Study Guide for Final
Math 6070
Spring 2013
1. Know how to derive various finite difference approximations for f ′ , f ′′ , including an expression
for their corresponding local truncation error terms. Be able to specify conditions on f that are
needed for the approximations and the error estimates to be valid.
Review the formulae derived in class, and problems 1–5 on HW 2.
2. Give a detailed analysis of what happens to the total error in the forward difference formula for
f ′ (x), as h approaches zero. Is there an optimal choice of h? What is it? Justify your answer.
Now answer the same questions about the central difference formula for f ′ (x).
3. Give a detailed analysis of what happens to the total error as h → 0 in the approximation
f ′′ (x) ≈
1
[f (x + h) − 2f (x) + f (x − h)]
h2
Is there an optimal choice of h? What is it? Justify your answer.
4. Consider the IVP x′ = f (t, x), x(t0 ) = x0 .
(a) Clearly state conditions under which a solution to this IVP is guaranteed over an interval.
How large is the interval? (See Theorems 11.1.1 and 11.1.3, p. 255.)
(b) Given a particular IVP, be able to determine the largest interval over which a solution is
guaranteed to exist (e.g., problem 6 on HW 2).
(c) Cleary state conditions under which a solution to this IVP is guaranteed to be unique. (See
Theorem 11.1.2, p.255.)
(d) Find two solutions to the IVP in Problem 3, p.295. Explain why this does not violate the
result on uniqueness of solutions.
5. If the initial condition of an IVP is perturbed, what can be said about the perturbation in the
solution? (Study Theorem 11.1.4, p. 256 of Greenbaum and Chartier.)
6. Give the general forms of
(a) an explicit k-step
(b) an implicit k-step
methods for solving an ODE with a given initial condition. Explain the difference between them.
Give an example of each.
7. Explain the following numerical methods for solving an ODE. For (a) – (d), be prepared to
calculate a few iterations by hand.
(a)
(b)
(c)
(d)
(e)
explicit Euler
explicit improved Euler
implicit improved Euler
Midpoint Method
Theta Methods
8. In the context of a numerical method for solving an ODE, define the following terms:
(a)
(b)
(c)
(d)
(e)
global error at a point.
local truncation error at a point.
a convergent method
the order of the method
a stable method.
Determine/discuss (a)–(e) for (explicit) Euler’s method.
9. Define what is meant by a Lipschitz function. Is a differentiable function Lipschitz? Is a Lipschitz
function differentiable? Is a Lipschitz function continuous? Explain
√ why or why not, in each case.
(Hint: Carefully examine the functions y = tan x, y = |x|, y = x over appropriate intervals.)
10. Describe the main idea behind Runge-Kutta methods for solving an IVP. Describe the general
format of Runge-Kutta Methods.
11. Show how 2–stage Runge Kutta methods of order 2 can be built from scratch. How many such
methods are there? Deduce that Improved Euler, the Midpoint method and Heun’s method are
all examples of order 2 Runge-Kutta methods.
12. For which n is it possible to build an n-stage explicit Runge-Kutta method of order n? How many
stages are needed for an order 5 explicit RK method? For order 6? For order 7?
Is it possible to get a 2–stage implicit Runge-Kutta method of order 3?
13. Adaptive methods: what are they, what are their advantages? What needs to be monitored in
order to design an adaptive method, and how is it accomplished?
14. What are the main features of the Runge-Kutta-Fehlberg Method? What are its advantages?
15. Show how a multi-step method can be constructed from a quadrature rule.
16. Explain the form of the 5-step implicit Adams-Moulton method, and the 5-step explicit AdamsBashforth method, and explain how these methods are derived. (No, you do not have to memorize
the constants.) Show how they can be used together to get a Predictor-Corrector method. What
is the advantage of using them together like this?
17. Consider the following claims:
(a) Implicit Euler has superior stability properties when compared with Explicit Euler.
(b) Implicit methods can be a boon when solving stiff systems.
Support these claims by comparing the performance of explicit vs. implicit Euler on an appropriate
linear system of ODEs.
18. How do we determine if a multi-step method is convergent? And oh, what do we mean when we
say a multi-step method is convergent?
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19. How does an IVP differ from a BVP? What numerical consequences ensue?
20. Give an example of a BVP whose DE has a solution, but whose boundary conditions are impossible
to satisfy.
21. Show how the BVP
x′′ = f (t, x),
x(a) = α,
x(b) = β)
can be tranformed into an equivalent BVP on the standard interval with homogenous boundary
conditions.
22. We discussed two theorems that gave sufficient conditions on f for a BVP of the form
x′′ = f (t, x),
x(0) = 0 = x(1)
to have a unique solution. Be able to use these criteria on specific examples.
23. Finite Difference Method
(a) What is the goal?
(b) What are the main ideas on which the method is based?
(c) What is the original problem transformed into? What important matrix shows up?
(d) What is the accuracy? Justify. You don’t have to be able to do the complete derivation, but
you should know the highlights, and the main results along the way.
24. Finite Element Method
(a) What is the goal?
(b) What are the main ideas on which the method is based? What key tools are used, and how?
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