## 2.274 Computational Incompressible Fluid Dynamics

Updated: 11/13/98

Schedule: Monday & Wednesday 9:00 - 10:30, Room 1-273
Instructor: Professor Anthony T. Patera , Rm. 3-266, x3-8122, patera@mit.edu . Office hours: Open
TA: John Weatherwax, Rm. 2-342, x3-7578, wax@math.mit.edu, office hours: Tuesday, 3:00-4:00 & Wednesday, 10:30-11:30.

Problem Set 4 (Due Dec 2):
• Exercise 21.12
• Problem 22.1

• Section 9.4, 9.5
• 20.6
• 21.1, 21.2, 21.3, 21.4

### Exercises:

• 21. {2, 3, 4, 9}

### Errata:

• Click here to see the page of errata from the book handout

### Class information:

Intended Audience: Students interested in formulating, understanding, analyzing, and implementing numerical methods, and in using existing numerical tools critically and effectively.  Material is centered around the partial differential equations of incompressible fluid flow and heat transfer, however the methods presented apply to a broad range of engineering problems.  The focus is mathematical, but not to the exclusion of physical arguments.

Prerequisites:  Basic knowledge of heat transfer and incompressible fluid mechanics; familiarity with ODEs, standard PDEs, and linear algebra; proficiency in C and MATLAB; modest exposure to elementary numerical procedures.

Course Content:  Most of the term will be spent on the formulation, analysis, and implementation of finite element spatial discretizations of linear problems:  symmetric coercive problems (heat conduction and, briefly, linear elasticity); nonsymmetric coercive problems (convection-diffusion phenomena); and saddle problems (creeping flow: the incompressible Stokes equations).  Additional topics to be discussed include: direct and iterative solution methods; nonlinear problems, in particular the Navier-Stokes equations; and finite-difference in time/finite-element in space treatment of unsteady problems.

Required Text:  Proto-Book Lecture Notes.

Class Format:  Students are expected to review relevant Lecture Notes prior to class.  Class time will be devoted to elaboration and clarification of the reading assignment, and interactive solution of associated examples and exercises.

Grading:  75% problem sets (involving both analysis and computation); 25% class participation.

Problem Sets:  The assignments involve some formulation, some analysis, and some programming and computation (in C and MATLAB).  In the latter case, the numerical method should be summarized (Lecture Notes or other texts may be referenced), and the results of the program displayed in tabular or graphical form.  Only minimal algebraic details should be included; do NOT submit a copy of your program as part of your problem set solution.
The problem sets will indicate explicitly which programs are to be written by the student, and which may be incorporated or derived from (legally accessed and properly referenced) third-party sources.  Students may collaborate on problem sets, but only to the extent of oral communication and untranscribed blackboard derivations.

Computer and Software Resources:  Students may use either their ?own? (workstation-equivalent) machines, or Project Athena public-cluster computers.  All third-party software required or recommended is available on Athena, in particular MATLAB (with the PDE Toolkit) and Numerical Recipes.  Any course-specific software to which students must directly link will be available in C or MATLAB on Athena.

Communication:  Any updates to the course material or the class schedule will be announced in class or distributed via email and archived on the course Web site http://web.mit.edu/2.274/www.

Office Hours:
Instructor:    open
T.A.:             Tuesday, 3:00-4:00 & Wednesday, 10:30-11:30.

Make-up Classes:  As needed, make-up classes will be held on Fridays, 9:00 - 10:30, in Room 1-273.

### Class Schedule and Syllabus:

 Date: TOPIC W Sept 9 Introduction M Sept 14 W Sept 16 Coercive Symmetric Problems: Heat Conduction, Strong Formulation M Sept 28  W Sept 30 Variational Formulation F Oct 2  M Oct 5 Abstract Finite Element Method W Oct 7 T Oct 13 Linear Elements in One Space Dimension W Oct 14 Higher-Order Elements F Oct 16  M Oct 19 Multi-Dimensional Problems W Oct 21 M Oct 26 Solution Methods W Oct 28 Linear Elasticity M Nov 2 W Nov 4 Saddle Problems: General Formulation M Nov 9 The Stokes Equations: Creeping Flow M Nov 16 W Nov 18 Finite Element Methods for Stokes Flow M Nov 23 Coercive Nonsymmetric Problems: Convection-Diffusion Formulation W Nov 25 M Nov 30 W Dec 2 Finite Element Methods for Convection-Diffusion M Dec 7 W Dec 9 Nonlinear Problems: The Navier-Stokes Equations (and Time-Dependent Problems)
Problem Sets will be due on:  Oct. 7; Oct 21; Nov. 2; Nov. 30; and Dec. 9.

