The final will last for only 2 hours, 8:30-10:30 on Thursday morning!
The form of important methods:
Euler's Method: wi+1 = wi + h*f(ti,wi)
Midpoint Method: wi+1 = wi + h*f(ti + h/2,wi+h/2f(ti,wi)
Trapezoidal Method: wi+1 = wi + h/2*(f(ti,wi) + f(ti+1,wi+1))
Ideas behind deriving Runge-Kutta Methods
Local truncation error; estimate the leading term
How to estimate the error using methods of order n and n+1
i.e. taui+1(h) = 1/h * (wi+1 - vi+1); Runge Kutta Fehlberg
Total error; i.e., errortot = SUM(h*taui)
Ideas behind adaptive timestep selection (do not remember specific formulars)
What is an explicit/implicit method ?
What is a multistep method ? (No tedious derivations in the exam; but something like homework 4, problem 1 is possible).
What is a predictor-corrector scheme ?
Region of stability ? What does this imply for the timestep for the model problem ?
Transform higher order ODE into system of first oder ODE's. Write this in vector/matrix notation.
Gauss Jacobi, Gauss Seidel, and SOR methods
Sufficient criteria for convergence for Gauss Jacobi, Gauss Seidel
Least Square fit to a straight line, or a polynomial of degree n.
Least square fit of a continuous function, or a discrete data set.
memorizing lenghty defintions/theorems.
Stability of the "general problem" (which we discussed in class right after the model problem).
Newton's Method to use Trapezoidal Method
Fast Fourier Transform (FFT)
Definitions or theorems about linear algebra. For example, you don't need to remember the conditions for a norm to be a norm (if there is a problem on the exam, I would give you the relevant definitions)