This syllabus may be slightly changed during the course, depending
on the pace.
Main textbook:
"An introduction to numerical analysis",
Kendall E. Atkinson, Second Edition,
Wiley, ISBN: 0471624896.
In addition, I used a number of other books:
by G. Whitham,
"Linear and nonlinear waves",
by J. Stoer and R. Bulirsch
"Intoduction to numerical analysis",
by Carslaw
"Introduction to the theory of Fourier series
and integrals"
Main emphasis of the course is on dynamic, time-dependent equations
with applications.
Main topics covered:
0. ODEs and PDEs - equations, notations, initial and boundary
conditions, non-dimensionalization, stability
analysis,
simple tranformations, simple solutions.
1. Numerical methods for ODEs:
Euler method as prototype of everything.
Different types of stability; various types
of converrgence;
consistency; estimation of accuracy; order
of accuracy;
Implicit methods; Explicit methods; multi-step
methods;
midpoint method; Adams methods; higher-order
methods.
convergence and stability for multi-step methods.
Runge-Kutta methods.
2. Numerical methods for PDEs
Finite-difference methods
Stability; CFL condition; advance
in time; examples.
3. Fourier series. Definitions, various forms.
Smoothness vs. convergence.
Sobolev spaces; proof of convergence.
Numerical examples: convergence of Fourier
series;
Gibbs phenomenon.
Spatial discretization; reductionto ODEs;
simulations.