### Reference Books:

(on reserve in Barker Library unless otherwise noted)
Acton, Numerical Methods That (Usually) Work, Harper & Row, 1970.
Ames, Numerical Methods for Partial Differential Equations, Academic Press, 1977.
Anderson, Tannehill, and Pletcher, Computational Fluid Mechanics and Heat Transfer, Hemisphere Publishing Corporation, 1984.
Bathe, Finite Element Procedures in Engineering Analysis, Prentice-Hall, 1982.
Bernardi and Maday, Approximations Spectrates de Problemes aux Limites Elliptiques, Springer-Verlag, 1992.
Brezzi and Fortin, Mixed & Hybrid Finite Element Methods, Springer-Verlag, 1991.
Canuto, Hussaini, Quateroni, and Zang, Spectral Methods in Fluid Dynamics, Springer-Verlag, 1987.
Ciarlet and Lions, eds., Handbook of Numerical Analysis, Elsevier (in Science Library).
Dahlquist, Bjorck, and Anderson, Numerical Methods, Prentice-Hall, 1974.
Dwoyer, Hussaini, and Voigt, Finite Elements: Theory and Application, Springer-Verlag, 1988.
Fletcher, Computational Galerkin Methods, Springer-Verlag, 1984.
Gear, Numerical Initial Value Problems in Ordinary Differential Equations, Prentice, Hall, 1971.
George and Liu, Computer Solution of Large Sparse Positive Definite Systems, Prentice-Hall, 1971.
Girault and Raviart, Finite Element Methods for Navier-Stokes Equations:  Theory and Algorithms, Springer-Verlag, 1986.
Golub and Van Loan, Matrix Computations, Johns Hopkins, 1989.
Gunzberger, Finite Element Methods for Viscous Incompressible Flows, Academic Press, 1989.
Johnson, Numerical Solution of Partial Differential Equations by the Finite Element Method, Cambridge University Press, 1987.
Kardestuncer and Norrie, eds., Finite Element Handbook, McGraw-Hill, 1987.
Kikuchi, Finite Element Methods in Mechanics, Cambridge University Press, 1986.
Naylor and Sell, Linear Operator Theory in Engineering and Science, Springer-Verlag, 1982.
Oden, Applied Functional Analysis:  A First Course for Students of Mechanics and Engineering Science, Prentice-Hall, 1977.
Oden and Reddy, An Introduction to the Mathematical Theory of Finite Elements, Wiley, 1976.
Peyret and Taylor, Computational Methods for Fluid Flow, Springer-Verlag, 1982.
Pironneau, Finite Element Methods for Fluids, Wiley & Sons, 1989.
Press et al., Numerical Recipes:  The Art of Scientific Computing (C version), Cambridge University Press, 1989.
Quarteroni and Valli, Numerical Approximation of Partial Differential Equations, Springer-Verlag, 1994.
Reddy, The Finite Element Method in Heat Transfer and Fluid Dynamics, CRC Press, 1995.
Richtmyer and Morton, Difference Methods for Initial Value Problems, Wiley, 1967.
Roache, Computational Fluid Dynamics, Hermosa, 1976.
Smith, Numerical Solution of Partial Differential Equations, Clarendon Press/Oxford, 1985.
Strang, Linear Algebra and Its Applications, Academic Press, 1980.
Strang, An Introduction to Applied Mathematics, Cambridge-Wellesly Press, 1986.
Strang and Fix, An Analysis of the Finite Element Method, Prentice-Hall, 1973.