This course aims to teach the basic
numerical methods in two important problems: numerical linear
algebra, and the resolution of PDEs. We will cover, in the first
case, LU and QR decomposition; eigenvalue search methods (QR
algorithm); Singular Value Decomposition and applications to Least
Square minimization; and iterative methods (Conjugate Gradient).
The student will be introduced to the LAPACK routines and their uses,
and will also be expected to develop their own algorithms. In the
second part, we begin by a short review of ODE methods, and then
explore three methods for the resolution of PDEs:
finite differences, spectral methods and finite element methods. The
student will be expected to implement one of these methods to solve a
standard prolem in fluid dynamics.
TEXTBOOKS
The instructor will hand out notes extracted from various textbooks
instead of using a single one. However, for reference, I am using the
following textbooks:
- For a good introduction to the necessary linear algebra, see Contemporary Linear Algebra by
Anton & Busby, Wiley pub.
- For a basic comprehensive textbook on many numerical methods, see
Numerical Recipes by
Press, Teukolsky, Vetterling & Flannery, published by Cambridge
University Press
- For more advanced textbooks on numerical linear algebra, see
- Numerical Linear Algebra,
by Trefethen & Bau, SIAM pub.
- An Introduction to Numerical
Analysis, by Atkinson, Wiley pub.
- Matrix computations,
by Golub & Van Loan
- For basic textbooks on ODE solution methods, see
- Scientific computing and
Differential Equations, by Golub & Ortega, Academic press
- Applied Numerical Analysis,
by Wheatley, Addison Wesley pub.
- For a good reference on the stability of numerical ODE methods
- An Introduction to Numerical
Analysis, by Atkinson, Wiley pub.
- For basic textbooks on PDE solution methods, see
- Applied Numerical Analysis,
by Wheatley, Addison Wesley pub.
- For reference of finite volume solutions, see
- Finite volume methods for
hyperbolic systems, R. Leveque, Cambridge University Press.
- For reference on Spectral methods
- For reference on Finite Element methods, see
- Numerical Partial
Differential Equations for Environmental Scientists and Engineers,
D. R. Lynch, Springer Verlag
- Numerical Analysis,
Burden & Faires, ITP Publishing.
EVALUATION
- Quizzes: there will be a few short quizzes based on the suggested
homework, counting for 20% of the grade
- Mid-term and final projects: The students will be expected to
solve two problems associated with the
course material: a data fitting problem (30% of the grade) and a fluid
dynamical problem (50% of the grade).
The evaluations will be carried out as take-home projects.
WEEK BY WEEK SYLLABUS
Week 1: Basic Fortran; data storage;
numerical
precision. Basic Linear Algebra; Gaussian Elimination.
Gaussian Pivot. ;
Week 2: LU decomposition; Cholesky decomposition
Householder matrices; QR decomposition
Week 3: Eigenvalue searches; QR algorithm
Week 4: Iterative methods for linear systems.
Conjugate gradient method;
Stability analysis for linear systems. Notions of parallel processing.
Week 5: ODEs (1) Initial Value Problems
ODEs (2) Two point boundary value methods; shooting
and Newton-Raphson method
Week 6: Finite difference methods for PDEs (1). Poisson
equation
Parabolic and hyperbolic problems
Week 7: Further concepts in finite differences.
Stability, multistep methods, ADI.
Semi implicit differencing.
Week 8: Introduction to finite volume methods; upwind
differencing
Spectral methods (1). General introduction
Week 9: Spectral methods (2). Fourier basis.
Spectral methods (3). Chebishev basis.
Week 10: Finite element methods (1). 1D tutorial
Finite element methods (2). 2D introduction.