## Midterm for Math 151B, May 11

### The midterm will be closed books, closed notes. Calculator is ok (but most likely not neccessary)

Things you should know for the midterm (and also for the final)

The form of important methods:

**Euler's Method**: w_{i+1} = w_{i} + h*f(t_{i},w_{i})

**Midpoint Method**: w_{i+1} = w_{i} + h*f(t_{i} + h/2,w_{i}+h/2f(t_{i},w_{i})

**Trapezoidal Method**: w_{i+1} = w_{i} + h/2*(f(t_{i},w_{i}) + f(t_{i+1},w_{i+1}))

Linear Iterations, Newton's Method to use Trapezoidal Method

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. tau_{i+1}(h) = 1/h * (w_{i+1} - v_{i+1}); Runge Kutta Fehlberg

How to estimate the error using different timesteps (step-doubling)

Total error; i.e., error_{tot} = SUM(h*tau_{i})

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.

**I will not ask the following:**

memorizing lenghty defintions/theorems.