MAT 221A-2: MATHEMATICAL FLUID DYNAMICS - FALL 2003
TR 3:10-4:40, Wellman 103
HOMEWORK PROBLEMS AND SOLUTIONS:
COURSE OUTLINE: This course will focus on the incompressible and
compressible Navier-Stokes and Euler equations. In particular, we shall merge
mathematical analysis, numerical analysis, and scientific computation to
understand the motion of fluids in physical, biological, and engineering
applications. Topics that will be covered in the first quarter include:
Weak and classical solutions for the Navier-Stokes and Euler equations
Basic energy laws and circulation theorems
Galerkin methods, approximation schemes, a priori estimates
Spectral schemes and finite element methods
Short-time existence and uniqueness for the Navier-Stokes and Euler
equations
Bounded domains and exterior problems
PDE methods for convergence of Galerkin schemes
Numerical methods: accuracy, stability, and consistency of numerical
schemes
Well-resolved and convergent schemes
Global weak (Leray) solutions for the 3D Navier-Stokes equations
Regularity and uniqueness in 2D
Runga-Kutta and time-stepping schemes
Projection methods
Lagrangian vs. Eulerian representations
Vorticity formulation and vortex methods
Flow over aerofoils
The second quarter course 221B
will deal with the important area of compressible flows,
moving interfaces, free boundaries, and shock waves.
INSTRUCTORS:
- Prof. Steve Shkoller
Off. Hrs: R 2:00-3:00 (Kerr 572, 752-3610)
shkoller@math.ucdavis.edu
http://www.math.ucdavis.edu/~shkoller
- Prof. Matt West
Off. Hrs: M 11:00-12:00 (Kerr 576, 754-9369)
mwest@math.ucdavis.edu
http://www.math.ucdavis.edu/~mwest
TEXTS (Optional):
Deville, M. O.; Fischer, P. F.; Mund, E. H.
High-order methods for incompressible fluid flow.
Cambridge Monographs on Applied and Computational Mathematics, 9.
Cambridge University Press, Cambridge, 2002.
Durran, Dale R.
Numerical methods for wave equations in geophysical fluid dynamics.
Texts in Applied Mathematics, 32.
Springer-Verlag, New York, 1999.
Acheson, D. J. Elementary fluid dynamics. Oxford Applied Mathematics and Computing Science Series. The Clarendon Press, Oxford University Press, New York, 1990.
Batchelor, G. K. An introduction to fluid dynamics. Second paperback edition. Cambridge Mathematical Library. Cambridge University Press, Cambridge, 1999.
Meyer, Richard E. Introduction to mathematical fluid dynamics. Corrected reprint of the 1971 original. Dover Publications, Inc., New York, 1982.
Lions, Pierre-Louis Mathematical topics in fluid mechanics. Vol. 1. Incompressible models. Oxford Lecture Series in Mathematics and its Applications, 3. Oxford Science Publications. The Clarendon Press, Oxford University Press, New York, 1996.
Van Dyke, M. Album of Fluid Motion. Parabolic Press, 1982.
PREREQUISITES: Undergraduate analysis; experience with MATLAB will be
helpful, but not mandatory.
ATTENDANCE: The first day of lecture is Thursday, September 25 and
the last day of lecture is Thursday, December 4. Regular attendance to
the lectures
is strongly advised. You will be responsible for
all the material presented in class, regardless of whether or not
you were present.
GRADING SCALE: Grades will be assigned
on the basis of your performance on homework and two in-class exams
as follows:
-
Homework: weekly, 60%
-
Exam #1 (Thursday, October 30, 2003 ): 20%
-
Exam #2 (Thursday, December 4, 2003 ): 20%
HOMEWORK AND EXAMS: Homework
assignments will be a marriage of analysis and
MATLAB code development. Homework will be assigned every Tuesday and
collected at the start of class the following Tuesday.
If you cannot attend class, then slip the homework under the door of the
instructor at least
20 minutes before class time (on the day it is due). No late homework will be
accepted, and there will be no makeup exams. In the case that an exam is missed
due to a medical emergency, it will not count toward the final grade.
You may discuss the homework assignments with classmates, but you are each
expected to do your own assignments which include the MATLAB programs that you
will be asked to design. Please write legibly. We strongly
recommend that you do your homework every day, and not wait until the
night before the assignment is due.