the subject matter, a foundation of extensive knowledge, and a skillful use of concepts and/or materials.
B: Good performance demonstrating capacity to use the appropriate concepts, a good understanding of the subject matter, and an ability to handle the problems and materials encountered in the subject.
C: Adequate performance demonstrating an adequate understanding of the subject matter, an ability to handle relatively simple problems, and adequate preparation for moving on to more advanced work in the field.
D: Minimally acceptable performance demonstrating at least partial familiarity with the subject matter and some capacity to deal with relatively simple problems, but also demonstrating deficiencies serious enough to make it inadvisable to proceed further in the field without additional work.
Textbooks
Notes will be distributed. Reference texts will be recommended for specific topics as needed.
Calendar
Calendar Table
LEC # 
TOPICS 
KEY DATES 
Ordinary Differential Equations 
1 
Numerical Integration of Ordinary Differential Equations: An Introduction 

2 
Convergence and Accuracy 

3 
Convergence of MultiStep Methods 

4 
Convergence of MultiStep Methods (cont.) 
Homework 1 due 
5 
Convergence of MultiStep Methods (cont.) 
Homework 2 due 
6 
Systems of ODE's and Eigenvalue Stability 
Homework 3 due 
7 
Stiffness and Implicit Methods 

8 
Stiffness and Implicit Methods (cont.) 
Homework 4 due 
9 
RungeKutta Methods 

Finite Volume/Difference Methods 
10 
Finite Volume Method 
Homework 5 due 
11 
Finite Volume Method (cont.) 
Homework 6 due 
12 
Finite Volume Method (cont.) 
Homework 7 due 
13 
Finite Difference Method 
Homework 8 due 
14 
Finite Difference Method (cont.) 

15 
Finite Difference Method (cont.) 

16 
Matrix Stability Analysis 
Project 1 due 
17 
Matrix Stability Analysis (cont.) 

18 
Fourier Stability Analysis 

19 
Fourier Stability Analysis (cont.) 

20 
Fourier Stability Analysis (cont.) 


Midterm Oral Exam 

Finite Element Methods 
21 
Method of Weighted Residuals 

22 
Method of Weighted Residuals (cont.) 
Homework 9 due 
23 
Finite Element Method for 1D Diffusion 

24 
Finite Element Method for 1D Diffusion (cont.) 
Homework 10 due 
25 
Finite Element Method for 1D Diffusion (cont.) 
Homework 11 due 
26 
Finite Element Method for 2D Diffusion 

27 
Finite Element Method for 2D Diffusion (cont.) 

28 
Finite Element Method for 2D Diffusion (cont.) 

29 
Higherorder Finite Element Method 

30 
Higherorder Finite Element Method (cont.) 
Project 2 due 
Probabilistic Simulation Techniques 
31 
Introduction to Monte Carlo Method 

32 
Introduction to Monte Carlo Method (cont.) 
Homework 12 due 
33 
Error Estimates for Monte Carlo Method 

34 
Error Estimates for Monte Carlo Method (cont.) 
Homework 13 due 
35 
Error Estimates for Monte Carlo Method (cont.) 

36 
Latin Hypercube Sampling 

37 

38 
Bootstrapping 
Project 3 due 
39 
Wrap Up 


Final Oral Exam 
