To solve differential equation problem: If f is “smooth enough”, then a solution will exist and be unique we will be able to approximate it accurately with a wide variety of method Two ways of expressing “smooth enough” Lipschitz continuity Smooth and uniformly monotone decreasing

1. 1

2. 6.1 The Initial Value Problem: Background
2

3. Example 6.1
3

4. Example 6.2
4

5. Example 6.3
5

6. Example 6.4
6

7. Methods
 To solve differential equation problem:
 If f is “smooth enough”, then
 a solution will exist and be unique
 we will be able to approximate it accurately with a wide
variety of method
 Two ways of expressing “smooth enough”
 Lipschitz continuity
 Smooth and uniformly monotone decreasing
7

8. Definitions 6.1 and 6.2
8

9. Example 6.5
9

10. Example 6.5 (con.)
10

11. Theorem 6.1
 Proof: See Goldstine’s book
11

12. 6.2 Euler’s Method
 We have treated Euler’s method in Chapter 2.
 There are two main derivations about Euler’s method
 Geometric derivation
 Analytic derivation
12

13. Geometric Derivation
13

14. 14

15. Analytic Derivation
15

16. Error Estimation for Euler’s
Method
16
 By the analytic derivation, we have
 The residual for Euler’s method:
 The truncation error:

17. Example 6.6
17

18. 18

19. 6.3 Analysis of Euler’s Method
19
 Proof: pp. 323-324 (You can study it by yourselves.)
O(h)

20. Theorem 6.4
 Proof: pp. 324-325 (You can study it by yourselves.)
20
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21. Discussion
 Both error theorems show that Euler’s method is only
first-order accurate (O(h)).
 If f is only Lipschitz continuous, then the constants
multiplying the initial error and the mash parameter can
be quite large, rapidly growing.
 If f is smooth and uniformly monotone decreasing in y,
then the constants in the error estimate are bounded for
all n.
21

22. Discussion
 How is the initial error affected by f ?
 If f is monotone decreasing in y, then the effect of the
initial error decreases rapidly as the computation
progresses.
 If f is only Lipschitz continuous, then any initial error that
is made could be amplified to something exponentially
large.
22

23. 6.4 Variants of Euler’s
Method
 Euler’s method is not the only or even the best scheme
for approximating solutions to initial value problems.
 Several ideas can be considered based on some simple
extensions of one derivation of Euler’s method.
23

24. Variants of Euler’s Method
24
 We start with the differential equation
And replace the derivative with the simple difference quotient derived in
(2.1)
 What happens if we use other approximations to the derivative?

25. Variants of Euler’s Method
 If we use
then we get the backward Euler method
 If we use
then we get the midpoint method
25
O(h)
O(h2
)
Method 1
Method 2

26. Variants of Euler’s Method
 If we use the methods based on interpolation (Section 4.5)
then we get two numerical methods
26
O(h2
)
O(h2
)
Method 4
Method 3

27. Variants of Euler’s Method
 By integrating the differential equation:
(6.23)
and apply the trapezoid rule to (6.23) to get
Thus
(6.25)
27
Method 5
O(h3
)

28. Variants of Euler’s Method
 We can use a midpoint rule approximation to integrating (6.23)
and get
28
O(h3
)
Method 6

29. Discussion
 What about these method? Are any of them
any good?
 Observations
 Methods 2, 3, and 4 are all based on derivative
approximations that are O(h2
), thus they are more
accurate than Euler method and method 1 (O(h)).
 Similarly, methods 5 and 6 are also more
accurate.
 Methods 2, 3, and 4 are not single-step methods,
but multistep methods. They depend on
information from more than one previous
approximate value of the unknown function.
29

30. Discussion
 Observations (con.)
 Concerning methods 1, 4, and 5, all of these formulas involve we
cannot explicitly solve for the new approximate values Thus these
methods are called implicit methods.
 Methods 2 and 3 are called explicit methods.
30

31. 6.4.1 The Residual and
Truncation Error
31

32. Definition 6.3
32

33. Example 6.7
33

34. Example 6.8
34

35. Definition 6.4
35

36. 6.4.2 Implicit Methods and Predictor-
Corrector Schemes
 How to get the value of yn+1? Using Newton’s method or
the secant method or a fixed point iteration.
36
Let y = yn+1

37. 37
F(y)
F’(y)
F(y)
h
F(y+h)-F(y)

38. Predictor-corrector idea
 Can we use a much cruder (coarse) means of estimating
yn+1?
38

39. Example 6.11
39

40. Example 6.12
40

41. 41

42. Discussion
 Generally speaking, unless the differential equation is
very sensitive to changes in the data, a simple
predictor-corrector method will be just as good as the
more time-consuming process of solving for the exact
values of yn+1 that satisfies the implicit recursion.
42

43. Discussion
 If the differential equation is linear, we can entirely
avoid the problem of implicitness.
 Write the general linear ODE as
43

44. 6.5 Single-step Method: Runge-Kutta
 The Runge-Kutta family of methods is one of the most
popular families of accurate solvers for initial value
problems.
44

45.  Consider the more general method:
45
Residual

46. 46

47.  Rewrite the formula of R, we get
47

48. 48
Solution 1
Solution 2

49. 49
Solution 3

50. Runge-Kutta Method
50

51. Example 6.16
51

52. Example 6.16 (con.)
52

53. 53
One major drawback of the
Runge-Kutta methods is that
they require more evaluations
of the function f than other
methods.