(a) For a

given ODE system, what are an initial value problem, a terminal value problem, and

a boundary value problem?

(b) What are

the basic differences between the numerical solution of ODEs and numerical integration?

(c)

Distinguish between the forward Euler and backward Euler methods.

(d) Define

local truncation error and order of accuracy and show that both the forward

Euler and the backward Euler methods are first order accurate.

(e) How does

the global error relate to the local truncation error?

(f) What is

an explicit RK method? Write down its general form.

(g) Define

convergence rate, or observed order.

(h) Name

three advantages that RK methods have over multistep methods and three

advantages that multistep methods have over RK methods.

(i) Why is

it difficult to apply error control and step size selection using the global

error?

(j) In what

sense are linear multistep methods linear?

(k) What are

the two Adams families of methods? What is the main distinguishing point between

them?

(l) Write

down the PECE method for the two-step formula pair consisting of the two-step Adams–Bashforth

method and the one-step, second order Adams–Moulton method.

(m) Define

region of absolute stability and explain its importance.

(n) What is

a stiff ODE problem? Why is this concept important in the numerical solution of

ODEs?

(o) Define

A-stability and L-stability and explain the difference between these concepts.